Lecture Notes in Mathematics Edited by A. Dold, Heidelberg and B. Eckmann, Z(Jrich
344 I
IIIIIIIIIIIIII
A. S. Troelstra (Editor) Universiteit van Amsterdam, Amsterdam/Nederland
Metamathematical Investigation of intuitionistic Arithmetic and Analysis
Springer-Verlag Berlin-Heidelberg- New York 1973
A M S Subject Classifications (1970): 02C15, 0 2 D 0 5 , 0 2 D 9 9 , 0 2 H 1 0
I S B N 3-540-06491-5 S p r i n g e r - V e r l a g B e r l i n • H e i d e l b e r g - N e w Y o r k I S B N 0-387-06491-5 S p r i n g e r - V e r l a g N e w Y o r k - H e i d e l b e r g • Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin - Heidelberg 1973. Library of Congress Catalog Card Number 73-14238. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Dedicated
to
GEORG K R E I S E L
who has c o n t r i b u t e d
so m u c h to the
s u b j e c t of this v o l u m e
Preface
The present volume found its origin in a course on functional and realizability interpretations on intuitionistic formal systems, presented at the Rijksuniversiteit Utrecht
(Netherlands) in the spring of 1970, and a course
on the metamathematics of intuitionlstic formal systems at the University of Amsterdam in 1971 - 1972.
The literature on the subject was widely scattered,
the connection between certain rules was often not made explicit in the literathre,
and some obvious questions were not answered there.
Therefore I thought it would be useful to give a coherent presentation of the principal methods for metamathematical investigation of intuitionistic formal systems and the results obtained by these methods, in the literature,
connecting results
filling gaps and adding some new material.
(for realizability and functional interpretations)
A first attempt
was made in Troelstra
1971, which , however, because of a rather terse style, was not readily assimilated by readers new to the field. ( I t
still provides a useful survey of
the applications to first-order systems however.) presentation,
Therefore a more elaborate
including other techniques of metamathematical research,
seemed
to be called for. Having learnt of the unpublished Ph.D. work of C.Smorynski on applications of Kripke-models to intuitionistic arithmetic,
and of Dr
Zucker's thesis on
the intuitienistic theory of higher-order generalized inductive definitions, subjects which both fitted very well into the scope of the planned volume, I asked them to contribute a chapter each ; their contributions appear as chapters V, and VI respectively°
The models for intuitionistic arithmetic
of finite type, functional and realizability interpretations,
and normaliza-
tion for natural deduction systems, and also the general editing of the volume I undertook myself. Finally, N.Ao Howard contributed an Appendix supplementing discussions in § 2.7 and § 3.5. The organization of the volume is primarily method-centered,
i.e. the
material presented is grouped mostly around methods and techniques, arranged according to the results obtained.
by different methods, appear at various places in the book. the
and not
Hence some results, obtainable This will enable
reader to compare the relative merits of the various methods. As regards intuitionistic arithmetic and closely related systems, the
treatment is almost wholly self-contained ;
some experience with classical
VI
metamathematics,
and the elements of intuitionism,
such as may be gleaned
from Kleene's Introduction to metamathematics and Heyting's book on Intuiticnism suffices°
The parts dealing with arithmetic can therefore be
used in a course for graduate students or a seminar. The sections dealing with analysis are not self-contained, more or less as a running commentary on the literature,
connecting and com-
paring various approaches and adding new results besides. thought of primarily as a help to the beginning researcher, find his way in the subject.
and serve
This part was to help him tc
For use in a seminar, these sections should
usually be supplemented by the reading of other papers. In keeping with this set-up, the listing of applications for intuitionistic arithmetic and closely related systems is rather extensive,
but in
the case of analysis we have often restricted ourselves to some typical examples ; further applications can easily be made by the reader himself once he has understood the method, and its applications to arithmetic. No special attention has been given to intuitionistic propositional logic and predicate logic, because as formal systems they exhibit many properties which do not generalize to arithmetic and analysis,
and therefore
would require a separate treatment. Speedy publication was thought more useful than final polish, to make the material outdated at the moment of its appearance.
so ~s not Hence also
the choice for publication in the "Lecture Notes in Mathematics".
Even
while refraining from a completely self-contained treatment of all parts, it was not possible to take all relevant work into account, not even on arithmetic ; for example, N. Goodman's work on the theory of constructions was left out altogether,
since it would not easily be fitted into the
framework of the other developments and so would consume too much space° We have no doubt that there are still many imperfections in this presentation ; it hardly needs saying that the authors will be grateful for errors, misprints,
additions to the bibliography being brought to their
attention. The contents of the present volume are primarily technical in character ; but it is to be hoped that the material will not inspire a thought- and mind-less multiplication of metamathematical results, without a thought spent on their possible significance for an analysis of intuitionistic basic notions and for foundations of mathematics in general.
On the other hand,
the "philosophical interest" of the subject is not promoted by uncritical analysis. property
(A single example : the interest of the well known disjunction ~A VB
= ~A
or
~ B , and the explicit definability for existen-
tial statements are frequently overrated,
especially as a criterion for the
VII
"constructive
character"
in Troelstra A.)
of the system considered.
As regards potential
tb me to be more promising for well-known
systems
results
(but also more difficult)
interest",
it seems
to look for new results
(possibly different in kind from the results dis-
cussed in this volume), and stronger systems.
See e.g. the discussion
"philosophical
instead of trying to extend known results to stronger Of course,
to be potentially
interesting,
the new
should also have a clear intuitive meaning in terms of the intended
interpretation
of the systems considered.
Directions for use. an analytical
In order to help the reader find his way, there is
table of contents at the beginning,
of notions and notations self-explanatory.
at the end.
§ 3.5 refers
Reference
a bibliography,
and lists
to the bibliography
(except in the appendix)
are
to chapter III,
§ 5, etc. The parts on arithmetic self-contained.
and closely related
11 ; chapter If, §§ I - 4 (2.4.18 excepted), of § 6 are used) ; chapter I I I , § § 4 (3.4.1- 14;
systems are more or less
As such we mention especially:
3.4.29),
Chapter I, §§ I - 8, §§ 10,
§ 5, § 7 (except where results
I (3.1.1 - 18), § 2 (3.2.1 - 28 ;
§ 5 (3.5.1- 11;
3.5.16
3o6.16), § 7 (3.7.1-s), § s (except 3.s.7), § 9;
3.2.33),
( i i i ) ) ; § 6 (3.6.1chapter IV, §§ I-4~
(i),
chapter V, §§ 1 - 6. Chapter I contains
all generalities,
and should usually be consulted
when needed only. Acknowledgements°
As regards my own contribution
especially indebted to G. Kreisel, material in his course notes by the dedication),
ical improvements undertook
expository and mathemat-
and to Miss Judith van Witsen,
the seemingly endless task of typing the manuscript.
acknowledgements
expressed
for his patient and careful reading of
suggesting many stylistic,
and corrections,
I am
the use of unpublished
(apart from the general indebtedness
to J.I. Zucker,
drafts of my chapters,
who permitted
to this volume,
who
Some other
have been made in footnotes.
Amsterdam,
June 1973.
A. S. Troelstra
TABLE OF CONTENTS
I°
INTUITIONISTIC
§I
FOR~L
SYSTE~
(A.S. Troelstra)
I n t U i t i o n i s t i c logic N o t a t i o n a l c o n v e n t i o n s (1,1.2) - Spector's system (1.1o3) - Oodel's s y s t e m (1.1~) - E q u i v a l e n c e of Spector's and Godel's system (1.1.5) - E q u i v a l e n c e of Spector's and K l e e n e ' s f o r m a l i z a t i o n (I°I°6) - A n a t u r a l d e d u c t i o n system (1.1.7 - 1.1.9) - D e d u c t i o n theorem for Spector's system ( 1 . 1 . 9 - 1.1.10) - E q u i v a l e n c e b e t w e e n natural d e d u c t i o n and Spector's system (1.1.11)
I
§
2
Conservative and d e f i n i t i o n a l extensions~ expansions D e f i n i t i o n of predicate logic with equality (1.2.1) D e f i n i t i o n of c o n s e r v a t i v e e x t e n s i o n (1.2.2) - E x p a n s i o n (I~2.3) - D e f i n i t i o n a l extension (1.2.4) A d d i t i o n of symbols for d e f i n a b l e p r e d i c a t e s (1.2.67 - A d d i t i o n of symbols for definable functions (1.2.7) - R e p l a c e m e n t of f u n c t i o n symbols by predicate symbols (1.2.8) - A d d i t i o n of d e f i n e d sorts of v a r i a b l e s ( 1 . 2 . 9 1.2.10)
14
§
3
I n t u i t i o n i s t i e first-order arithmetic Language of HA (1.7.2) - A x i o m s and rules of HA (1.3.3) - D e f i n i n g axioms for p r i m i t i v e recursive functions (1.3.4) - Rule and a x i o m schema of i n d u c t i o n (1.3.5) - N a t u r a l d e d u c t i o n variant Of HA (1.3.6) - E l i m i n a b i l i t y of d i s j u n c t i o n in s y s t e m s ~ o n t a i n i n g a r i t h m e t i c (1.3.7) - F o r m u l a t i o n of H~ without function symbols (1.3.8) - N o t a t i o n a l conventions (pairing, coding of finite sequences, proof predicates, godelnumbers, godel- and r o s s e r s e n t e n c e s , numerals) (1.3.9) - F o r m a l i z a t i o n of e l e m e n t a r y r e c u r s i o n theory (1.3.10)
18
§
4
Inductive d e f i n i t i o n s in HA D e f i n i t i o n of class F ( I ~ . 2 ) - Normal form for elements of F ( 1 . 4 . 5 - 1.4.4) - E x p l i c i t d e f i n a b i l i t y of p r e d i c a t e s i n t r o d u c e d as closed u n d e r a c o n d i t i o n
28
-
from
r
(1.4.5)
Partial r e f l e c t i o n p r i n c i p l e s C ~ d e l n u m b e r i n g of f u n c t i o n constants and terms (1.5.2) - E v a l u a t i o n of closed terms (1.5.3) - C o n s t r u c t i o n of partial truth d e f i n i t i o n s (1.5.4) - Partial r e f l e c t i o n p r i n c i p l e s (1.5.5 - 1.5.6) - R e m a r k on r e f i n e m e n t s (1.5.7) - R e m a r k on q u a n t i f i e r - f r e e systems (1.5.8) - R e f l e c t i o n principle for qf-H~A ( 1 . 5 . 9 - 1.5.10). I n t u i t i o n i s t i c arithmetic in all finite types Type structure ~ (1.6.2) - D e s c r i p t i o n of N - I q A w ( 1 . 6 o 3 - 1.6.7) - D e f i n i t i o n of the k - operator ~ . 6 . 8 ) - HA as a subsystem of N - H A m (1.6.9) - Intensional identity or equality (1.6o10) - D e s c r i p t i o n of ~ _ ~ w (1.6.11) - D e s c r i p t i o n of ~ _ ~ w ~_HA m~ (1.6.12) qf - D e s c r i p t i o n of qf c ~ qf-~ -H~' (1.6o13) - q f - ~-__H~A~, q f - W E - H ~ A as equational calculi (1.6.14) - The systems ~, qf - I ~ w (1.6.15)
33
-
39 -
-
X
S i m u l t a n e o u s r e c u r s i o n and p a i r i n g ; a c o m p a r i s o n of v a r i o u s t r e a t m e n t s (1.6.16) - P a i r i n g operators in q f - W ~ E ~ - H ~ ~ (1.6.17) - Historical notes, v a r i a n t s in the l i t e r a t u r e (1.6.18)
-
§
7
Induction and simultaneous recursion S i m u l t a n e o u s r e c u r s i o n in q f _ N _ y ~ W (1.7.2 - 1.7.7) - The i n d u c t i o n lemma for qf_~_~w ( 1 . 7 . 8 - 1.7.10) - R e p l a c e m e n t of r e c u r s o r by iterator (1.7.11) - S i m u l t a n e o u s r e c u r s i o n and the induction lemma in qf-~ (1.7.12)
51
§
8
~ore about N-HA ~ Cartesian product types and pairing operators (1.8.2) - The X - operator as a p r i m i t i v e n o t i o n (1.8.4) - R e d u c t i o n to pure types ( 1 . 8 . 5 - 1.8.8) - R e d u c t i o n to numerical types in q f - W E - ~ ~ (1.8.9)
6o
§
9
E x t e n s i o n s of a r i t h m e t i c E x t e n s i o n s of a r i t h m e t i c expressed in ~(HA) or ~(H~A) extended by relation constants (reflection principles, g e n e r a l i z e d inductive definitions) (1.9.2) - Language of ~S o (1.9.3) - C o m p r e h e n s i o n principles (1.9.4) - E x t e n s i o n a ! i t y (1.9. 5 - 1.9.7) - HAS o + EXT + ACA is c o n s e r v a t i v e over HA (19.8) - F o r m u l a t i o n of HAS with X - terms (1.9.9) - D e s c r i p t i o n of EL (1.9.10) - Some n o t a t i o n s and c o n v e n t i o n s (1.9.11) - F o r m a l i z a t i o n of e l e m e n t a r y r e c u r s i o n theory in EL ( 1 . 9 o 1 2 - 1.9.16) - D e f i n i t i o n s of A°x, A~x, A ° ~ , ~ ' ~ (1o9.17) - Systems of i n t u i t i c n i s t i c analysis based on the concept of a lawlike sequence ; IDB (1.9.18) - Systems of i n t u i t i o n istic analysis based on a concept of choice sequence I].9.19) - Bar i n d u c t i o n (1.9.20) E x t e n d e d bar induction k- . 9 . 2 1 - 1.9.23) - Fan theorem (I.~o24) - E x t e n s i o n s of N - H A ~ : IDB w (1.9.25) - Theories with bar r e c u r s i o n of
66
Jigger typ~E ~ - ~ + funetionals § 10
§ 11
BR
(1.9.26) - Girard's theory of
(1.9.27)
Relations b e t w e e n classical and i n t u i t i o n i s t i c systems : t r a n s l a t i o n into the negative fragment D e f i n i t i o n of the m a p p i n g ' (1.10.2) - D e f i n i t i o n of H a r r o p formula, and strictly positive part (s.p.p.) (1.10.5) - D e f i n i t i o n of n e g a t i v e formula (1.10.6) - P r o p e r t i e s of the m a p p i n g ' ( 1 . 1 0 . 9 - 1.10.13) General d i s c u s s i o n of various s c h e m a t a and prooftheoretic closure conditions D e f i n i t i o n of admissible rule, and intended i n t u i t i o n istic i n t e r p r e t a t i o n of the logical constants (1.11.1) D i s j u n c t i o n and explicit d e f i n a b i l i t y p r o p e r t y (1.11.2) - The schema Vx(A V B x ) ~ A V VxBx (1.11.3) - The schema Vx ~ A ~ ~oVxA (1.11.4) - ~ r k o v ' s schema and rule 11.5) - Independence of premiss schemata and rules , . 1 1 6) Church,s thesis and ~ l e (1.11;7).
85
9o
XI
MODELS A N D C O M P U T A B I L I T Y
I.
(A.S. Troelstra)
§
I
D e f i n i t i o n s by i n d u c t i o n over the type structure D e f i n i t i o n over the type structure (applicative set, type level) (2.1.1) - E s t a b l i s h i n g properties for a p p l i c a t i v e sets of terms (2.1.2) - D e f i n a b i l i t y aspects (2.1.3) - Sets of terms closed u n d e r ~ - a b s t r a c t i o n (2.1.4)
97
§
2
C o m p u t a b i l i t y of terms in N - H A w 100 D e f i n i t i o n of r e d u c t i o n and--standard r e d u c t i o n for terms of N - H A w (2.2.2) - C o m p a r i s o n of standard and strict r e d u C t i O n (2.2.3) - A l t e r n a t i v e d e f i n i t i o n of ~ (2.2.4) - D e f i n i t i o n of computability, strict -, standard (2.2.5) - All terms of N _ ~ w are standard c o m p u t a b l e ( 2 . 2 . 6 - 9) - ~ - H A ~ W c o n s e r v a t i v e over its i n d u c t i o n - f r e e part for equations b e t w e e n closed terms (2.2.10) - Strong c o m p u t a b i l i t y and strong n o r m a l i z a t i o n ( 2 . 2 . 1 2 - 19) - U n i q u e n e s s of n o r m a l form ( 2 . 2 . 2 0 - 29) - C o m p u t a b i l i t y and strong c o m p u t a b i l i t y for k - b a s e d theories ( 2 . 2 . 3 0 - 34) - D i s c u s s i o n and c o m p a r i s o n of proofs of c o m p u t a b i l i t y for terms of H~Aw in the l i t e r a t u r e (2.2.35)
§
3
More about c o m ~ u t a b i l i t ~ 116 C o m p u t a b i l i t y in ~ - ! ~ w + IE o (2.3.1 - 5) - The equality axioms IE I (2.3.6) - Standard c o m p u t a b i l i t y of terms in languages with Cartesian product type (2.3.7) - Computability relative to assignment of functions ( 2 . 3 . 8 - 10) A r i t h m e t i z a t i o n of c o m p u t a b i l i t y ( 2 . 3 . 1 1 - 13)
§
4
Models b a s e d on p a r t i a l r e c u r s i v e f u n c t i o n a p p l i c a t i o n : HE0, HE0 125 Models : normal, extensional models (2.4oi) - Submodel, homomorphism, e m b e d d i n g (2.4.3) - C o n s t r u c t i o n of inner e x t e n s i o n a l models from a r b i t r a r y models of ~-~H~ w (2.4.5) - The s e t - t h e o r e t i c a l model of E - H A w (2.4.6) - D e s c r i p t i o n of HR0 (2.4.8) - The f e r m a l ~ h e o r i ~ HR0, HR0(2.4.10) - D e s c r i p t i o n of HE0 (2.4.11) - HE0 and h~a inner e x t e n s i o n a l model of HR0 are different (2.4.12) - P r o v a b l e f a i t h f u l n e s s of HRO , u n i f o r m l y in type 0 variables ( 2 . 4 . 1 3 - 14) - Closed type I terms of N _ ~ w are ~ p r o v a b l y recursive (2.4.15) - Sketch of a variant of HR0 satisfying ~ - c o n v e r s i o n (2.4.18) - P a i r i n g in HR0, HE0 (2.4.19)
§
5
Term models of N - H ~ A w D e f i n i t i o n of CTM, CTNF, CTM', CTNF' ( 2 . 5 . 1 - 2 ) Some p r o p e r t i e s of CTM, CTNF, CTM', CTNF' ( ) 2 . 5 . 3 - CTNF I is i s o m o r p h i c to a submodel of HRO for a suitable v e r s i o n of HRO (2.5.5) - A l t e r n a t i v e proof of u n i q u e n e s s of normal form (2.5.6) - HRO can be made into a model for .~w + IE I (2.5.8) - E x a m p l e s of versions of HR0 where distinct normal terms are r e p r e s e n t e d by the same element (2.5.9) - IE o is weaker than IE I (2.5.10)
132
§
6
M o d e l s b a s e d on continuous f u n c t i o n a p p l i c a t i o n : ICF, ECF D e f i n i t i o n of ICF(Z~) (2.6.2) - In ICF a m o d u l u s - o f c o n t i n u i t y functional exists (2.6.3) - ICF(~I) contains a f a n - f u n c t i o n a l if ~ satisfies FAN (2.6.4) H e r e d i t a r i l y continuous functionals ECF(~) (2.6.5) ECF(iL) contains a f a n - f u n c t i o n a l if LL satisfies FAN (2.~.6) - E C F does not contain a modulus of c o n t i n u i t y
138
XII f u n c t i o n a l (2.6.7) - A r e c u r s i v e l y well-founded, but not w e l l - f o u n d e d tree (2.6.9) - Provable f a i t h f u l n e s s of ICF u n i f o r m l y in type I v a r i a b l e s (2.6.11 - 12) - The equivalence b e t w e e n ECF(~) and HRO ( 2 . 6 . 1 3 - 21) - KLS holds in H A + M p R (2.6.15 - 17) - Basis t h e o r e m (2.6.19~ - QF-S£~ T holds for ECF (2.6.20) - The models ECFr(U) and ICFrlU) ( 2 . 6 . 2 2 ) - A v a r i a n t of ICF and E C F (2.6.23) - P a i r i n g operators in ICF, ECF, ICF*, ECF*
§
7
E x t e n s i o n a l i t y and c o n t i n u i t y in N - H ~ w 155 E x t e n s i o n a l i t y and h e r e d i t a r y e x t e n s i o n a l i t y ( 2 . 7 . 2 - 4 ) - Derived rules of e x t e n s i o n a l i t y (2.7.5) - C o u n t e r e x a m p l e to the rule of e x t e n s i o n a l i t y when variables of type level > I are present (2.7.6) - Closed type 3 terms of N - H A w are not extensional in every model (2.7,7) - Provable modulus of c o n t i n u i t y for type 2 terms of N - H ~ w (2.7.8) P r o d u c t topology (2.7.9) - "Floating product topology" (2.7.10) Other models of N - HA w The schemata $I ~ 5 9 ~ ( 2 . 8 . 2 2.8.4) - S o a r p e l l i n i ' s models (2.8.5) - Compact and h e r e d i t a r i l y m a j o r i z a b l e f u n c t i o n a l s (2.8.6)
§
9
166 C o m p u t a b i l i t y and models for extensions of N-HA~ w E x t e n s i o n of c o m p u t a b i l i t y t'o' f u n c t i o n a l s of ~ - IDD~w and related theories (2.9.2) - C o m p u t a b i l i t y for barrecursive functionals (2.9.3) - C o m p u t a b i l i t y for Girard's system of f u n c t i o n a l s (2.9.4) - E x t n e s i o n s of HR0, HE0 to models for other systems (2.9.5) - A p p l i c a t i o n of K - HR0 : C o m p u t a b i l i t y of closed terms of ~ - ID~BBw (2.9.6) - E x t e n s i o n of HE0, HEO to Girard's system of f u n c t i o n a l s (2.9.~) - Similarly for ICF, E C F (2.9.8) - Nodels for - ~ + BR (2.9. 9 - 1 2 ) ,
I. R E A L I Z A B I L I T Y A N D F U N C T I O N A L I N T E R P R E T A T I O N S
§i
162
(AoS. Troelstra)
A theme with v a r i a t i o n s : Kleene's FIC 175 D e f i n i t i o n of F|C (3.1.2) - Soundness theorem (3.1.4) - E x i s t e n c e and d i s j u n c t i o n u n d e r i m p l i c a t i o n (3.1.5) IPR e for H~ (3.1.7) - C h a r a c t e r i z a t i o n of CIC by d e d u c i b i l i t y conditions (3.1.8) - CIC respects logical equivalence, and CIC holds for Harrop formulae (3.1.9) - CIC holds also for f o r m u l a e w h i c h are not e q u i v a l e n t tO a Harrop formula (3.1.10) - IP~ is not derivable in H~ (3@Io11) - D i s j u n c t i o n and explicit d e f i n a b i l i t y p r o p e r t y for H A + M p R (3.1.12) - A v a r i a n t of FIC (3.1.13)- IPR for H~ (3°1.15) - A method of d e a l i n g with variables u s i n g partial r e f l e c t i o n p r i n c i p l e s (3.1.16) - Closure u n d e r Church's rule (3.1.18) - E x t e n s i o n and g e n e r a l i z a t i o n of FIC to h i g h e r - o r d e r systems (3.1.19) - FIC for HAS o + P C A , w i t h a p p l i c a t i o n s (3.1.20) - E x t e n s i o n to HAS (3.1.21 - 23) - E x t e n s i o n of M o s c h o v a k i s ' s methods to ID~B, ID~BI (3.1.2%)
XIII
§
2
Realizabilit~
notions b a s e d on partial recursive
function application 188 D e f i n i t i o n of ~ p - r e a l i z a b i l i t y (3.2.2) - E x a m p l e s (3.2.4) - Soundness theorem (3.2.4) - A n a l y s i s of ~ - r e a l i z a b i l i t y ( 3 . 2 . 9 - 19) - The r8le of almost negative formulae ( 5 . 2 . 9 3.2.15) - The schema ECT~ ( 3 . 2 . 1 4 - 15) - I d e m p o t e n c y of r e a l i z a b i l i t y (3.2.16) - ~ h a r a c t e r i z a t i o n of H ~ - ~ - r e a l -
izability (3.2.18- 19) - C o r o n a r i e s
(3.2.20)
-
Realizability
for M a r k o v ' s schema (3.2.21 - 22) - R e a l i z a b i l i t y for TI(<) ( 3 . 2 . 2 3 - 24) - C h a r a c t e r i z a t i o n of HA c - r - r e a l i z a b i l i t y
~ . ~.-2. 28) - Extensions to othe$ systems ( 3 . 2 . 2 9 )
-
R e a l i z a b i l i t y for IDB (3.2.30) - H A S + CT o + U P is consistent relative to HAS (3.2.31) - Some g e n e r a l i z a t i o n s
-
(3.2.32)
- Comparison
ef-~-
realizability with
~-
real-
izability ( 3 . 2 . 3 3 ) §
§
3
4
R e a l i z a b i l i t y notions based on continuous f u n c t i o n application I D e f i n i t i o n of ~ p - r e a l i z a b i l i t y (3.3°2) - Soundness (3.3.3) - Special instances of soundness (3.3.4) - Soundness for IDB (3.5.6) - The g e n e r a l i z e d continuity schema GC (3°3.9) - C h a r a c t e r i z a t i o n of r I - realizability (3.3.13) =
206
Modified r e a l i z a b i l i t y D e f i n i t i o n of m r p - r e a l i z a b i l i t y (3.4.2) - Examples (3.4.3) - Sound~ess t h e o r e m (3.4.5) - C h a r a c t e r i z a t i o n of m r - r e a l i z a b i l i t y ( 3 . 4 . 7 - 8) - Inessential (but convenient) variants of m r - r e a l i z a b i l i t y (3.4.9) - C o m p a r i s o n of H R O - m r r e a l i z a b i l i t y and r - realizability (3.4.11) - m r - r e a l i z a b i l i t y and n o n - r e a l i z a b i l i t y of various s c h e m a t a ( 3 . 4 . 1 2 25) - m r - r e a l i z a b i l i t y of MpR , CT, CTo (3.4.12-13) HA+CT~TZECTo (3.4.14) - WCT is ICF r - m r - r e a l i z a b l e (3.4.15) - m ~ - reali z a b i l i t y of FAN ~ n d WC-N ( 3 . 4 . 1 6 - 19) - m ~ - r e a l i z a b i l i t y of BI M ( 3 . 4 . 2 0 - 21) - m r - r e a l i z a b i l i t y of + TI(<) (3.4.22 - 25) - M o d i f i e d r e a l i z a b i l i t y for HAS 3 . 4 . 2 7 - 28) - C h a r a c t e r i z a t i o n of provably recursive f u n c t i o n s (3.4.29)
213
~
§
5
The D i a l e c t i c a i n t e r p r e t a t i o n and t r a n s l a t i o n D e f i n i t i o n of the D i a l e c t i c a t r a n s l a t i o n (3.5.2) - M o t i v a t i o n (3.5.3) - Soundness t h e o r e m (3.5.4) _ N_~W is not D i a l e c t i c a i n t e r p r e t a b l e in itself ; d e c i d a b i l i t y of prime formulae (3.5.6) - A x i e m a t i z a t i o n of D i a l e c t i c a i n t e r p r e t a b i l i t y ( 3 . 5 . 7 - 11) - The interp r e t a b i l i t y of the e x t e n s i o n a l i t y a x i o m ( 3 . 5 . 1 2 - 15) - D i a l e c t i c a i n t e r p r e t a b i l i t y of CT, CTo, C - N , FAN, IP (3.5.16) - The D i l l e r - N a h m variant of the D i s l e c t i c a i n t e r p r e t a t i o n (3.5.17) - Shoenfield's variant (3.5.18) - E x t e n d i n g the D i a l e c t i c a i n t e r p r e t a t i o n to stronger systems ( 3 . 5 . 1 9 - 21) - Church's thesis and bar r e c u r s i o n (3.5.20) - Girard's e x t e n s i o n (3.5.21)
230
§
6
Applications: c o n s i s t e n c y and c o n s e r v a t i v e extension r e s u ~ Conservative extension results ( 3 . 6 . 2 - 9) - A x i o m s of choice for HRO, HEO ( 3 . 6 . 1 0 - 16) - E x t e n s i o n s to analysis ( 3 . 6 . 1 7 - 20)
250
XIV
IV.
§
8
Markov's schema and ~ r k o v ' s rule Forms of Narkov's schema and rule (3.8.1) - Not all negations of almost negative formulae are ~ e g a t i v e (3.8.2) - HA + ~ M p R is consistent and closed under Nl°pR (3.8.3) - Characterization of MpR (3.8.4) - Validity of MR ~ , MR in various systems (3.8.5) - HA and HA c have the same provably recursive functions (3.~.6) M~"~I'~ for systems stronger than arithmetic [3.8.7)
263
§
9
Applications of ~ - r ealizabilitF Definition of ? - realizability (5.9.2) - Soundness theorem (3.9.4) ~IJ~A ~vLKLSI ( 3 . 9 . 5 - 12) - Some other results on KLS and the corresponding rules (3.9.12).
267
N O R N A L I Z A T I O N THEOREMS (A.S. Troelstra)
FOR SYSTENS
OF NATURAL
DEDUCTION
§
I
The strong normalization theorem for HA Notational conventions about proof trees (4.1.2) - Description of the reduction processes (4.1.3) - Definitions of thread, segment, maximal segment, normal form, strictly normal form, reduction sequence, reduction tree (4.1.4) - Remarks on reductions, normal deductions (4.1.6) - Strong normalization for ~ ( 4 . 1 . 7 - 18) - Definition of strong validity (4.1.9) Each strongly valid deduction has a finite reduction tree (4.1.13) - Definition of strong validity under substitution (4.1.15) - All deductions are strongly valid under substitution (4.1.16 - 17) - Uniqueness of normal form of deductions ( 4 . 1 . 1 9 - 21)
275
§
2
Appliqations of the n o r m a l i z a t i o n theorem 299 Definition of path and spine ( 4 . 2 . 2 - 3) - Structure of path and spine in normal deductions ( 4 . 2 . 4 - 8) Disjunction and explicit definability property ( 4 . 2 . 9 - 12) - The rule IPR without parameters (4.2.13) - Narkov's rule NRpR (4.2.14) - Disjunction and explicit definability property for H ~ + N p R ( 4 . 2 . 1 5 - 17) - Conservative extensions over quantifier-free fragments ( 4 . 2 . 1 8 - 19) - Reflection principle for closed Z ~ - formulae (4.2.20)
§
3
N o r m a l i z a t i o n for I~ + IP with applications Strong n o r m a l i z a t i o ~ f o r H A + IP 14.3.1 - 2) - Definition of spine (4.3.3) - Structure of spine in normal deductions ( 4 . 3 . 4 - 5) - Disjunction and explicit definability property for ~ + IP (4.3.6)
§
4
Formalization of the normalization theorem~ with applications Formal definition of strong validity (4.4.2) - Theorem on arithmetization of normalization (4.4.3) - Closure under IPR with parameters (4.4.5) - Closure under Church's rule (4.4.6)
307
311
XV
§
5
Normalization for second order logic and arithmetic The system M2(S ) (4.5.2) - Formalizing t~e proof of the normalization theorem (4.5.3) - Construction of a satisfaction relation ( 4 . 5 . 4 - 6) - Properties of the satisfaction relation ( 4 . 5 . 7 - 9) - Partial reflection principle (4.5o10) - Applications (4.5°11) - HAS is closed under a rule of choice from numbers to species
315
(4.5.12). V.
APPLICATIONS
OF KRiPKE
MODELS
(CoA.
Smorynski)
§
I
Kripke models Discussion (5o1.1) - Definition of Kripke models (5.1.2.) - Some basic properties of Kripke models (5.1.3) Examples (5.1o4) - The completeness theorem ( 5 . 1 . 5 - 11) - The Aczel slash ( 5 . 1 . 1 2 - 18) - The operation ( ) ~ (Z)' ( 5 . 1 . 1 9 - 21) - Nodels with equality ( 5 . 1 . 2 2 - 23) Function symbols (5.1o24) - I n t u i t i o n i s m ? (5.1.26)
524
§
2
The treatment of Heytin~'s arithmetic The class of models of H~ is closed under the operation ( ) ~ ( Z ) ' ; disjunction and explicit definability property (5.2.1 - 4) - Applications of the operation ( ) ~ (Z)' ( 5 . 2 . 5 - 8) - Formulae preserved under
339
§
3
( ) ~ (~)' (5.2.9- 12) - Examples. Reflection principles and transfinite induction ( 5 . 2 . 1 3 - 23) Additional information from ( ) ~ (z)': de Jonah's theorem Statement of de Jongh's theorem (5.3.1 - 2) - Preliminaries on the propositional calculus ( 5 . 3 . 3 - 8) - Lemma on Jaskowski's trees (5.3.9) - The G o d e l - R o s s e r M o s t o w s k i - Kripke-;~Tyhill theorem ( 5 . 3 . 1 0 - 12) - de Jongh's theorem for one propositional variable (5.3.16 - 19) - Another theorem of de Jongh ( 5 . 3 . 2 0 - 22) - Further results on de Jongh's theorem (5.3.23)
348
§
4
Markov's schema Markov's schema (5.4.1 - 5) - Independence of MP ( 5 . 4 . 4 - 6) - A comment on proof-theoretic closure properties ( 5 . 4 . 7 - 9) - The operation ( ) ~ (Z + w)' ( 5 . 4 . 1 0 - 14) - The class of models of ~ + ~ is preserved under ( ) ~ (Z + ~)' (5o4.11) - HA +NDP possesses ED, DP (5.4.12) - Closure properties of the class of sets F such that validity of ~.~+ F is preserved by ( ) ~ (Z + w)' (5°4.13)
360
§
5
The schema IP~ Proof-theoretic closure results ( 5 . 5 . 2 - 3) - Mutual independence of ~P and IP~ (5.5.4 - 7) - Final comments on I P ~ (5.5.8) - Uniform independence of
369
iP~, ~ §
6
(5°5.9)
Definability of models of ~ c : appliqations The operation ( ) ~ (Z)* (5.6.1) - Definability ( 5 . 6 . 2 - 7) - The H i l b e r t - B e r n a y s completeness theorem ( 5 . 6 . 8 - 9) - The O o d e l - R o s s e r - M o s t o w s k i - K r l p k e ~lyhill theorem revisited ( 5 . 6 . 1 0 - 12) - Z,-° substitution instances in de Jongh's theorem (5.6.13 - 16) -
372
XVI
- de Jongh's theorem for MP (5.6.20- 22) - Other applications (to systems with RFN(~), T I ( < ) ) (5.6.23- 26) §
VI°
7
Other systems Subsystems of Heyting's arithmetic (5.7.1 - 2) - Extensions of HA: Theory of species (5.7°3) - Other set-theoretic approaches (5.7.4)°
388
ITERATED INDUCTIVE DEFINITIONS, TREES AND ORDINALS (J.I° Zucker) §
I
§
2
Introduction
The systems
392
~2(A)
397
Inductively defined sets of numbers (6~2°I) - The class ; the theory ID2(A ) for A E ~ ; c definition of the ordinals II,~D2I,qI,...,..D~I, II~II, I~,1 (6.2.2) §
3
The theor# ~2 An intuitionistic theory of trees of the first 3 number or tree classes
401
§
4
Computability of closed terms of ~2 Definitions (6.4.1) - Uniqueness of normal form (6.4.3) - Standard computability (6.4.4) - Proof of computability of closed terms (6°4.5 - 7), and hence of their normal-
404
izabiiity (6°4.8) (6.4~o)
Definition of the ordinal
ItlC
§
5
Strong computability Definitions (6.5oi) - All closed terms of ~2 are strongly computable (6.5.2- 11), and hence strongly normalizable (6o5.12) - Hence all terms of T 2 (not necessarily closed) are normalizable (6.5.13~
408
§
6
Models of T2 ; modelling ~2 in ID2(~) Definitions~(6.6- I - 3) - Exampl~s o ~ w e l l - f o u n d e d models : ~2, HR02, HE02 and CTNF 2 (6.6.4) - Extensionality: some general remarks (6.6.5) - Distinctions between well-founded models of ~I and T2 (6.6.6) - Definition of the ordinal I ~ 2 1 (6.6.7) - Theorem: I~21 ~ II~D21
413
(6.~.8)
§
7
Functional interyretation of I~D2(A) Introduction ; definition of modified realizability (mr-)interpretation, the theories T g [ P ] , E - ~2~ e t c . ( < 7 . 1 ) - Pairing functions (6°7.2)-~ Norm~l forms for translations of A E G (6°7.3) - Axioms for PI and P2 (6°7.4) - "Soundness theorems" for interpretation (6.7.5-7) - Theorem: 1~21 < Im21, ( 6 . 7 . 9 ) - Hence main result: |L~al = I~21 (L7.I0~ - Note on extensionality axioms (6.7.11)
421
§
8
Functional interpretations of classical systems IDa( ) ~nd I ~ (m) Introduction ; definitions of ~v((~), ID#((~), ID~(O) (v = 1,2) and the ordinal I~II (6"8"1T- Func~onal interpretations (modified realizability and Dialectica)
435
of
I~DI(O )
and I.,,,D~(O') ( 6 . 8 . 2 - 3) -
XVII
-
Proof
/ IDol = 1 ~ i '
that
technique
(6.8o~-'~)
usin~ a majorizin~
- Historical
survey:
other
methods of characterizating liDS1 (6.s.6) Ftmctional interpretations of ~ 2 ( O ) and I D#(O) ;
un~o~
§
whether
II~l : I~21 (~.8.7- 11)
9 Extensions to
IDa(A) and IDa(A) ~ v> 2. Equivalences with some subsystems of classical analysis Generalization of result to II.Dgl =tier I for certain > 2 ; equivalence of various I~D~(A) with subsystems of classical analysis ; unknown whether II-~D~I = I~vl for ~ > I (6.9.1) - Positive result : c £ II~D<wl = IID<wl (6.9.2) - Theory W - IDw(A ) equivalent tO classical analysis + H~- CA (6.9.3)
APPENDIX : HEREDITARILY (W°A. Howard)
}~JORIZABLE
FUNCTIONALS
OF FINITE TYPE 454
§
I
§
2
Hereditarily
§
3
Primitive recursive functionals
§ 4
45o
Extensionality majorizable
Discussion of
VyE3(y)
functionals
455 457 460 462
BIBLIOGRAPHY INDEX I.
II.
List of symbols A)
Formal systems
476
B)
Schemata and rules
476
C)
Syntactical
478
D)
Other symbols
List of notions
variables
478 482
Chapter I INTUIT!ONISTiC F O ~ A L
SYSTEMS
§ I. ~ ! ~ ~ = ! ~ " 1.1.1.
Contents.
In the present section we describe logical notation and
systems for intuitionistic predicate logic to be used in the sequel.
The
reader already acquainted with these subjects may skip them and use them for reference only.
We shall presuppose acquaintance with classical predicate
logic and the treatment ef "elementary" metamathematics of Classical systems, and elementary recursion theory (for example, as found in Kleene 1952).
In
the sequel, proofs in formal systems are usually not set out in a completely formalized form, but we compromise between readability and rigour ; i.e. the proof is described in sufficient detail so as to make full formalization a routine matter ; but we try to avoid an excess of detail which obscures the underlying idea. In view of this aim, we usually freely employ theorems and rules of intuitionistic predicate logic, whose proof is not to be found in this monograph. For the reader with little previous acquaintance with intuitionistic reasoning, we recommend Heyting 1956, Chapter VII and Troelstra 1969, § 2 for intuitive background,
and Kleene 1952 for formal details.
~remark
here only
that in order to convert an intuitive proof of an intuitionistic logical theorem into a formal argument,
the system of natural deduction described in
1.1.7 is usually very convenient° In agreement with the attitude towards formalization described above, the description of formal systems for intuitionistic predicate logic below does not serve as a basis for deductions in the formal systems to be studied, but as a reference for metamathematical arguments proceeding by induction on the length of deductions.
Nevertheless,
the discussion
is fairly detailed,
to
enable a reader without experience with intuitionistic formal systems to get accustomed to them.
In later sections and chapters the development gradually
becomes more condensed. Ioi.2. (i)
Some notational conventions° As logical symbols we use
&, V, ~, V, --,~
mathematical abbreviations we use Definitions
~, ~, [, ~, 6, ~ ,
(or abbreviating expressions)
acter are usually indicated by
(absurdity) ; as metaetc.
of a more or less permanent char-
~def ; ~ .is used for definitions or abbre-
viations of a more local character (i.e. within a certain argument), express syntactical
identity.
and to
(ii)
x, y, z, u, v, w
(provided with sub- or superscripts if necessary)
will be used as syntactical variables for variables ; in systems containing arithmetic they are usually reserved for numerical variables.
In the sequel
we shall often have to introduce other categories of symbols as syntactical variables for certain sorts of variables in the formal systems studied. Usually we do not use separate sots of symbols for free and bound variables,
with the exception of systems of natural deduction, where we feel
the notational distinction between variables)
(bound)
to be a definite advantage.
a, b, c, ...
variables and parameters
(free
In this case, lower case letters
from the beginning of the alphabet are used to indicate para-
meters. (iii)
Capitals
(primarily from the beginning of the alphabet)
are used as syntactical variables for formulae,
t, s
denote terms (with an exception in chapter IV, where
A, B, C, ...
will be used to s
is reserved for
successor). Variables
x, y, z, ...
a term
are indicated by the use of square brackets
(iv)
t
occurring free (perhaps only as dummy variables)
In all categories of variables introduced,
in
t[x], t[x,y] , etc.
gub- and superscripts may
be added to create more variables of the same category. (v)
Syntactical descriptions of the classes of formulae and terms, in
our various formal systems, are as usual ; if we wish, we may assume the actual notation to be bracket-free in godelization)
("Polish" notation, which is convenient
and think of the usual notations with brackets as "abbre-
viations" for better readability.
Our bracketing conventions are us usal :
unary operators bind stronger than binary ones, ~.
V, &
bind stronger than
In general, we shall omit brackets whenever we can do so without impair-
ing readability. (vi)
Some abbreviations :
~A ~defA~A,
=def ( A ~ ) & (3~A),
A~
VXI"''XnA ~def VxIVx2"''VXn A ' VxEA(B)
~def Vx(Ax ~ B) ,
~x1~''XnA ~def ~x1"''~XnA ' ~xEA(B)
Also in formulae, we sometimes use unary predicates (vii)
x EQ
~def ~ x ( A x ~ B ) . as an alternative to
For substitution of a term
sort) in an expression
t
Quite frequently,
for a variable
x
(generally a term or a formula)
(especially in chapter IV) we writs
[x/tiE,
where
(t, x
for
of the same
or in a deduction
E
is the expression.
when there is no danger of confusion, we shall also
use the more imprecise convention that whenever an expression been introduced,
Qx,
Q.
E(t)
we use square brackets:
denotes t[x,y],
[x/t]E.
E(x)
has
For variables occurring in terms
etc. ; in contrast,
if
~
is a function,
q0t
or
~(t)
(viii)
stand for
~
applied to the armament
t o
Formal systems will be indicated by capitals or combination of
capitals with a wavy underlining (e.g. The language of a formal system formed formulae by
Fm(H)
or
H
FmH,
H~,
qf-~-N~A ~,
is denoted by
~
etc.).
~(H) , the set of well-
the set of theorems by
Thm(H)
or
ThmHo Deducibility in AE H
H
is indicated by
means the same as
H ~A ,
also interpreted as usual. either If in
HU F ~
~,
or
constants of
or, rarely, as
~H
A
is a theorem of
H.
If we add to
H
a set of axioms
F,
o H~HI
is
we write
H + F.
denotes a given language, and we write
H ~
i.e.
~P]
P
a predicate letter not occurring
for the language obtained by adding
P
to the
~°
Similarly, if
H
is a formal system, presented by giving a set of rules,
axioms and axiom schemata,
~[P]
is the system with language
~(~)[P] ,
with the same rules, axioms and axiom schemata (i.e. the schematic letters in an axiom schema now stand for formulae of the extended language). (L~Church's
k - n o t a t i o n will sometimes be used informally to indicate
functions or predicates.
1.1.3. Spector's s~stem. The systems for intuitionistic predicate logic described in this and the next section are "Hilbert-type systems", i.e. based on logical axioms and inference rules. his
The present system, taken from Spector 1962 (leaving out
A2 , in view of footnote 7 on page 10 of Spector 1962), is given by the
following axioms and rules : PL I)
A-~A
PL 2)
A,A-~B = B
PL3)
A-~B, B-~C ~ A - ~ C
PL4)
A &B-~A,
A &B-~ B,
A-~A VB,
PL5)
A-*C,
B-~C = A V B - ~ C
PLg)
A-~B,
A-~C ~ A - ~ B & C
PLS)
A~(B--C) =A~B--C
PL 9)
A-~A °
and for predicate logic ( x C
not containing
x
free)
Q Ii)
B--A(x) ~ B - - W A ( x )
Q 21 ) Q 3i)
VxJ~x-~At At-~ XxAx
B-~A V B
a variable of sort
i ~ t
a term of sort
i ,
Q%i)
Ax~B
= SxAx~Bo
In applications assumptions
of
Q Ii
comtaining
and Q 4 i x
the premiss is supposed not to depend on
free, i.e. has been derived without the use of
such assumptions. 1.1.4.
G~del's systemo
For the purpose of verifying interpretation
the soundness
of the so-called Dialectica
(see chapter III), Godel suggested another system, based on
QI-Q4,
PL2,
3, 7, 8, 9 and
PLI0)
AVA~A~
A~A&A
PL11)
A~A
VB,
A&B~A
PL12)
AVB
~ BVA,
PLY3)
A~B
~ C VA ~ CVB.
A&B
~ B&A
This system has the advantage
of keeping complexities
in the rules and axioms there appear fewer logical previous 1.1.5.
down to a minimum
(i.e.
symbols than in the
system). E~uivalence
of Spector's and G~del's
In Hilbert- type systems, tions be represented
we may either suppose deductions from assump-
as finite
sequence being an axiom,
system.
sequences of formulae,
an assumption,
each formula of the
or obtained from formulae appearing
earlier in the sequence by means of a rule of the system.
(This form is
often quite convenient for actual arithmetization ; however, noted that in some cases it is more natural volved to be indicated pictori~ representation
it should be
to suppose the rule or axiom in-
explicitly at each element of the sequence).
A more
is by means of deduction trees, which we shall use
below. It is perhaps useful to remark already here, themselves
that in case the proof trees
are objects of study (as in chapter IV) we must think of them as
being completely presented by a tree of formulae together with an indication which rule or axiom is applied at each node.
However,
in presenting proof
trees pictorially below, we shall not always explicitly indicate used,
the rules
so as not to encumber typography.
A proof given as a sequence may be thought of as being obtained by consistently extending the partial order of the tree to a linear order. If F
~g
denotes deducibility
a set of assumption formulae,
in Spector's
system,
~G
in G~del's
sense that r~A
~
F~GA
(for first- and higher-order
system,
then the two systems are equivalent in the
languages,
o n , or many-sorted)
F ~sA PL I)
(F~ 3)
follows by the f o l l o w i n g d e d u c t i o n s :
A-~A & A
(PLIO)
A &A-~A
(PL11)
A-~A
PL 4)
A&B~A
(PL 3)
is the second half of
PL11
(PL12) B&A~B
(PLll)
A&B~B&A A&B~B
(PL 3)
A~AVB
is the first half of
B~BVA
(PL11)
BVA~AVB
PL11 (PL12)
B~AVB
PL
5)
(PL13) A-~C (ass) (ps 3)CVA-~CVC Cvc-~C (PZlo) (PL13) B-~C (ass)(PL12) AVC-~CVA CVA~C (PL 3) Av~-~AvC AVC-~ C (pT, 3) AVB
-~ C
(PL I
P~ 6)
deduction)
B&C~B&C (C~B&C) A ~ (C~B&C) (PL (PL12) C & A ~ A & C A&C ~ B&C (PL C&A ~ B & C (PL 7) (ass) ~ c C~(A~B&C) ( ~ 3) A~ (A~B&C) (PL 8) (PLIO) A ~ A & A A&A--B&C (PL 3) (ass)
A ~ B
B~
(p~ 7) (PL 3) 8) 3)
A ~ B&C Conversely,
PLIO)
we v e r i f y that A -" A
~$A
A -~ A
= ~ ~GA
(PL 5)
AVA--A PL11)
is part of
PL12)
(PL 5) B - ~ A V ~
by the f o l l o w i n g d e d u c t i o n s :
A~ A A -~ A & A
PL 4.
BVA
(P~ 4)
A-~AVB
(P~ 4)
A&B~A
(PL 4)
-* A V B
(P~ 6) A & ~
(PL 4) A&B
-~ B & A
A ~ A
(PL 6)
PL13)
(ass) A--B B--CVB (PL4) (PL 3] A - CvB (PL 5)
(PL4) C-- CVB
C VA
1.1.6.
~
C VB
Equivalence of Spector's and Kleene's formalization.
Yet another Hilbert-type system is described in Kleene 1952, chapters IV, V.
The equivalence with Spector's Warning.
system is proved in Specter 1962.
In one respect our conventions differ from Kleene's : Kleene
1952 permits application of Q I, Q 4 assumption formulae
also when the variable occurs in
(i.e. the assumption formulae are treated as if they
are universally closed) ; if Kleene wishes to indicate that certain variables are to be treated as parameters,
and hence are not permitted as proper
variables of an application of Q I, Q 4 the notation 1.1.7.
("variables held constant") he uses
~x1°''Xn
A natural deduction system.
We now distinguish between parameters and (bound) variables. shall use
a, b, c, ...
for the parameters,
x, y, z, .o.
Below, we
for the variables.
We describe the system briefly (a detailed and rigorous description is in Prawitz
1965 ; more briefly in Prawitz
1971).
The rules may be schematically described as follows : AI)
A
B
&El)
A&B
~r)
A AVB
--I)
¢
A&B A
AEr)
B
~l)
A&B B
VE)
~ AVB
~ vB
C
C
c "E)
A
A'B
B A..-, B
VxAx
At
gx.Ax
~
ZxAx ZxAx
C C
mz
~--A
In the explanations below, it should be taken into account that we are primarily concerned with formula ~ccurrenees
(foVs), i.e. a formula together
with a position in a tree-like arrangement of formulas. rence
A "
means "an occurrence of the formula
A " .
"A formula occur-
We shall sometimes
loosely use "formula"~ when~ as is apparent from the context, formula occurrences are meant.
The I - r u l e s rules,
are called introduction rules, the
elimination
since a logical constant is introduced in the conclusion,
ly eliminated from a premiss. .4 I"
E-rules
So we sometimes write . 4
respective-
introduction " for
etc.
We suppose deductions to be represented in tree form ! the top formulas of the tree are then the assumptions,
and the (uniquely determined)
formula (occurrence) is the conclusion of the deduction.
end
Each formula occur-
rence is either a top formula, or the conclusion of an application of one of the inference rules, its immediate predecessors being the premisses in the application of the rule.
At applications of
VE, ~E, ~I
certain assumptions
(of the form indicated by the formulae crossed out in our list of rules)are discharged ; a discharged assumption is said to be closed (by the inference). Only assumptions which have not been discharged previously of the proof tree above the one considered)
(i.e. at a node
can be discharged.
also be stressed that not necessarily all assumptions cf the same form occurring above the application of
It should
(possibly even none) VE, ~E, ~I
are dis-
charged. We shall think of the assumptions
to be grouped into assumption classes;
each assumption class consists of a number of occurrences of the same formula. All formulas of an assumption class are always treated simultaneously,
i.e.
at each application of a rule in the deduction either all formulas of the class are discharged or none of them is discharged. A formula occurrence above
A
A
is said to depend on the assumptions
that have not been closed by some inference above
In the applications of tions containing
a,
VI
the premiss
Aa
and in an application of
the upper occurrence of
C
containing the parameter
standing
A .
must not depend on assump~E
of the form
~xAx
must not depend on assumptions other t h a ~ a.
In applications of
~,
~I
a
C , Aa
is called the
proper parameter of the inference ; a parameter is a proper parameter of a deduction
if it is used as the proper parameter of an
VI, ~E
inference.
The open assumptions of a deduction are the assumptions on which the end formula of the deduction depends.
A deduction is said to be closed if there
are no open assumptions° We shall always assume a completely described deduction to have specified at each node which rule is being applied, and for the assumptions malae), at which inference
(top for-
(if any) they are discharged.
With respect to the rules, we wish to introduce some further terminology. In an application cf an
E-rule,
the constant to be eliminated, the other premisses,
the premiss containing the occurrence of
is called the major premiss of the inference ;
if any, are called minor premisses.
So, in our list of
inferences above, &E, VE, ~E,
A&B,
VE, ZE
A VB, A ~ B ,
respectively.
application of an
I - rule or ~ I
VxAx, ~mAx
are the major premisses of
It is convenient to call any premiss of an also a major premiss.
It will be obvious that deductions which only differ in the naming of their proper parameters may be regarded as assentially the same.
It is easy to
verify that we may always select our proper parameters so as to satisfy the following requirements, (cf. Prawitz (i)
for a given deduction
[
The proper parameter of an application H
(ii)
of
A
from assumptions
~
of
VI
in
H
only in formula occurrences above the consequence of
The proper parameter of an application
~
of
ZE
in
(iii) Every proper parameter in application of the Examples.
H
Vl-rule
occurs in ~o
H
only in formula occurrences mbove the minor premiss of
1.1.8.
F
1965 , chapter I, § 3).
occurs in ~.
is a proper parameter of exactly one or the
ZE-rule
in
H.
We give some examples of deductions in the system of
natural deduction ; the theorems derived will be used later on (§ 10). In the examples,
"(ass)" marks an assumption which is not discharged
"open") in the deduction. number "(I)",'<2)", number.
Assumptions which are discharged are marked by a
etc., all assumptions in the same class getting the same
This number is then repeated at the application of a rule where the
assumption is discharged. I)
(i.e.
(I) A
A -- B (ass) B
B~A
(I)
If)
WA~(1 ) (2) -~ A a
Aa
-~E
A (3) ~ ~ VxAx
~ VxAx A -1~Aa
-i (I) -~E
- I (2) vl
Vx -7-IAx -~ I ~
VxAx -~ Vx -~-~Ax
(3)
zzx)
(1) A
(2)
~A
A (3) m m ~
A
Since also
A--B
z,,
(2) (-1)
"mA ~
"7-7-qA
it follows that mmmA
°
(I) (3)
= (A-~)
(4) A I~I~B
(3) (2)
A -~ -~-7B
-~
(2)
" ~ - q -q A
mA ~
-~B
(I) A
7mA
(2)
-7-~mA -~ m A
(2) A
mi
-.-i--1 A
__A (~) mA (3)
IV)
(1)
(4)
(2)
~k B
(I) _ _
m (A-* B)
A-~B
(~)
(4)
]k
B
-7mA
(2)
mmA
-~ m - 7 B
(5)
x,(3)
-~-~ (A- B) (4)
of
~I
(4)
mB
-TmB
An application
-~ (A- B)
A-*B
(5)
.
in (I) y i e l d s
(~A -- ( T m A
~ ~mB) -- m m B )
(III)
and thus
V)
-7-7(A&B)
-~ m m A
& m-IB
may be proved very similarly
to II.
10
(I) A
Further
A&B
(~) ~ A & B (4) ~-~A & ~ B
B (2)
A
(~)
-m-mA
A :B
& -~-~B ( 4 )
~A
(2)
-~'q B
A
(3) (4)
Hence I. I. 9.
The following schemata and rules are derivable in Spector's
Lemma°
system : (a)
A-
(b)
A = B - A
(B--A)
(o) A ~ B , A--(B--C)=oA--C (d) [(A-- (~--C)) & (A-~B)] & A--0 (e) ( A - (B--C)) -- [(A--B) -- (A-O)I (f) [ ( A - - B ) - (A~C)]-- (A--(B--0)) (~) A - - ( B ~ O ) , A - (c-D) = A - - ( B - D ) (h) #-- (B--C) = B - - (A--C) (i) (A-c) & (B-c) - (A vB ~ o) (j) (A-~) - [(A-o) - ( A - - ( ~ C ) ) ] (k) A -- (B& O--D) = A -- (B--(C--D)) (l) A-- ( B ~ ( C - - D ) ) = A - (~& c-D)) . Proof. (a)
From
A&B
(b)
I=ediate
(o)
A-A
~ A
(PL4), A ~ (B--A)
from (e) with
(PLI)
A-~
by (PL7).
PL2o
(ass)
A~A&B
A--(B~C)
ass
A&B--C A~C
(d)
~e put
B ~ [A -- (B ~ C ) ] & ( A ~ B ) ,
{~Lq) :., ~
B -- (A--B)
A
B (PL4)
: -- ( A ~ ( B ~ C ) )
(PL4)
Here we have used an abbreviated notation for the proof tree ~ (0) next to a horizontal line indicates that the line represents a part of the
11
proof tree of the same f o r m as for the part of (c) above. words, we use
(c) as a derived rule to a b b r e v i a t e
Similar conventions
(e) (f)
our p r o o f trees.
are u s e d below.
A p p l y PL7 twice to (d). We put
D m (A--B) -- (A--C),
E m (D&A)
E -- D (PL4, PL4, PL3)
E -- A (PL4, PL4, PL3) E --
C
A - ( B - c) (ass)
(A-B) -
(PL4)
(c)
(f) is obtained by two a p p l i c a t i o n s
(g)
& B°
D -- (A--C)
E -- (A--C)
of PL7 to
A - (c-D)
(A--C) ( ( e ) ,
PL2)
(A--B) -- (A--D)
E-- C .
(~ss)
(A--C) -- (A--D) ( ( e ) , P~2) (PL3)
(f)
A - (B-D) (h)
B&A
-~ A B&A
B&A
A~
-~ B
-~ A & B
(B~C)
A&B~
B&A~
ass
C (P~3)
C
B - ( A - c) (i)
Let
D
m (A ~ C )
& (B~C) ,
D - ( A - c) (h) A - ( D - c)
E ~ A VB ~ C
D - ( B - c) (h)
B-
( D - c)
A v B -- (D-- c) (h) D--E
(J)
Let
D ~ (A-,B) & ( A - - C ) ,
-- A
~ -, (A-, B) (e)
~ -= D&Ao ~ --, A
E-~B
E -, (A-, C) (e)
E-~C E-~ B&C
]) --, (A -, B & c )
(A-, ~) --, [C.A--' c) -, (A-, B&C) ] (k)
By r e p e a t e d use of PL4, PL3
(A&S) & C --' A ~ ( B ~ C)
A -- (B & C -, D) ( ~ s ~ ) (A&B)
& C -- A & ( B & C ) (A&B)
& C ~ D
(A ~ B ) -- ( C ~ D)
A & (]~&C) --, ])
In other
12
(1)
Similar to (k), but arguing in the inverse direction.
1.1.10. Deduction Proof. of
theorem for Spector's
We write simply
deductions
for
kS"
A~B
from
F, A ~ E B
B
from
F U {AI
itself and therefore
either
I n the f i r s t
ease ,
Fb
B ,
In the second case,
F ~
AHA
by PLI .
In the third case ,
F ~
A--B
by 1.1. 9 (b).
~p.
of length
~k,
can be transformed
F o
The deduction has length I ; then the deduction
Induction
= F ~sA~B"
We show, by induction on the length
, that a deduction of
into a deduction of Basis.
~
system.
B 6 ~,
or
B ~ A , or
B
hence by 1 . 1 . 9 (b)
consists of
B
is an axiom.
F ~ A~B.
Assume the assertion to have been proved for all deductions and let
At, ..., Ak, B
thesis, we have already
shown
be a deduction.
F ~ A~A i ,
If
B
is an axiom, or belongs
If
B
is obtained from the
to
A
By induction hypo-
I < i < k.
F U {A},
we proceed as for the basis step.
by application
of a rule, we must distin-
1
guish various Case PL2 : (1.1.9
Aj
(e))
Case PL3 :
cases.
~A l --S
A i ~ A~A
r b (AM(A''A''))' r~
"
~e have
F~A--Ai,
r~A--
(Ai--B);
r ~ [A--(Ai--B)] -- [(A--Ai)-- (A--B)] , hence '' , A.$ ~ A " ~ A ''' ,
r b
(A-- (A"--A'")) ;
B ~ A'--A'" .
application
of
also
F ~ A--B. ~e have 1.1.9 (g)
yields
A'(A'--A'") o
Case PL5:
Assume,
by hypothesis
~hen r ~ A - - [(B--D)~(C--D)] W[(B'D)~(C'D)]'[(BVC)'D]
r ~ A -- (B--D) ,
~ ~ A -- (C--D) .
(1.1.9 ( j ) ) , a~d (1.1.9 ( i ) ) . so
rbA--[(SvC)--B]
(P~3). Case PL6 :
Similarly,
Case PL7 :
Use 1.1.9 (k)o
Case PL8:
Use 1.1.9 (i).
Case QI
:
Use PLS:
Assume F~
A&C
using 1.1.9 (j).
F ~ A -- ( C ~ B x ) ,
F,
A
-- Bx ; then apply Q1 :
not containing
F ~ A&C
~ VxBx,
x
free.
and thus by PL7
r ~ A - (C--VxSx) Case Q4 : apply Q4 :
Assume F~
F ~ A -- (Bx--C) o
~Bx--
Use 1 . 1 . 9 ( h ) ,
so
(A--C) ; apply 1.1. 9 (h) a g a i n to o b t a i n
r ~ A - (~Sx-- C). 1oto11. Theorem. system.)
P ~N A i f f
from assumptions Proof.
(Equivalence
F
F ~5 A
between natural deduction and Spector's
( F ~NA
indicates
in the natural deduction
First we show that if
F~gA
,
then
that
A
can be deduced
system). F~NA
matter : we have to show that (a) for the axioms
A
.
This is a routine of Spector's
system,
13
~ N A , and (b) that for an instance of a rule system there is a deduction deductions
in
~N
FI,F2 ~ N F 3
FI,F 2 = F3
in Spector's
(in fact, as we shall see, the
for axioms and rules are uniform in the formula variables
used to describe the axiom (schemata)
and rules).
For example, A
is represented by the natural deduction proof A
A ~ B V C
and the rule
Vl
v B
A-~AVB
(PL5) by
A
A-- C..E
AVB
B
etc., etc.
B-~ e
C
C
VE
C AvB-~ It remains
-~I
C
to be shown that
ffF.~NA
,
then
have to show that each rule of the natural to a derived rule in Spectorts
system.
F~sA
°
To see this, we
deduction calculus
For example,
for
vE
corresponds we have to
show : If
F,A
By the deduction
rksAvB-c,
~S C,
F, B ~ S C,
theorem,
F ~S A M C ,
sowithP~
The only crucial case is
then
F,AVB
F ~s B~C,
~S C° and with PL5,
r,AV~sC.
~ I ,
but this is provided by the deduction
theorem. 1.1.12. Remark on the e~uivalence
proofs in 1.1.11 and !°Io5 under addition-
al axioms. The equivalence
proofs remain obviously valid if we add further axioms
(in the case of the natural deduction
system,
formulas but are not counted as assumptions). rules, however,
the equivalence
of the deduction theorem for Spectorls
1.1°13.
system,
essential
in 1.1.11, has to
of further cases corresponding
to the
rules.
Sequent calculi.
We do not make use of Gentzen's (cf. Gentzen 1935 , Kleene is referred to Prawitz 1.1.14.
In the case of additional
proofs have to be extended ; e.g. the proof
be extended with the consideration additional
axioms may appear as top
Convention
istic system).
If
1952 , § 77) ; for an equivalence proof the reader
the classical
is any formal
sorted) predicate logic, al (many-sorted)
and its variants
1965 , Appendix A.
(for indicating ~
calculus of sequents
predicate
~c
equivalent
of an intuition-
system based on intultionistic
denotes the corresponding
logic (,c,, for "classical").
(many-
system with classic-
14
§ 2.
~
1.2.1.
2
~
7
~
=
~
~
=
~
Contents of the section.
some theorems on definitional text becks,
~
=
~
~
"
In this section we have brought
extensions,
together
which are not emphasized
but which will be used quite frequently in this volume,
explicitly or implicitly.
For the proofs,
Under an intuitionistic shall understand
(many-sorted)
we must refer to Kleene
predicate
a system of intuitionistic
in most either 1952, §74.
calculus with equality we
predicate
logic with equality =
satisfying the axiom
x)
Vx(x =
and the schema x=y
~
(Ax~Ay) .
It readily follows that
=
is symmetric
and transitive.
(In what follows equality need not be given for all sorts of variables, one makes some obvious formulation
(many-sorted) additional 1.2.2.
a formal
system
logic with equality, predicate
if
logic,
~ ~
intuitionistic
H', H
predicate
logic,
(the non-logical
be formal and let
is an extension of the language of
H,
contained in the set of theorems of
Thm(H') Let F
F~Fm(H) o (or w.r.t,
H
is said to be based on intuitionistic
the equality axiom and schema, and possibly
Let
servative extension of
situation to the reader.)
is based on the rules of intuitionistic
axioms and axiom schemata
Definition.
if
in the theorems ;but we shall leave the
of the theorems in this more general
In this section, predicate
stipulations
axioms).
systems based on (many-sorted)
H~'
(i.e. the language of
and the set of theorems of
H' ).
(or conservative
Then
H'
over
H),
~
~' is
is said to be a conif
n Fm(H) = Thm(H) . Then
r)
~'
is said to be conservative
over
H
relative
to
if
Thm(~') n r ~ Thm(~) n r (we s h a l l 1.2.3.
Definition.
tuitionistic of
sometimes abbreviate
predicate
Let
H',
logic,
~ ~ if there is a mapping
this H
as
i'
be f o r m a l
and l e t ~
~F = HnF1. s y s t e m s b a s e d on m a n y - s o r t e d
H~H,
Ht
is
of those formulae of
only free variables are of sorts occurring in
Fm(H) ,
in-
s a i d to be an e x p a n s i o n Fm(~') such that
where the
15
(i) (ii)
[, b A~-~ ~ ['WA - ~W~A
(iii)
~
We say that
~ A
for
[i
A 6 Fm(~) . relative to
is an expansion of
F,
F~Fm(H)
if (iii)
is weakened to (iii)' ~A ~ A 1.2.4.
for
Definition.
A 6 F. Let
H', H
be formal systems based on many-sorted
predicate logic, and let the language of
H'
be obtained by adding non-
logical constants (i.e. constants assumed to be in the range of certain sorts of variables, and predicate constants). al extension of
H,
Then
H'
if there exists a mapping
is said to be a definition-
~
such that (i), (ii), (iii)
hold and (iv)
~(~) = ~ ,
(v)
~((Q~)A) ~ ( Q x ) ~
(i.e.
~
1.2.5.
~(Ao B) = ~ o for
~B
for
o ~ V, a,
Q ~ V, ~.
i s a h o m o m o r p h i s m w.r°t, logical operations) Remark.
A definitional extension is an expansion, and an expansion
is a conservative extension. 1.2.6.
Theorem
(Addition of symbols for definable predicates).
be
Let
any theory based on (many-sorted) intuitionistic predicate logic, and let A ( x I ....
,xn) ~ Fm(~),
Xl, ..o, x n o
Let
where
~
all the free v a r i a b l e s
of
A
are a m o n g
be obtained by addition of a predicate symbol
M,
with axiom A(x1,..o,Xn) ~--~M(Xl ..... Xn) . Then
H'
Proof°
is a definitional extension of
H.
Trivial ; see Kleene 1952 , lec. cir.
scribe the mapping P
~o
(Kleene 1952 ' § 74, Example I) For future reference we de-
required by the definition of definitional extension :
(a)
If
is a prime formula, not of the form
(b)
~o(Mt1...tn) ~ A(tl,..o,tn)
(c)
~o
Mtl...t n,
~o P ~ P
is a hemomerphism W.rot. the logical operations.
1.2.7o
Theorem
Let
be a theory based on (many-sorted) intuitionistic predicate calculus
H
with equality.
(Addition of symbols for definable functions).
Assume, for a formula
A
containing free only
Xl,...,Xn,Y :
~ ~ y A ( x I .... ,xn, Y) where Let
~yBy H'
abbreviates, as usual,
be obtained from
together with an axiom
H
~y[By&Vz(Bz--z=y)].
by addition of a new function symbol
f ,
16
A ( x 1 , . . . , x n, f(x I . . . . ,Xn)) and extension of the axiom schemata to formulae in the extended language. Let
~I
be the mapping of
Fm(~)
into
Fm(~)
Let us call a term not containing occurrences of the form
ftl..°t n , an
is a plain
f - term ; if
defined as follows. f , f - less ; and a term
ti,...,$ n
are
f-less,
ftl...t n
f - term.
We define occurrences Assume
~I of
for prime formulae f - terms in
q> 0
for
term in
P ; let
ed from
P
v
P .
P
by induction on the number
~I P ~ P,
if
q=O
ftl...t n
be a variable not occurring in ft1.o.t n
of
of term occur-
be the first occurrence
by replacing the occurrence
q
.
P ; let (on some standard enumeration
rences in prime formulae)
Then
of
of a plain
P ; let by
C(v)
f-
be obtain-
Vo
~I P ~ ~v[F(t1°..tn, v) & ~IC(V)] °
(The variable ~I
v
may be assumed to be chosen in a standard manner.)
is defined for logically
is a homomorphism Wor.t. transform of
compound formulas by the requirement
the logical operations.
~1 A
that it
is called the
f-less
A .
Then the assertion of the theorem is as follows: ~'
is a definitional
extension of
the definition of definitional
~
(with
extension),
~I
as the mapping required by
provided the additional
axiom
schemata satisfy the condition (a)
if
A
Proof.
is an axiom by an additional
Kleene
1.2o8°
Theorem
Let
be a lanEuage
~
(Replacement
(n+1) - a r y predicate of
~
of 1.2.6), and ~, ~'
(i)
~(~)
[
(iii) If
of
F .
~I(A)
(resp.
Let
~o,~I
such that
is the
~ ~I
A .
in
f-less
systems,
be mappings
~o(A) A
symbol
by
f
symbols). and an
of the formulae
is obtained by replacing f(t1,oo.,tn) = t
transform of
A
(cf. proof
(see 1.2.7).
based on intuitionistic
predicate logic
such that from
~
by
omitting
f,
~(~')
~s obtained by
F ;
contains an axiom A
~
n - ary function
F(tl,...,tn,t )
is obtained
omitting
(ii)
symbol
be two formal
with equality,
then
of function s2mbols by predicate
containing an
into the formulae of
every occurrence
Let
axiom schema,
1952, § 74 (Theorem 42).
H~y F(x I ..... xn, Y)
is a non-logical
axiom of
~
(resp. of
~ ~I(A)).
Then
~b~1%( A)~A,
H'W%~I(A) eeA,
~' ) then
~' ~ 0 o ( A )
17
+rbA= Note that
~,+rbA~
~'+~or ~ %A, ~+~Ir~I A
if
the
H"
is
common e x t e n s i o n
of
H
and
H'
containing the axioms and axiom schemata of both, then al extension of
H
as well as
in H"
the language
~,
is a definition-
H' .
Proof.
See Kleene 1952, § 74, theorem 45.
1.2.9.
Theorem
Let
H
be a system based on intuitionistic
let
M(x)
(Addition of defined sorts of variables).
be a formula of
(many-sorted) predicate logic ;
Fm(~) , containing free only
x,
and
~ ~M(x). Let
~'
be obtained by addition of a new sort of variables
(say
~,~,~,...I,
with rules for term and formula construction also extended to the new variables, with the axiom schemata and rules of
H
(but where in an axiom or
axiom schema involving quantified variables,
the axiom or axiom schema is not
to be generalized by replacing quantification over the original variables by quantifiers over the new variables),
and with the new axiom and schemata and
rules ~,
Mt ~ ( V ~ A _ ~ A t ) , Mt ~ ( ~ t ~ A ~ )
Then
H'
Proof.
is an expansion of
,
H.
(K!eene 1952, § 74, Example 13).
as follows.
Let
A
We define a mapping
We describe the correlation
~2
be a formula containing a set
V
~V
by induction on their
for the subformulae of
A
of free variables.
complexity : ~ ( P ( ~ I ..... ~ n ) ~ P[RI'''°'Yn)
for prime formulae
~v(~lO ~2) ~ ~v(B1 ) o ~v(B2) ~V(QX)B) (Qx) ~V(B)
for
P,
o ~ ~, v, &,
for Q ~ v, ~,
~v(V~Bx.i) ~ V Y i ( ~ y i ~ ¥ B ~ i ~
Here
y1,...,y n
are variables not in
V,
and
~i,..,,~ n
be a complete list of the new variables occurring in Then we put
may be assumed to
A o
~2A = ~NA .
1.2.10. Alternative approach ~0 defined In 1.2.9, the defined sort
sorts of variables.
of variables was treated as a subset of the
original set of individual variables,
with respect to the formation rules.
If we wish afterwards to introduce symbols for functions defined on of elements of
{x I N x l ,
as completely disjoint, and state the formation rules segarately. We then need axioms Vx ~ y ( ~ = y ) (with a primitive or defined Vx6 M ~
(x=z) -
n - tuples
it is preferable to treat the new sort of variables
Cf. e.g. K ~ e i s e l - T r o e l s t r a
1970, 3.3.4).
= ) ,
18
§ 3.
Intuitionistic f i r s t - order arithmetic.
======================================
1.3.1.
Contents of the section.
In this section we describe intuitionistic
first-order arithmetic and notational
conventions,
choice of pairing and
sequence codings, and the formalization of elementary recursion theory.
1.3.2.
Language of
I~.
The language of Heyting's arithmetic
first-order language, with logical constants be identified with a constant
O
O = I ),
numerical variables
(indicated by
(zero), a unary function constant
S
=
is a
(which may x, y, z, u, v, w ) ,
(successor),
function symbols for all primitive recursive functions and a single binary predicate constant
~
V, Z, ~, &, V, ~
constant
(see below in Io3o4),
(equality between numbers).
Terms
and formulae are defined as usual. 1.3.3° H~
Axioms and rules of
HA.
is based on intuitionistic first-order predicate logic and contains
in addition the following axioms : "X
=
X
x = y
&
z=y-~x=z
x i = x'11 -* ~(Xq . . . . . F~
for any
SxjO
Xi .....
Xn) = ~ ( x I . . . . .
n - ary function constant
x~ . . . . .
Xn)
I< i < n
,
Sx=Sy-~
x=y
,
the definining axioms for the primitive recursive functions
(see 1.3.4) and
all instances of the schema of induction I~O
A0 & Vx(Ax--A(Sx)) -+ V ~ x .
1.3.4.
Defininin~ axioms for primitive recursive functions°
The precise selection of initial functions and defining schemata is not relevant to our discussion of intuitionistic arithmetic in this volume. A very simple set is as follows : Initial functions are the zero-place n-place
projection function
In i'
0 ,
1~i~n,
I - place successor for all
n,
S , and the
satis[~ng
l~(x I .... ,xi,...,Xn) = x i ° Our defining schemata are composition and recursion. Composition is described as follows. and
~
an
If
~I' "°'' ~m
are
n-place
functions
m - place function which have been defined before, then we may
introduce a new
n-place
function
g
with axiom
19
~(x~ . . . .
'~n ) = * ( ~ ( ~
. . . . '~n ) . . . .
' %(~1 .....
is said to be defined by composition from Recursion : if
~
is an
n - place function and
which have been defined before, function
~
~,
Xn))"
~1,.o.,~m. @
a
n + 2 - place function
then we may introduce a new
(n+1) - place
with axioms
f ~(0, x 1 ' . . . . Xn) = ~ ( X l ' ' ' ' ' X n ) , x n)
~(Sy,Xl,In this case, Io3.5.
Rule
~
is said to be defined by recursion from
•
~, ~.
and axiom schema of induction.
Instead of using
IND , we might also have added the rule of induction
Rule_IND
(x
$ ( ~ ( Y , X 1 ..... X n ) , Y,X I .... , x n)
BO , Bx ~ B(Sx) ~ By
not occurring in assumptions on whieh
Bx ~ B(Sx)
depends).
A minor variant : BO,
Vx(Bx~B(Sx))
It is obvious that
IND
~ By.
implies
R u l e - IND o
For the converse, we must
apply the rule to Bx ~ AO & Vy(Ay--A(Sy)) - - A x .
1.3.6.
Natural deduction variant of
The description in 1.3.2- 1.3.4 of
HA. ~
is independent of the particular
formalization of intuitionistic predicate logic which is used. obtain a natural deduction variant of
~
However,
to
which is especially suited to the
proof-theoretic researches in chapter V, we have to make some changes in the non-logical part also. As in the discussion of intuitionistic predicate logic, we distinguish between parameters and variables.
We have one individual constant,
single unary function constant, denoted by cate constant constants
=
(equality),
FI, F2, F3,
...
S
(successor), a binary predi-
and a denumerable sequence of predicate
for the graphs of primitive recursive functions.
To the rules of predicate logic we add rules (the basic rules)
t= t
i
t
t i = t!
= t'
0 , a
t = t'
t'=
Fktl...tioO.tn
Fktl-..t~...t n
t"
~0
O = St
t=t'
St=St'
#k
St=St'
t = t'
and, for example,
parallel
to the first set of initial functions
schemata given in 1.3.4, we introduce
Fk'S
and defining
such that
FkX I --. x i ... XnX i and if Fk
Fko , Fkl , ..., Fkm
have already been introduced,
we introduce
an
(k ~ ko, kl, ..., km)
Fk2tl.--tnt:,-..,F!~mtl-O.tnt~,Fkot~.*.t~
Fklt1"''tnt ~
t
F k t1°--tn t and if
Fm, F n
have already been introduced,
we introduce
an
Fk
(k> m,n)
with two rules : F m tl...t n t
Fktot I" ..t n t
FkOt1.°-tnt (Thus to each F
Fk(Sto)tl...tnt'
n - ary primitive recursive
such that~ intuitively,
Finally,
Fnttotl...tnt'
function
~
there corresponds
~(Xl,...,Xn) =y~--~ FXl...XnY
an
o)
we add a rule of induction * in the form o
l.o'
A(?a)
Ao ~t
where
a
is a parameter not occurring in assumptions
except of the form a
on which
A(Sa)
depends
Aa.
is called the proper parameter of the
IND - application.
A proper parameter of a deduction may now be a proper parameter of an application
of
VI, ~E, IND.
We shall also agree to call the premisses premisses ; the premiss of the form premiss of the form
A(Sa)
AO
of an application
of
IND
is called the zero premiss,
the inductive
premiss ; the premisses
minor the
of the
basic rules are regarded as major premisseso The conditions formulating
on variables given in 1.1.7 may now be sharpened by re-
(iii) as
(iii) Every proper parameter in application
of
VI, ~E
or
H
is a proper parameter of exactly one IND
in
H~
and adding
• ) In the sequel, IND will be used indiscriminately lation in 1.3.3 and the one given here.
to refer to the formu-
21
(iv)
The proper parameter of an application of INO
in
~
only in formula occurrences above the inductive premiss of The present formulation of
~
(say
Nat -~..)
described in 1 . 3 o 2 - 1.3.4 (to be called simply sense : obviously,
Nat -li~
is equivalent to
HA
below), in the following
Nat-HA*
obtained by reand
IND'
by the
Addition of symbols for the primitive recursive functions H
with their axioms is then an expansion expansion of
IND.
is equivalent to the one
placing the basic rules by corresponding implications, schema of 1.3o5.
ff
occurs in
HA
of
Nat - H A * ;
also,
So there exist mappings
(of. 1.2o7, Io2.8).
H
is an
~, ~'
such
that
(I)
~A
o~at-~
~A
=
I~ ~ ~'A
~ff ff (~ +~A) ~ (~,A +~A). 1.3. 7 .
Eliminability of disjunction in systems containing arithmetic.
In intuitionistic arithmetic,
A VB~-~ ~ [ ( x = O ~ A )
we have
& (x/0~B)].
In order to show that this may be taken as a definition for show that the axioms and rules for
V
V,
we have to
are derivable for this defined connec-
tive from the rules and axioms for the other logical operators. In order to see this for the natural deductive formulation, introduce the following notational sequence of) deductions, where Z
of the form
A.
If
H
convention : [~]
[A]
let us first
stands for a (finite
indicates a set of open assumptions in
is a deduction with conclusion
A ,
II [A] Z denotes the result of substituting in
for the occurrences of the class [A]
Z.
Now
vl
can be shown to be a derived rule as follows :
(I) o~o A 0=0 -~ A
--.I
o=o
sQ,,So
0/0 -~ B
is obtained as follows :
so=o
A
A_ Az s (1)
( 0 = 0 ~ A ) & (0~0-* B) ~ [ ( x = O - - A ) ~ (~/0 -- S)] VE
H
SO=O ~ A
SO/O ~ B
(S0=0 -- A) & (S0/0 ~ B) ~[(~=0
~ A) ~ ( ~ / 0 -- S ) ]
22
[A] Assume
Z
[B] ,
ZI
to be given. b=O O=b
b=Sa
O=Sa
(b=0"A)~(biO"
~)
A
(b=0--A)~(b~0-- ~)
b7 o
b=0 -- A
b=0
blO -
[A] Z
[B] Z'
C
C
b=O -* C
b=Sa -~ C
b=b ~[(x=0--A)
b=b ~ C
& (xl0-B)]
C
Further we note that we obtain corresponding
results for the Spector system
and the Godel system since the proof of equivalence also applies if we restrict ourselves I__nnpractice,
however,
we shall usually treat
this requires only very little additional many developments
between these systems
to the fragments not involving V
as a primitive,
effort in our proofs,
V o since
and moreover
then apply to predicate logic also, almost without change
or additions° 1.3o8.
Formulation
of
~
It is of course possible in favour of predicate
without any function
constants
natural deduction formulation by a binary predicate
x=y
symbols.
to carry the step of eliminating
S(x,y)
function
symbols
one step further than has been done in the
of
~,
and to replace the successor function
such that
~ s(~,~) - s(y,~)
s(x,z) ~ s(y,~) - ~ = y s(x,y) ~ o ~ y and the inductive
clause for a predicate
introduced by the recursion
constant
F
representing
a function
schema now appears as
FkXo...XnX & FnXXo°,oXnY & S(Xo,Z ) ~ FkZXoo°.XnY (where
Fn
has been introduced before
F k)°
Induction appears as A(0) & Vxy(Ax & Sxy -- Ay) This formulation is equivalent
-- V ~ x .
to our original formulations
(again by using
§ 1.2) in the same manner as indicated for Nat - H A in 1.3.6 (formulas A formulation of this type is used in chapter V°
(I)).
23
1.3.9.
Notational
and a b b r e v i a t i o n s
conventions.
We list a n u m b e r of n o t a t i o n a l
to be used in the sequel.
A) For f r e q u e n t l y used primitive notations
conventions
in c o m m o n use,
recursive
functions
and p r e d i c a t e s
such as "prd" for the p r e d e c e s s o r
we adopt
function,
satis-
fying prd 0 = 0 , cut-off
subtraction
prd(Sx) = x ; is d e n o t e d by
x " O=x, signature :
"sg",
x I Sy
~,
= prd(x ~y)
;
satisfies
sg 0 = 0 ,
sg(Sx) = S O ;
absolute d i f f e r e n c e
" I • - " I" :
Ix-yl = ( x ~ y ) + ( y ~ x ) ~ m a x i m u m and m i n i m u m "max",
"min"
max(x,y)
= x + (ylx)
min(x,y)
= max(x,y) %
For Kleene's
T - predicate
Ix-yl
-
and the result - e x t r a c t ~ u g
f u n c t i o n we use
T, U
respectively. B) Pairing° J' it' J2
are assumed to be a p a i r i n g f u n c t i o n f r o m
N×
N
onto
N , with
its i n v e r s e s : (I)
jlj(x,y ) = x ,
j2J(x,y ) = y ,
The use of a p a i r i n g f u n c t i o n
cases,
that
2x3 y
J' J1' J2
2j(x,y)
are p r i m i t i v e recursive.
= (x+y)(x+y+1)+2x
The p a i r i n g f u n c t i o n r e p r e s e n t e d (3)
x < j(x,y) ,
(4)
x < x,-
(x,y) . However,
N.
In n e a r l y all
w h i c h m a t t e r are given by (I), t o g e t h e r with the
may fix on some definite p a i r i n g function, (2)
is not essential ;
to encode the pair
to assume the p a i r i n g to be onto
the only p r o p e r t i e s
information
j2 z) = z.
onto the n a t u r a l numbers
e.g. we might have used Kleene's it is often convenient
j(jlz,
F o r definiteness,
o
by (2) has the additional
y < j(x,y)
O(x,y) < j(x,,y),
w h i c h we shall assume from n o w on.
we
e.g. by r e q u i r i n g
if
X+ y > 0 ;
y < y,-
advantages
j(O,O) = 0
O(x,y) < j(~,y')
that
24
C) ~ d ~ n g numbers
of_fin~e
~equ~ce~.
we also prefer
Troelstra
Of course, variables
to have a coding ontq
1970 , 2.5.3,
is to r a n g e o w z c o d e
running
of sequences
only
over finite
sequences
since they can be coded by natural
and
and V e s l e ~
a separate
1965).
sort of
a conservative
extension,
numbers ; cf. 1.2.9) ; but this is rather resources,
we may assume
the standard
as f o l l o w s :
function
of natural
that a certain variable
(as in Kleene
(obviously
For the sake of definiteness, pairing
sequences
(as used in Kreisel
solution would be to introduce
a heavy draw on our typographical
of
N
in order not to have to specify
numbers
an elegant
For a coding of finite
which N~e wish to avoid. our coding
to be constructed
we first introduce
codings
from
vu
u - tuples v1(x ) = x v2(xl,x2) = j(xl,x2) ~u+1(Xo,Xl, .... x u) = j(~e,~u(Xl .... ,Xu))~
there
.U
exist inverses
Ji
such that .U
J~u(~1 .... 'Xu) " ~ i ' now we fix our coding < >=0,
,U
~u(J1 = .... '~u ~)= ~
of finite
sequences
by
<Xo> ~def Sj(O'Xo) '
<x o .... ,Xu~ = Sj(U,~u+1(Xe, .... xu)), where
<Xo,..~,Xu>
denotes
number for the empty
the code number for
sequence.
The present
x u < <Xo,...,Xu,...> <Xo,.o.,Xu> As an abbreviation ~
ith(n)
def
is used
denotes
the code
choice of coding i m p l i e s :
'
< <xo,...,Xu,...,Xu+v>
for
v> 0 .
we introduce
to denote ,
the length
of the sequence
coded by
n ,
so
lth<xo,..o,Xu_1>=u°
the concatenation,
~Xo~oo.,Xu_1~
< >
<x>.
Ith< > = 0 *
Xo,...,x u,
so
* ~Xu,°o.,Xu+v~
= ~Xo,...,Xu+v~.
We put n~m
=def ~ n ' ( n , n '
n ~m
~def n ~ m
( ~, ~
are used in different
partial
ordering
for an arbitrary
between
= m)
& n~m
o meanings
finite
primitive
in this volume :
sequences,
recursive
I ° ) as the natural
as just defined ; 2 ° ) as a symbol
well-ordering,
in the discussion
of the
25 principle
of transfinite
for "reduces to". Let us write for
induction
TI(<) ;
3 ° ) as a metamathematical
symbol
In all cases the meaning will be clear from the context°)
(n)x
for a primitive recursive function of
n, x
such that,
n = (Xo,...,Xu_1>
tl(n)
(n)i=x i
for
(n)i=o
for i~u°
("tail of
n"
i
is a primitive recursive
tl(O) = 0 ,
tl(~) = 0 ,
The derivation of elementary
function
such that
tl(~* n) = n o
properties
of the codings
(a tedious affair,
which the reader might wish to skip) can be found in Kreisel and Troelstra 1970, § 2.
• e shall frequently have to use formalized proof-predicates provability.
ProofH(x,y ) ,
or
ProofR(x,y )
for ar~v "canonical"
proof-predicate
interpretation : x
is the g~delnumber
number
y.
"ProofH"
"Canonical" ditions,
for the formal
so as to make it possible
~ , with intuitive
of a proof of a formula with g~delrecursive.
some natural
to prove Godel's
(For derivability
II, pp.
system
may be assumed to be primitive
will mean that it satisfies
theorem for them. 1970, Vol.
and formalized
We use
conditions,
derivability
con-
second incompleteness
see Hilbert and Bernays
294- 295 , where they are described in detail.)
We put
Pr~(y) %of ~x Proof~(x,y). "PrH"
is the "provability"- predicate,
form.
(In our versions
of
~,
and may be assumed to be of
"Proof(x,y)"
may be represented
o ZI -
by a prime
formula.) A ~delsentence
for a system
(i.e. of the form
VxAx ,
of consistency
~,
~m
of
~
(containing,
say
primitive recursive)
~VxAx
,
s nd on assumption
~
)
is a
H oI - sentence
such that on assumption of
w- consistency of
~,
VxAx . A rosser-sentence
of
Ax
H
is described like a g~delsentence,
is sufficient for ~
mVxAx.
times called sentences independent In our standard formulation section),
Ax
Numerals. and especially
of
but now consistency
G~del- or tosser-sentences w.r.t. HA
are some-
~.
(the first one described
in this
may be supposed to be a prime formula. As syntactical fi, •
variables for numerals we use
(in chapter V
~, y, z, ~, ~,
there is a deviating local convention
26 concerning numerals). If
A
then
is a formula,
t
rA~
denotes its g~delnumber ; if
denotes its godelnumber.
are among
x1,°..,x n,
wise ~ that
we shall use the convention
A(~ I ..... ~n)
t
If all the free variables
is a term, of
A(Xl,...,x ~
(unless indicated other-
stands for the g~delnumber
of
A(~ I .... ,~n )
as
s(CA(x1,. .. 'Xn) 'Yl , .... Yn ) a function of Xl,..°,x n . is the godelnumber of the formula obtained by substitution of the numerals (More precisely,
~I'"
°o
'Yn
for
--
Xl,...,x n
= rA(~1,o.o,Xn)~ ) o contexts,
in
A , then
represented
VA(~I, ..°, ~ n) 9 by formulae
Note that in view of the preceding
ProofH~(y, WA(x I ..... -Xn) ~ ) , etc. are in
containing
x1~...,x n
HA
free°
of elementary recursion t heor2o
For some of our researches, (chapter I I I , §
,X n ) , y l , . ° . , y n ) =
.°
The notation may cause problems in more complicate~
1°3. I0. Formalization
as the
s(rA(x1,°
but suffices for our purposes.
conventions,
if
notably in the case of formalized
2) it is necessary
to know that the principal
s-m-n- theorem and the recursion
The reader may take this on faith,
realizability
theorems
theorem can be formalized
in
such HA.
or better rely on the detailed formaliza-
tion of recursion theory in Part I of Kleene 1969 ; by omitting there everything pertaining
to function arguments
recursive fun ctionals) theory in
H~A°
(The use of different
finite sequences is completely Kreisel
and Troelstra
formalized
of elementary recursion
pairing functions and encodings
irrelevant
of
in this context ; cf. remarks in
1970 , 2.4.15 , 2°5.3.)
Below we shall list the principal About the
(Kleene 1969 discusses
one obtains a formalization
facts needed.
T - predicate we may assume ff T(x, y, z) & T(x, y, z') ~ z = z' .
In our first version of functions are present, formula
HA ~ where symbols for the primitive recursive we may suppose
Txyz
to be represented
by a prime
CTXYZ = 0 o
We shall freely use Kleene-brackets cursive functions
{° I
as a notation for partial re-
and partially defined terms,
between partially defined terms ; in Kleene detail how these notations
and also the equality
1969, Part I it is shown in great
can be used as systematic
Let us, following Kleene,
denote as
abbreviations.
p - terms the class of expressions
satisfying the formation rules for terms and in addition : If
to, tl, .... t n
are
p-terms
then so is
It In(tl .... ,tn) . O
(Actually,
we have no need for Kleene-brackets
but it is no trouble including them°)
with more than one argument,
Instead of
{tol~(tl)
we usually
27
simply It O}(tl) ° Each p - term r e p r e s e n t s b l e s . We use f u r t h e r m o r e
write
a partial
~t ~def ~ x ( t ~ x ) Note that The
Jt
and
s-m-n-
primitive
t= s
,
recursive
t = s =def
!t
&
can be expressed
!s
as
function
&
free varia-
t ~ s
Z o1 _ formulae°
theorem
recursive
of i t s
may now be stated as follows: m function o ouch that n
There
exists a
l z l m + n ( x l , . . . , x m + n) = t s ~ ( z , x 1 , o . . , x m) }n(x, m+1 , ° . . , x m + n) This makes
it easy
to prove
the recursion
theorem:
n+1 (Consider
~~xl~n+l-(sntu,u),z1,.oO,Zn) t,
Ivln+1(U,Zl .... ,Zn) ~
then
; we can find a
Ix}n+1(s(u,u),zl,...,Zn)
v
; now take
such that y = s(v,v) ,
I~In(zl .... 'Zn) ~ Is(v'v) In(z1""'Zn ) ~ Iv}n+1(v'zl ..... Zn)
= {x}n+1(s(v,v),zl ~e
n
. . . . ,z n) = I x } n + l ( y , Z l , o . o , Z n
shall make frequent A convenient
use of the recursion
abbreviation
) .)
theorem.
is
Itl(tl .... 'tn) ~def I°'°IIt}(tl)l(t2)°'° l(tn) ° (Ackually, Ix}n(yl, .... yn) can be defined in such a way that
IxP(y I... yn ) ~ Ixl(y~ oo. yn) .)
28
§ 4.
Inductive = =definitions in = = = = = = =
HA
= = = = = = = = =
1.4.1.
We intend to show in this section how certain inductive definitions
of sets of natural numbers may be replaced by explicit definitions. result is used repeatedly, especially in § 4.4.
The
The reading of this section
may be postponed until needed.
1.4.2.
Definition.
Let
~X]
~ £(KA)[X]
denote the language of
extended by a single additional unary predicate symbol F
is the least class of formulae of
(i)
the formulae of
(ii)
formulae
(iii) if (iv)
KA
Xt' , t'
A, B E F
if
A E F,
t
a term of
then
X.
such that
are contained in a term of
then
~X]
P;
KA , are in
P ;
A & B , A VB E F ~ x A E P,
Vx i t A E P
(t
not containing
x
free,
H~A).
Our next aim is to show that formulae
A(X, x) E ~ X ]
equivalent to a certain type of standard formula of
can be shown to be ~[X]
(see the statement
of 1.4.4 below) ; we first need a simple lemma : 1.4.3.
Lemma.
Vx o i to[Y] Vx I i
Vx ~ t[y] A(~o(X,y), ~1(x,y), y) , recursive in Proof.
for suitable
is equivalent to
t, ~o' °I'
primitive
x, y, to, t I .
For our standard pairing function
with its inverses y
t1[Xo,Y] A(x o, x I, Y)
J1' J2
we shall assume
j
onto the natural numbers, and x<x'
-- j(x,y) < j(x',y),
-- j(x,y) < j(x,y,) .
We define $(y)
=
t(y)
supIt1[xo,Y] I X o ~ t o [ Y ] l J(to[Y], ¢(y))
~o(X,y)
= { jl x [ 0
if J2 x ! t1[JlX,Y] & Jl x ! to[Y] otherwise ,
~l(x,y)
= ~ j2 x L0
otherwise .
if
J2 x i tl[JlX,Y] & Jl x i to[Y]
Now assume
(1)
Vx° ~ to~Y]Vx I ~ tl[Xo,Y ] A(Xo, Xl, y)
x!t[y]
and let
Now either
%(x,y)
jl x ~ to[Y ] , j2 x ~ t1[JlX , y]
,
and then
~o(X, y) = jl x ,
J 2 x , and by (1) A(%(~,y), % ( x , y ) , y ) , or j l x > to[Y ] v j2x > t l [ J l x , y ] ; in t h i s case, %(~,y) = %(xw) = O,
and
=
since by
(I)
Conversely, let
A(O,O,~) ,
once again
A(~o(x,y), ~ 1 ( x , y ) , y ) .
29
(2)
Vx
and assume
%(x,y)
x it[y]~
1.4.4.
X o ~ t o [ Y ], xl!t1[Xo,Y].
Lemma.
H~A[~] (i.e.
= %,
~l(x,y)
Each formula ~
=x I, A(X,z)
If we put
J(Xo,Xl) = x,
so by (2)
A(x o , x 1 , y ) .
of
F
extended to the language
then
is provably equivalent in ~[X] ) to a formula of the
following general form (I) Proof.
~xVyit(x,z ) [P(x, y, z) V (Q(x, y, z) & Xt'[x, y, z])] . To prove the lemma we have to show that formulae equivalent to &
formula of type (I) satisfy the closure conditions (i) - (iv) in the definition of F. Below we shall omit all variables which are redundant in the context of the argument.
(i)
Let
Pz
be an arbitrary formula of
P~ ~ (x,y (ii)
Xt'
HA.
Then obviously
~vy!o (Pz v [0= I ~ xo])
are assumed not to occur free in
P).
is equivalent to
~Vy<_O (o~ 1 v [o=o & xt,]) (t' (iv)
does not contain
x,y).
We note the following equivalences
(2)
SxyVz<_t[x,y] A(x,y,z) ~
(3)
Vx~tZy A(x,y) ~
( t not containing The closure of
F
~xVzAt[JlX, j2 x] A(JlX, j2 x, z)
~nVx~t i(x, (n)x)
x ). under existential quantification is immediate by (2).
The closure under bounded universal quantificatio~ follows from (5) and the previous lemma : Vu~t* Sy Vxit[u,y ] A(u, x, y) SnVu~t* Vx~t[u, (n)u ] A(u, x, (n)u) <---> *-9 ~nVv~t'[n]
A(~o(V,n), ~1(v,n), (n)$o(V,n)) .
( i i i ) Let A - ~xVyito(X)
[Pc v ( % ~ Xt~)],
B m ~xVyit1(x ) [Pq V (QI & Xt~)]. After contraction of two existential quantifiers, we obtain A&B ~XVYo~to[JlX]VY1~t1[J2x ] l[Po(Jlx,Y) V(Qo(JlX,y)&Xt~[JlX,Y]) ] & &[P1(J2x,Y) V (Ql(J2x,Y) & Nt~[J2x,Y])]} .
3o
We put
P(x,y, u) Q(x, y, u) t[~, u] t,[x,y,u]
~ ~ ~ m
(Po(JlX,y,u) &u=O) V (P1(J2x,y,u)&u~O) (Qo(jlx,y,u)&u=O) V (Q1(J2x,y,u) &u/O) ( l ~ U ) t o [ X ] . s~(u), t l [ ~ ] ( l ~ u ) t ~ [ x , y ] + sg(u).t~[x,y].
Then
A&~
~ ~x Vuil vySt[~,u ] [P
v
(Q&xt,)].
By the previous lemma, this is equivalent to a formula of type (I).
~
AVB
~x IVyito[X ] (Po V ( Q o & X t ~ ) ) V V V y ~ t l [ x ] (P1 V ( Q I & X t l ) ) I "
We put
P,(x,y,u) Q,(x,y,u) c"[x,y,~] t,[x,u] Then
A VB
~
~ ~ ~ ~
(Po~UoO) v (P1~u/°) (Qo&U=°) v ( % & u / O ) (l~)t, + s d u ) . t~ 0 (1~u)t ° . s~(u).t I
Zx~uVy~t"[x,u]
[P' V (Q' &Xt"')] ;
by (2) this is equivalent
to a formula of the form (I). 1.4.5.
Theorem.
Let
A(X,z)
be a formula of the class
an arithmetical predicate (i.e. definable in (I)
A(PA,Z ) ~
PA z
F;
then there is
such that
PA z
and for each arithmetical predicate
(2)
H~)
Q
Vz[A(Q,z) -- az I -- Vz[PAz -- Qz]
are derivable in Proof.
HA.
By lemma 1.4.4, we may restrict our attention to a predicate
A(X,z)
of the form (3)
~Vy~t[x,z][P(x,y,z)
A proof that proof
H
z
V (Q(x,y,z) & Xt'[x,y,z])].
satisfies (3) may be supposed to be in tree form : the
provides an
x , and for each
which either establishes
P(x,y,z)
y~t[x,z]
ff contains a sub-proof
(and then represents an end node of
Z
the tree
T
associated with
in the latter case, with
H
Z
H)
or establishes
Q(x,y,z)
is associated a subtree
T
Z
and of
Xt'[x,y,z] ; T,
which
establishes Zx'Vy'!t[x',t'[x,y,z]](P(x',y',t'[x,y,z])
V
V (Q(x',y',t'[x,y,z])&Xt'[x',y',t'[x,y,z]])) , and so on. It is this intuitive idea which suggests the explicit definition which will
3~
be given below. Any natural number may be supposed to code a finite tree ; < > empty tree, and if
n = <Xo,...,Xu> , then the tree
Tn
codes the
represented by
n
has the structure
where
So,...,S u
Xo, .. .,x u,
are the trees coded by
Below we adept the following notations: [n]m
inductively on [n]o=n,
Now we put for
lth (m)
we write
nu
respectively. for
(n) u , and define
by
[n]
= (n)u=nu,
[n]m * = [[n]m] = (~n]m)u"
PA ~ :
PA z ~ S n B m S p [ n / O & m o= z&VuVy~t[Pu,mu]I(In]u,
& ( [ n ] u . < y >= O& [n]u/O ~ P(pu,Y,mu)) t] ~!
I.
We have to show
~xVy~t[x,z][P(x,y,z) where
A(PA,Z ) -- PA z .
V (Q(x,y,z) & ~InBmBp B(n,m,p,t'[x,y,z]))] ,
~n~m~p B(n,m,p,z)
~ PA z , implies that for a suitable
Vy~t[Xo,Z][P(Xo,Y,Z ) V (Q(Xo,Y,Z)&SnBmBp Hence we can find
w, n, m, p
x°
B(n,m,p,t'[Xo,Y,Z])) ] •
such that
Vy~t[Xo,Zl[(Wy / 0 -- P(xo,Y,Z)) & & (Wy= 0 -- Q(Xo,Y,Z ) &B(ny,my,py,t'[Xo,Y,Z]))] Now we define
n', m', p'
such that
n' = (no ..... nt(xo,Z) > with
ny = 0
for
Wy/O,
~y~ny m ! O
m'
satisfies
p'
satisfies
W y = O.
z
~
(mY)u
!
Then obviously
=
for
=
X
Po o P' * u = (Py)u
•
52
n' / 0 & [mg= z & VuVy<__t(pg,z) [([n']u.<j>/O- ~ ~(Pu I 'y'm ' , m 'u] = mu*) & u) & t l [pu,y, & ([n']u~O & [n']u.: 0 B(n', m', p', z) , which implies Part II.
(Pu,Y,mu)) }] PA z .
Assume
(4) i.e. ~xVy~t[x,z][P(x,y,z) V (Q(x,y,z) & Rt'[x,y,z])] " Rz. We shall prove by induction on (5)
n
VmVpVz[B(n,m,p,z)--Rz] .
Basis : For n= 0 (5) is trivially fulfilled, because the antecedent is then false. Induction step : Now assume (5) to have been proved for all n< n' , hence for all x such that [n']<x>/O. We may rewrite B(n',m,p,z) as n' / 0 & m ° = z
&
& VwVuVy~t[P<w>.u,m<w>.u~l([n']<w>.u. ~ 0 a(P<w>, u, Y, m<w>. u) & t'[P<w>, u, Y, m<w>. u] = m<w>.u.) & ([n']<w>.u.= O & [n']<w>.u ~ O -- P(P<w>*u' y ' m<w>*u))l & & VyAt[Po,mo]{([n' ] / 0 -- Q(Po'Y'mo) & t'[Po'Y'mo] = m) & & ([n'] = O & [n']o/O -- P(Po' y' mo)) i°
"
It follows that [n']<w>/ O -- B([n']<w> , m, p, m<w>) hence
(6)
[n']<w> ~ O ~ R(m<w>)
(induction hypothesis)
and
(7)
i Vy~t[P°'Z]I([n']/O " Q(po,y,z) & t'[po,y,z ] = m) & & ([n']=O & [n']o/O ~ P(po,y,z))l,
therefore, combining (6) and (7) : Vy!t[Po,Z](F(po,y,z ) V (Q(po,y,z) &Rt'[po,y,z]) ) . By assumption (4),
Rz .
33
§ 5.
Partial reflection principles. =============================
Io5.1.
Contents of the section.
Let
Proof (x,rA ~)
indicate
the proof-
n
predicate of
H~
(e.go for Spector's
taining formulae
of logical complexity
logical complexity complexities
is
~n
restricted
only.
to derivations
In other words,
con-
if the
of a proof is described as the maximum of the logical
of the formulae occurring in it, then
ProofHA(X,~A ~) x
system),
and the logical
complexity
Proofn(x,rA~ )
holds iff
of the deduction represented
by
< n °
The principal
aim of this section is to establish i__nn HA
principles
for the subsystems
complexity
of the formulae considered,
(1)
~
b ~
of
Pr°ofn(x''A~)
HA
obtained by putting a bound on the i.e.
~ A .
The main step towards establishing
of a "valuation-
(I) is the construction
function" which assigns to (the g~delnumber value
reflection
of) a closed term its intended
(and which can be shown to do so i__nn ~ ) o
1.5.2~
Godelnumbering
For definiteness
of function constants and terms.
in the formal description,
we specify some details of the
godelnumbering. Ne put
~
for the code number of the function constant
~,
where
m <0> ~ ~i ~<2, n, i> n
~ < ~ ' ~ ' ~ I .... '~m >
if
~
is defined by composition from
~ <4, }, ~>
if
~
is defined by recursion from
The godelnumber
of an arbitrary
Each closed term is of the form function constant (Note that
O
closed term is defined as follows.
gtl~..t n
of our language,
~'%'''''~m" ~, ~.
~e put
(n
possibly gtl...t n
as a function constant has number
0 ), where
~
= <~, t I , ...,
is a tn ) o
, as a term number
<<0>> o ) The sequence of g~delnumbers is to denote the g~delnumber v0 = <<0>>, 1.5.3.
v(Sx)
Evaluation
Any closed term
= <,
of numerals
is primitive
assigned to the numeral
recursive ; if
~ , v
vx
is given by
vx>.
of closed terms. t
in
HA
has a standard deduction of may be described as follows.
can be evaluated by a standard procedure, t =~
for a numeral
~
in
HA.
and
This procedure
34
A contractible introduced
term is a term of the form
by composition
For closed occurrence
terms
of
10 )
0
2° )
rcso(St)
or recursion~
t , we define
t"
o~ a
~
a constant
projcc[{o~.
the "ri6ht most contractible
(shbreviated : rcso(t) )
does not have an
~I.O.~ n ,
imduct~vely
subterm
as follows:
rcso(t) ;
is the occurrence
in
St
corresponding
to
rcso(t) ,
~
constant,
if this exists; 30 )
if
t m ~(tj,...,ti,~i+1,.°.,~n)
a numeral, 4°)
if
rcso(t)
t m ~(~1,.o.,~n)
rcso(t)
is
A contraction form
then
t
,
a function
~
a function
constant,
if
,
~
of
defined
I~Xj.o.~ n
by
¢(~I .... '~n )
A "standard tI ~ t ,
tn
applied
to the
the construction is obviously
n= O,
<2)
z
a)
For
~ m 0 ,
For
~ m S
c)
Suppose
t"
ti+ I
~def W 1 ° ' ' x Val(~)
SRED(~
is defined
ti
by
by recursion
from
from
if
tl, oo., t n
ti
by a contraction
tn ~,
a value
for
t.
is u n i q u e l y
determined,
the
predicate
expressing :
the
constant
sequence
to the term with
~: v(~(xl
'Xn)))
rodef S R E D ( C ~ ~, v(~)) o
for each function
constant
~
this is immediate° also,
from the definition
is defined
by composition
by recursion
~ in
is defined I~A.
In the verification sequence
for
y,
~I "''~ ~
~
quences
~' ~1' "''' ~m'
is a sequence
results
function
Suppose
reduction
~i ' or of a term of the from
has a standard reduction
~hen we e~sily verify
VaI(x)
S, 0 , then
~ Val<~) o
b)
d)
not
rcso(ti) ;
X(~(~,~ 2 ..... ~n),y, x 2 ..... ~n )
Z ol - arithmetical
n-ary
we may put
~
for
ti+ fl out of
be the
term with godelnumber
Now we establish,
~
~nd by
We then call
of
godelnumber z'. Let us write, for each
For
ti to
unique.
S R E D ( z , z')
Val(~)
sequence
such that
rcso(ti) .
Since
(I)
~I ~ ~'
if
reduction
a numeral,
value Let
not
by
by composition
~(~I(~I .... ,~n) , ..o, ~m(~1 .... ,~n)) , and if
~, x,
corresponding
itself.
is the replacement
~(~l,O.O,~n)
,
is the occurrence
of
V o
from
¢' 91' "''' ~m'
Val(~)
~e now establish
from
9
and
X,
(I) by induction
and assume on
of (a) - (d) we have to use repeatedly for
tl, ..., tn o
~tlo..t n
and assume
always
"contains"
Val(9),
XI . that a standard
standard
reduction
se-
55
We have established (2), and now we readily prove, by induction on the logical complexity of ~mong
X1,
(~)
o~
x
n
t , for any term
whose free variables are
:
" ' ~ n] "
~bs~(%[~,..
1.5.4.
t[x1,...,Xn]
' ~t[=~,.
..
°
,xn])
Construction of partial trut h definitions°
By induction on for all formulae
(I)
n A
we construct truth definitions
T n , such that in
~HA,
of logical complexity _
HA ~- T n ( ~ A ( = I '
(the variables free in
"' A
m) ) ~
A(=I' "'~m)
are among
x1~...,x m)o
For prime formulae we put
To( ~ t ( ~ , . . . ,~n) = s(~ t . . . . . ~n)~--~ ~y[ SR~D( ~t(~
....
SR~D(%(~ 1 .... Assume
Tn
to have been defined.
A, B, (Qx)Cx
~
(2)
closed (where
Q
Then we define
stands for
Z
)~
j y ~)
)~JYD]
Tn+ I
or
•
such that for
V)
Tn+I(~AoB~) ~--~Tn(~A~) oTn(~B I)
(for
o ~-~, a~, V)
Tn+l('-(Qx)Cx -~) ~
(for
Q ~ Z, V)
(QX)Tn("C~- ~)
Such a definition is possible,
since for the usual standard godelnumberings
it is primitive recursively decidable what lhe main logical opera~or'~s ; so we may define
Tn+ I
by cases in agreement with (2).
proved by induction over
n°
For
n=O,
Now (I) is readily
(I) is immediate by the result (3)
of 1.5.3, and the induction step is given by (2)° 1.5o5o
Lemmao
(a)
(For Spector's or aSdel's version of
(b)
(For the natural deduction version of
~o)
~ PrOOfn(Y, %(x I .... ,Xn)" ) ~ ~=1oo.X n Tn(rA(~ I ..... ~n )I) . H~.)
Let us write
F(a I .... ,an) = A(a I .... ,an) , where F = IC1(a I ..... an) .... , Cp(a I .... an)}, for : A(a1,°..,an) can be deduced from assumptions F ; let al, o.., a n be a list containing all parameters free in ~
Proofn(x, rHal,...,an)
F, A . Then
~ A(a I ..... an)")
Vx1.°.Xn(Tn(%1(~ I ,.. . ,~ n) ~ ) ~ ,o . ~Tndcp(= - ~ ,° -o,~n >~) Tn(~A(~I, .... ~n)")° Proof.
~e consider case (a) ; the treatment of case (b) is entirely similar.
Let us write
~(z,x)
for the function which is such that
~(%(Vio,...,Vim)~,
=) = ~A(~Tio . . . . . (=)----~mf°
3~ Then we prove by induction on
lthx
that
Proofn(X,Z) ~ VyTn(~(z,y))
o
To give an example of the argument for the induction
step, assume
VxVv < u(Ith(x) = v & Proofn(X,Z ) -- Vy Tn(~(z,y)) o Now assume
lth(x)
=u &
Proofn(X,Z)
.
~e have to distinguish
various cases,
the proof with number
x.
depending on the last rule applied in
For example,
proof obtained by application
of
assume
x
to be the number of a
Q I to some subproof
xI
of
x
of an
assertion of the form
A(Vio , ... ,Vim ) Note that cursively from
~ B(Vil ,...
xl ~ rA ~, rB~ ' rC~
,v.im )
~ C(Vio
and the numbers
,vii ,.
io,.O.,i m
(in fact, for the usual g~delnumberings,
x .
.. ,Vim
) .
can be found re-
primitive recursively)
So by induction hypothesis Vy Tn(~(rA~ , y))
in
I~A ,
which
implies
Vy{Tn(rB((Y)il
in
turn (in
....
HA) :
, (Y)im)~)
-- Tn(rC((Y)io,
(Y)il ..... --
~im)" )t,
i.e.
VY~oi~n<%<<~Til .....
(~#m))
-- Tn(%<~ o, (Y)il .... ' (Y)im)~)}
hence
VY{~n(%<
VyITn(~(rB~,y)) i.e. by the properties
~ Tn(~(rVVioC~,y))} of
T
,
n
Vy ~n(~(% - VVioChY)) as desired. 1.5.6. (a)
(Partial reflection
principles).
For Godel's or Speeter's formulation
~b (b)
Theorem
of
~:
Pr°°fn(x,~A(~l, . . . . ~n )~) " A<~I' . . . . xn)"
For the natural deduction formulation
of
~
:
Proof.
b Pr°°fn(X, ~ A ( ~ l , ' ' ' , ~ n ) ~ ) ~ A(Xl . . . . . Xn) . ~e consider (~)~ (b) i s t r e a t e d s i m i l a r l y .
Assume
Proofn(X, rA(~ I .... ,~n )~) , then by 1.5.5
1
)},
3?
:~n('-'A (~I,...,
~n)-' )
1, ,5.~r
and therefore by
A(x1,-..,x n) . Q.e.d. 1.5.7.
R e m a r k on refinements.
W e may introduce
complexity,
e.g. by c o n t r a c t i n g
occurrences
of the same type of quantifierso
d(rA ~) = 0
d(rA~B I) d(rA1o.°,
conjunctions,
= ~x(d(~A1),
oA~
A I o o.. c A n
or e v e r y w h e r e
d(~1))
where are
A
1.5.8.
AI,..o,A n
• ..
(QXn)A(xl
(Qy)B,
is e v e r y w h e r e
Q
Tn
&,
BOB'. x n) ~
and e i t h e r all of
o
) + I,
are
V
or all
Q
to this new measure.
s,ystems. systems discussed here and in the sequel, restricted
and the systems
d e s c r i b e d in 1 . 6 . 1 5 - 1 . 6 ° 1 5 )
qf
to q u a n t i f i e r - f r e e
N-~w
qf-WE-HA
w
there are two possible
in the f o r m u l a t i o n :
(i) W e state axioms
then be stated as
such as e.g.
x=x
with free variables,
of terms for free v a r i a b l e s ;
AO , A(x) -* A(sx)
and f o r m u l a t e i n d u c t i o n as substitution
and have a
the i n d u c t i o n rule may
= A(y) , or
(ii) we state the axioms for a r b i t r a r y
terms
A(O) , A ( x ) - * A ( S x )
(as schemata), = A(t) ;
e°g.
t = t,
then we may emit the
rule.
Theorem. R u l e - IND
Let
qf-HA
be
~
restricted
instead of induction.
W e may r e f o r m u l a t e
quantifier-free
qf - ! ~
formula c o r r e s p o n d s
to q u a n t i f i e r - f r e e
"
A(x1'''''Xn) °
as an e q u a t i o n a l to an equation
calculus t = s ,
what amounts to the same, we can adapt the d e f i n i t i o n as in 1.5.6 the present
theorem.
formulae,
Then
b P r ° O f q f _ ~ A ( X ' ~ A ( X l .... '~n )~) Proof°
A I,...,A n by in-
(either
.
w i t h a rule of induction,
rule of substitution
with
o
b e i n g d e s c r i b e d as a r i t h m e t i c
qf - I - HA w , q f - HA w
Io5.9.
+ I
. .) .= ~('-A(:,: . . .I ,xn)-'
,.
quantifier-free
qf- P~,
formulae,
variants
, d(rAf))
not b e i n g of the f o r m
R e m a r k on q u a n t i f i e r - f r e e
In the various
by
°o
It is easy to adapt the d e f i n i t i o n
(namely
d
I
.
7
is not of the f o r m
~.
+
) = max(d(rA~),
and the b i n a r y o p e r a t o r
V) ,
d('~(Qxl)
I.e. we define a degree
stands for any f o r m u l a obtained from
sertion of b r a c k e t s
and successive
for prime formulae
n
where
a more r e f i n e d m e a s u r e of
disjunctions
of
(i.e. each
cf. 1.6 ~@), To .
or
We then prove
~8
Io5.10o Corollary to 1.5.3.
(i) (ii)
Let
~
be
qf-HA
with
R u l e - IN~
left out.
]HA ~ t [ x 1 , . . . , x n ] = y - - P r t t ( r t [ [ 1 . . . . . i n ] = [ 1 ) o There e x i s t s a ( p r i m i t i v e ) r e c u r s i v e ~t such t h a t
i~ Proof.
< = I .... 'Xn] =Y ~ P~°°f~(~t(=1"'''xn'Y)'%[~1
..... ~n] = Y"l.
( i ) i s immediate from (3) making use of
~b (ii) A ~o~
sR~D( rt ~, ~t'~) ~ Pr~(~t=t'~) • detailed
inspectie~
of the proof
of (2),
(~) i n
1.5.3 y i e l d s
(ii).
One firs~ establishes
(4)
~
PrOOfqf_HA(k[(Xl . . . . . xn'Y ) , r ~(x~,. ,. ,~n)=y')
b [(x fl' . . . . xn ) =Y
for the constants
[
of
H~,
suitable recursive
(ii) by induction on the complexity of
k[~
and then establishes
t o
For example~ in the proof of (4), we have to consider the case %hat defined by recursion from
b X(xl'''°,x ½
X, ~,
[
is
and assume as induction hypotheses
n) =Y " Pr°°fH(kX(X 1 . . . . . Xn ' y ) ' r x ( ~ 1 . . . . . ~n ) = ~ )
b ¢ ( ~ ' ~ o , % ..... Xn) = Y ~ ~r°°f!~(X¢(~'% ..... ~n 'y)'
r¢(~,~o . . . . . ~n ) = ~ l )
.
Now we note
~(0, x 1 . . . . . hence
=y ~ x(= 1 . . . . . Xn) = y ,
~)
P r o o f H ( k x ( x l . . . . . Xn ' y ) , r X(£I . . . . ,~n) = y l )
There exists a primitive recursive a proof of
X(X I ..... Xn) = y
~I
o
which transforms the g~delnumber of
into a number of a proof of
(since the ne~ proof is obtained by addin¢
[(0, xl .... 'in ) = ~
[(O,= I ..... ~n) = × ( ~ ..... i n)
(instan%iation of an axiom) and applying the equality axioms).
x~(o, x ~ , . . o , = n, y) Suppose Let
k~(z~ x1,°o.,Xn~ y)
[(Sz, xl,oo.,Xn) = y °
Now take
= ~ x ( = ~ .... ,Xn, y). to be defined°
Then
¢([(~, ~ , .... Xn), z, ~ ..... x n) =
and Gbbreviatin¢
¢(z,x~ ..... =n) as
Proo~(~¢([(~,~,ooO,Xn), Proo~(~¢(~,=~,...,=,¢(~, Combining the proofs and the instantiation
of
~)
~, x~ ..... = n, y), ~ ¢(~, ~, ~,...,~n ) =y~) =~ .... ,=n )),
@(~' z' Xl ..... xn ) = y
~ [ ( [ ~ , ~ .... ,~n ) = [ ~ ° and of
$ ( [ ' [1 ..... xn ) = ~
~(S~,~ I ..... Xn ) = ~(~'~'~I ..... Xn) , we obtain, primitive
recursively in %he numbers of these proofs, the proof number
k{(Sz, = I ' . . . . Xn' y)
of a proof of
{(S~, x1,ooO,Xn) = y ,
etc~ etc.
39
1.6.1.
Contents of the section.
This section deals with extensions
to theories involving objects of finite type. inductively
3 - 7 describe
~
defined
("E" from "extensional", free fragments Subsection
of
"WE" from "weakly extensional")
15 describes
8 intro-
9 it is shown
to an intensional
variants
q f - N - H ~ A ~, q f - ~ - H A ~, q f - W E - ~ a weak system
HA~,
~
variant
W E - K A ~, E-H~A m and the quantifier-
are described.
with its quantifier-free
in connection with the Dialectica
The reader who is not interested
(§ 3.5) may decide to skip this section, dealing with Subsection
part
16 discusses
and subsection
of combinators Subsection Directions
in the Dialectica
inter-
interpretation
and also the material
in § 1.7
HA w . pairing operators
17 describes
as a primitive.
for use.
skip §§ 1.6, 1.7,
qf-W~-~
with the
recursion m,
X- operator instead
on these subjects in § 1.7.
The reader who is primarily interested in
with the intuitionistic
in
which can
notes and a discussion of the literature.
1.8 altogether.
H~A, may
If the reader has no previous acquaintance
theory of f~nite types, he may find it enlightening,
after a brief glance at subsection of these theories described
I - 14, to have a look at the models
HR0,
in chapter II.
~.
Type structure
The t y p e s t r u c t u r e
N-~W
~ore material
18 gives historical
and simultaneous
a pairing for
even be used in a suitable version of
1.6.2.
N -HA w
and extensional
The system is of interest
pretation.
HEO
Subsection
in subsection
in a language with only equations between type zero terms as prime
formulae.
general,
of intuitionistic
N - H ~ ~.
10 - 14 the extensions
("I" from "intensional")
q f - ~HA w ,
N-~
~- operator,
is properly contained in
In subsections ~-HA ~
the basic system
in all finite types ("N" from "neutral")°
duces the combinatorially that
HA
in 1.6.2.
Subsections arithmetic
of
The type structure is defined
T
is
defined
inductively
by t h e f o l l o w i n g
two
clauses : TI)
0 E T
T2)
~,T 6 T =
Remarks. represents
(~)T E T.
(i) Intuitively,
each type represents
the natural numbers,
and if
~, T
a class of mappings from objects of type
~
(ii) There are many alternative
for
as
(o,~), ~ ,
~,
notations
a class of objects:
are types,
then
(G)~
to objects of type (~)T
type
•.
in the literature,
such
(~)~, etc.
(iii) Each ~ E ~ is of the form induction over I"
0
represents
(al)...(~n)O , as is readily verified by
4o
1.6.3- 1.6.7. 1.6.3.
Description of the neutral
Language of
for each type
N-~.
N - H A w.
The language contains variables ~ E T.
There is a symbol
theory
(indicated by
(Type superscripts
=
x ~, y~, z~, u~, v~, wa )
are often omitted.)
for equality between objects of type
~,
for each
E T ; we usually omit the type subscript. Furthermore, 0
of type
Z p,O~T
0
, Ra
there are constants (zero) ; S
for all
As logical
for objects of certain types : a constant
of type
p,a,v E T,
constants we use
(O)0
(successor),
and constants
H
a~T
'
whose types will be described below. &, -% v, Vx ~, ~x a
(for each variable
x~ ,
~ E T). 1.6.4. Let
Terms. Tm
denote the class of terms of type
Terms are defined inductively
Constants and variables of type
Tm 2)
t E Tm(a)T , t' E Tm a Notational
=
s, t, T
t1(t2t3)
So if
SXl...SXnA
as
tlt2t3...t n.
stands for
and provided
So
tlt2t 3
abbre-
(t1(t2t})) . for finite
x - (Xl,...,Xn) , VxA,= ~xA=
will be used to denote finite
(possibly empty)
abbreviate
(possibly empty)
that a term is of type
a,
strings of terms.
we often simply write
We shall often omit type superscripts ! but it is always assumed in
writing down an expression, are fitting).
that the terms are well-formed
SO, for example,
y E a, u E T, z E 0,
then
if we write
~ ~def slt1"''tm'
concatenation
however,
stands for
and if we assume
s i E (T1)...(Vm)~i , t
3
E T.
3
"''' Snt1"''tm"
Immediately after variable-binding
~ (Yl ..... Ym )
xyz = u,
(i.e. the types
x E (a)(p)T.
Let s ~ Sl,...,Sn, t -= tl,...,tm, = = (1~i~n, I< j (m) , then
indicates
if necessa-
respectively.
If we wish to indicate t E ~.
for terms,
to create more variables,
x, y, z, u, v, w, X, Y, Z, U, V, W
strings of variables.
s t t, T
variables
for clarity.
(...((tlt2)t3)...tn)
((tlt2)t}) , but
VXl...VXnA,
Tm
.
as syntactical
ry provided with primes or subscripts
We shall use
belong to
(tt,) E Tm
with a type superscript ~ s~, ta
viates
~
conventions.
We will reserve
We abbreviate
I ~ E ~} .
as follows :
Tm I)
1.6.5.
a , Tm = U I T m
so
operators V~
(V, ~
where
Vxi...Vx n VY1...Vy m
~)
juxtaposition~indeed
x= ~ (Xl,...,Xn) , etc.
41 1.6.6.
Formulae.
Let us denote the class of formulae as sions of the form
t~ =
s~ .
Fm
Fm.
Prime formulae
are expres-
is defined as usual by two inductive
clauses : Fm I)
Prime formulae b e l o n g to
Fm
Fm 2)
If
(A&B),
A,B ~ F m ,
(VxeA),
then also
(A--B) ,
(~x=A) .
In b r a c k e t i n g we f o l l o w the usual 1.6.7.
(A V B ) ,
conventions.
A x i o m s and rules.
(a) A x i o m s a~d rules for m a n y - s o r t e d (b) A x i o m s
intuitionistic
predicate logic.
for equality : x
= x
x
= y
& z
x
=y
~z
=y
~ x x
= z
,
= z
,
X ( ~ ) T = y ( ~ ) T . X(~)TZ ~ = y ( ~ ) T Z 0 , and the usual
s~
equality axioms for
Sx ° / O,
x o ~ yO @-9 Sx ° = Sy ° .
(c) The rule or a x i o m schema of i n d u c t i o n
(for arbitrary
formulae
of the
Ianguage ). (d) D e f i n i n g axioms for
H0,S'
~
0,~x 0y~ = x p ,
r~
O,o'
x~(y~)
x 6
(O)(c,)~,
y
~
(9)a,
z 6
O,
((O)(a),)((P)~)(O)~. R xy 0
= x
~x~(Sz) 1.6.8.
Theorem
(Definition
To each term
tT[x ~]
of the
(kx~.tT[xa])(t ') = tT[t ']
(ii)
Ix~ . tx a = t
for
x ~ / t',t" ,
R
t
k - operator).
kxa.t
not c o n t a i n i n g
(a)
x a ~ t ~ = kx~.t ~ ~def ff~,~ t~
Xxa.x %~ za,(o)~, n ,(o)ana, ° x~ / t ;
x a 6 t , or
then
such that
x~ ,
y
a variable
by i n d u c t i o n on the c o m p l e x i t y
(b) (d)
~X~tTIx a]
then
is defined
(c)
a term
(t' 6 ~) ,
t' = t" -- kx~Iy/t'It = k x ~ . i y / t " ] t , Proof.
z 6 0 ,
~ (~)((~)(o)~)(o)~.
we can construct
(i)
(iii) if
x ~ a, y ~ (~)(0)~,
I
y(Rxyz)~
=
different of
kxe'tx~ ~def t (x ~ E t I
Z(kx~.t)(kx~.t,)
.
and
t' / x ~) ;
then
kxa.tt v
def
t :
from
X .
42
Now (ii) is immediate
by clause
proved
by induction
simultaneously
As an example,
we prove
(a)
Let
x ~ ~ t~ .
(b)
Z ,(o)~,
(c)
Let
x ~ ~ t,
(d)
Let
x a E t , or
on the complexity = HT,
()~x~.tT)(t~)
n ,o
x
then
~, (o)
=
Xa(n
t~tl = t T ,
'° x a ) = x~ .
(x a E t'
and
t' / x a) ;
= Z(kx~.t[x~])(kxC.t,[x~])t"
=
In combinatoc~
with square brackets : X-notation
defined
and primitive
HA
the defined
for
kx.t.
X- operators,
~x.t , where
kx1.(kx2.(... 1.6.9.
logic, [x]t
X- operator
We find it more
instead ; but if one has to discuss
Abbreviation=
there
•
as a subsystem
of
is usually
written
suggestive
in a single
to use
context distinction.
stands for
N - I ~ m.
to each function
constant
~
of
~,
a term
T°
of
as f o l l o w s .
TO E 0 , TS ~ S , and if U~(Xl,...,Xn) If
@
Uin
= x i , we put
is explicitly
defined
is the function
T@ ~ kXl,..x n . T % ( T If
~
is defined
such that
T i ~ kx1"''Xn'X i • Un from $o,~I,...,~m
~ ( x 1 . . . . . Xn) = ¢ o ( ~ 1 ( x 1 . . . . ,Xn) . . . . ,~m(Xl . . . . .
(iii)
hypothesis).
should be a notational
x ~ (Xl,...,Xn) ,
(XXn.t)...))
Let us associate - I ~ ~,
(ii)
(induction
etc.
Remark.
(i)
t.
of
= (kxa.t[xa])t '' ((kx~.t'[x~])t ") = tit"It'[t"]
the
can be
(kx~.tx~)(t ') = tt' o
(kx¢.t[x~]t,[xC])t"
Etc.,
and ( i i i )
(i).
Then
~ ,(o)
(i)
(c) of the definition.
IXl...Xn)...(T
by primitive
such that
Xn) ,
we p u t
mX1...Xn) .
recursion
from
@I'
@2
such that
(0,x I .... ,x n) = ¢I(Xi ..... x n) • (Sz,x l ....
we p u t Then,
T
for any
like"
(iv)
A~ ~ T O
= ¢2(~(Z,X 1 ....
of
~
constant
More precisely, by induction
for each function
A~(t I ..... tn) ~ (A~)(Atl)...(Atn) A(t = s) ~ (At = As) ,
(vii)
A
t~ that
is a homomorphism translates A
under
is hi-unique.
w.r.t, A
~
of
HA,
if we define
constant
(v)
then
,x n)
"behaves
T Xl...x n a mapping
A
on the complexity :
(vi)
Note
,Xn)Z,X 1 ....
~ R T ¢ l ( k ~ y z . T¢2(yff)zff) .
n - ary function
~(Xl,..o,Xn).
and formulae
, x n)
~
of
HA,
Ax ° E x ° ,
,
logical
operators,
into a subsystem
of
~-~.
on terms
43
1.6.10.
Intensional identity or equalitF.
A basic feature of intuitionism is that we have to deal with mathematical objects as they are given to us ; for example, a species (set) of natural numbers is ~iven by a description
(definition) of a property of natural
numbers ; the extension of the set (in the classical sense) may then be conceived either as a mode of speech, to avoid speaking about equivalence w.r.t, membership,
or as an equivalence class, i.e. a species of higher type;
the latter point of view makes sense if we accept the concept of a power species. Similarly~
a (lawlike) function is given as a rule;
in the classical sense is a derived notion.
its extension (graph)
From a foundational point of
view, it is therefore natural to pay attention to the concept of definitional or intensional equality : two objects are said to be definitionally or intensionally equal, if they are given to us as the same oBject. We do not a priori suppose the concept of "definition" or "description" to he restricted to definition in a given language. Whatever the precise content of the concept of intensional identity,
it
seems clear that it should be decidable whether two objects are definitionally equal or not.
This is expressed in the extension
I-HA ~
of
N-HA w
described below. 1.6.11. In
Description of
I - H~ ~.
I - HA ~ , the intended interpretation of
ity between objects of type a constant
E
e (~)(a)O
E xay ~ Ecx~y~
a ; I-HA ~
for each type
0 e-~ x a
in~e~sional equal-
is :
~ 6 =T
N-HA ~
by adding
such that
y~
0 V Eaxa y s
=
=
is obtained from
=
I
~
which implies decidability of equality at all types : x 1.6.12.
= y
V xa/y ~ .
Description of
E-~,
WEE-H~A w.
Another way of interpreting equality in extensional equality: argument of type
(I)
~
two objects of type
(a)T
they yield equal values.
vz (~)~ u(~)~(z=u
Adding (1) to
N-HAm
N-HA m
is assuming it to be are equal if for every
This amounts to
*-~ Vya(~y°uy)) .
yields an extensional version of intuitionistic
arithmetic in all finite types, which we may denote by
E - H A ~. ~O
For some purposes (notably the study of the D i a l e c t i c a - i n t e r p r e t a t i o n in Chapter III) it is more convenient to use another version of be denoted by
E - H A w.
Here equality between objects of type
E - H A~ o ~ 0
' to is the only
44
primitive concept, equality for all other types is a defined notion (inductively on the complexity of the types)
~(~)~
u(~)~ -=def VYa(~y:uy) "
:
The axioms for equality of redundant, such as
N-~
are retained (but some of them become
z = u -~ zy = uy,
which now holds by definition) ; the full
force of extensionality is now in
(2)
x ~ = y~
Note that
~
E-HA m
~(=)~= : ~(~)~y. may be interpreted as a definitional extension of
obtained by addition of new symbols to
E- ~ ® ~-~®
is
obtained
from
=
E-t% ~
E-~m
for definable equality of type
weakening
(2)
to the f o l l o w i n g
rule
o f ext ensionalit,y
EX~-~.
t- t ~ l . . . x n
(A(t) in
quantifier-free,
A, t, s , such that
tion
~F
t-A(t)
: sx~...Xn,
x1,...,x n txl...x n
indicates that
F
= ffA(s)
a sequence of variables not occurring
and
sxl..oX n
are of type
0 o
The nota-
has been derived without assumptions.)
A
slightly stronger rule is EXT-R'. ( P
~- P -- t x 1 . . . x n : s x 1 . . . x n ,
ffA(t)
= f f P -- A ( s ) .
quantifier-free, other conditions as before.)
~ - ~
cf. Io9.25- 1.9.27 ) Note also that
(3)
EXT-R
EXT- R
~- t x l . . . X n
where
In certain analogues of
(theories with bar recursion of higher type, or induction over trees;
Fix a]
must be replaced by
EXT-R' .
is equivalent to the following rule
= SXl..-x n
~
ffF[t]
:FIe]
is a term in the language of
formula, we can always find a term
FA[X ~]
,
E-~
~.
For, if
such that
A[x ~]
is a
FA[X~]=O*--~A(xa)
(see 1.6oI~) ; therefore, from (3)
tx1..oXn=
sx1...x n
=
f f F A [ t ] = FA[S ] ,
hence
~txl...~ Conversely, apply viously
holds,
and
: s~l...x EXT-R
n
ffFA[t]
with
A(s) ~ F[t]
= F[x ~]
~
ffFA[S]
for
: o
A(xa) , then
A(t)
ob-
= F[s].
If we would have formulated (4)
F[t]
: 0
A(t), tXloooX n = s x l . . . x
EXT- R
n
as
~ A(s)
(conditions as for
EXT-R,
tx1.o.X n = sX1.o.X n
depends), then the resulting system does not satisfy the
deduction theorem.
For (4) implies the axiom of extensionality (2),~nce (4)
yields
but
Xl,...,x n
not free in assumptions on which
45
x :y=, z (~)Tx a : z(=)~x a
b z(=)~x = : ~(=)~y=
hence the deduction theorem would yield
(2).
From results in Howard
however,
it follows that (2) cannot be proved in
EXT-R)
and that therefore
deduction
W~-~
w
of
qf-N-HA
The quantifier-free qf-N-HA
w
~,
part of
qf_ ~_~w
~ - I ~ ~,
qf_WE_H~
~ - H ~ w,
WE-HA~,
etc., may be obtained from the corresponding
quantifiers
From logic we drop quantifier rules and - axioms. in
of type (ii)
q f - W ~ E - H A w,
t ° = sa
taxl ...x n = s~ xl..°x n,
abbreviation for t, s
denoted as theories with
as follows.
deductions
in
(with (4) replacing
in this version does not satisfy the
theorem.
1.6.13. Description
(i)
WE-HA w
or (open) assumptions
In discussing
is then to be conceived as an Xl,..o,X n
of the deduction,
variables not occurring such that
tx1.o.X n
The induction
schema is replaced by the induction rule :
A(O), A(x) ~ A(Sx °) = Ax ° , for in assumptions
(iii) Ax s ~ At ~ , if
1.6.14.
x
quantifier free,
qf-WE-~
~
systems in
with
EXT-R'
q f - ~ - H A ~,
qf-~.-H~A w
qf_ ~_~w
propositional
not occurring free
qf-WE-~
I 58 ) instead of
theorem holds even if we liberalize
Since in
x
does not occur in (open) assumptions°
on quantifier-free
Note that for deduction
A
of the deduction.
(Cf. our remark
EXT - R'
as equational w
formulae are decidable,
E X T - R,
the
by deleting
~.
calculi°
prime formulae are decidable,
all
and therefore the propositional
ators may be represented by certain constant
terms expressing
oper-
the classical
truth functions. We consider the case of recursive functions
qf- ~-HAW:
(of type
Let
(0)(0)0)
con(Sx,y) = con(x,Sy) = SO,
con(O,0)
=
dis(S~,Sy)
=
imp(Sx,y)
= imp(x,0)
imp(0,Sx)
= S0o
dis(O,x)
Now we construct, T A = 0*--*A Tt= s
=
o ,
= 0
,
for any propositional
con, dis, imp
be primitive
such that
disCo,x)
For
is
0 o
=
O, so,
A , a term
TA
such that
: we take
Ets.
TA& B ~ con(TA,TB) ; TAV B ~ diS(TA,TB) ;
TA~ B ~ imp(TA,TB). Similarly for q f - W ~ E . - HA w ; here we only need to put
Tt= s ~ It-s I .
4g
1.6.15. The szstems
~w
qf_H~Wo
In connection with the Dialectica interpretation subsystem
~w
of
~-HA ~
is of interest.
ed in the Dialectica interpretation
can omit this section,
till he arrives at studying the Dialeetica The only prime formulae of the constant s are those of type
0
as a primitive.
predicate
logic°
x
0
= x
0
Sx ° % 0 substitutivity
x
~
x°
= z
0
0
& y
0
=z
0
~ x
0
=y
0
y° ,
~S
~ o fly ]
=
(which may be viewed as very special instances
of
t[~]
] :
t[xz(yz)] fix],
tERxy(Sz)]
is clearly a subsystem of over
the
rule), for all t E 0 :
t[RxyO]
conservative
schema,
objects :
schemata
lt[nxY
to
intuitionistic
axioms consist of the induction
y°~-cSx°
=
%t[Zxyz]
m ~
0
except that we now only need equality of
~N-HA~,
tie]
the extensionality
are equations between terms of type
equality and successor
for type
and the following
or postpone it
interpretation.)
The logical basis is many-sorted
0 0
,
xo = y e
SUB
HA ~
The nonlogical
usual axioms for type
in § 3.5, the following
(The reader who is not interest-
HA Wo)
~-~o
qf_~w
= t[y(R~yz)Z]o
(~e do not know ~hether
~-~
is defined in the obvious way,
is
similarly
qf - K - m--~"
Remarks~
(i) The schemata
SUB
t [ ( k x ~ o t ' ) t ''] =~[[x~/t"]t '] for
give us for the defined
(te O) , and especially
X- operator :
(>~.t)t'
= [x~/t']t
t E 0.
(ii) If we use F[tt]
t ~I = t2
for all type
0
x~)°-¢~I = x(¢)°-¢~2 ° versely,
taking for
as a metamathematical terms For
abbreviation for
F[x ~] , then we see that
~F[tl]
t I = t2
t I¢ = t 2¢ implies the latter assertion,
x(~)° :
Xx~. F[x~] , we obtain
~F[%1]
=
amounts to and con-
: F[t2] , in
view of the preceding remark. (iii) 1.6.16.
HA
is of course also a subsystem of
HAlo
Simul..t..aneous recursion and pairin~ : a comparison of various
treatments. We have formulated
our system
For certain applications, simultaneous ¢i6 T
for
recursion: I < i
(to be abbreviated
~_~w
however,
for each sequence of types
one requires as
with a primitive
Re.
or
R)
R
for each
(cf. § 3.4, 3.5) one requires
a sequence satisfying
~* ~ <~1,...,~n > ,
of constants
R~.,
c.
constants for
Rn
4? (~)
~zo
~(Sz)
=~,
=~(~z)~
where ~ ~*~ ~ = (YI'''"Yn)' Yi ~ i ~ (~I)'"(%)(0)%' Ri ~. c (~I)'°'(%)(~I)'"(~n)(°)~i, liiin. I% has been shown by Schutte
(see Hindley- Lereher- Seldin 1972, p.156)
that constants for simultaneous
-HA m
(1.7.7)
fier-free
;
the p r o o f
fragment of
reoursion
that
qf-~-HA~
recursion
satisfying
t h e y satisfy ( i )
(1) can be defined in
can be given in t h e q u a n t i -
(1°5°13)o
A more complicated way of constructing course-of-values
l
is described
constants
R , via simultaneous
in Diller- Schutte
1971o
If one wishes to avoid the fairly long and ad hoc argument needed to obtain R =
from
R
, there are various possibilities°
(A) One may i n c l u d e scriptions
constants
of
~-}~9o
satisfying
Since q u i t e
and t e r m s can be d e a l t
variables
~.
with
(1) a s p r i m i t i v e s
consistently
in notation now and then.
followed in Troelstra
sequences o f v a r i a b l e s
difficulties
apart from a
(This alternative has been
1971.)
(B) One may extend the type structure Cartesian-product
de-
i n complete analogy to the t r e a t m e n t of s i n g l e
and terms, this causes no particular
certain awkwardness
in the
types,
T
adding to
to a structure TJ , T 2
T'
including
a third closure condition
( X binds stronger than application) : T3) : ~,T e T' = ~ Xm e T' (and replacing
T
by
T'
and to the set of constants with inverses
D' ~T
in T1 , T2
).
e ( ~ ) ( ~ ) ~ x~
one adds pairing operators
E (~ ×T)~,
(~×~)~
D"
(for a l l
~,,cT')
wit~
~T
axioms
(2)
D,(~xy)
= x,
(3)
D(D,~)(D"~)
~,,(Dxy) = y , = ~.
It is shown in 1.8.2 that this enlargement expansion
(and a f o r t i o r i
a conservative
In the presence of pairing operators (3))~ we are able to define operators a* = (~l,OOO,an)
N_HA m
extension)
("p " for "pairing) of
~_~mo
satisfying (2) (but not necessarily i D . , D ~. for n - tuples
such that
Di(Dxlo°.Xn ) = xi ,
1 < i < no
By a standard trick well known from recursion theory, eursion operators may then reoursiOno
be defined ; we illustrate
simultaneous
T ~ R%~2(Dxlx2)(Xuz. prove by induction
D(Yl(D'u)z)(Y2(D'u)z))
re-
the process for double
We put
Thee ~ c r e a d i l y
is an
.
48
D'(TO)
= x I,
D"(T@=
~2
D'(T(Sz)) = YI(D'(Tz))z, and
therefore
we may
D"T(Sz) = Y2(D"(Tz))z
take
R ~I,~ I 2 ~ kxlx2YlY2Z • D'(Tz) ,
~ KXlX2YlY2Z. D"(Tz) R ~I,~2 2
•
It has been shown in B arendre~t A that it is impossible to define in
types
~ XT 6 T ,
and o p e r a t o r s
D 6 (~)(T)~ Xr,
D' 6 (~ X T ) ~ ,
~-~w
D" 6 ( ~ X r ) T
satisfying (2). In suitable versions of is possible to construct
~-HA w
with the
a × T ~ D, D', D"
k - o p e r a t o r s as a primitive it such that (2) is satisfied
end of 1.6.17) ; then, of course~ the constants A fortiori~ in the extensional system
R=
qf-~E-H~
(see
can be defined. w,
it is possible to
define product types and pairing operators such that (2) is satisfied (of. 1.6.17), so in extensional contexts simultaneous recursion does not cause any problems. It should be noted that the methods o~ Soh~tte from H i n d l e y - L e r c h e r Seldin 1972 , or of Diller and Schutte 1971 do not extend automatically to other schemata for defining functionals which have "simultaneous" and "single" versions, such as bar-recursion, or definition by induction over well-founded trees;
in such cases, we are forced to fall back on the methods of treatment
described under (A) and (B).
For this reason, we have e.g. in § 2.3 indicated
a treatment of computability including pairing operators. 1.6.1~ Pairing operators in
qf-WE-~
w.
A pairing operator for the extensional theory, with inverses, is implicit in the~Y-pure types (1.8.5- Io8.8), since in the description of the reduction pairing operators with inverses for the pure types are given. Assume a product type
0 X 0 6 T , and operators
Do,o, D' D" O,O' 0,0
to be
given such that
(1)
D~ ,o(Do, ° x yo
o)
=x, o
D"
(Do,oX°Y°) = y o
O~O
Then product types D~, r 6 (~ XT)T
~ XT,
and operators
D~,~ 6 (~)(T)~ XT,
(satisfying 1.6.16 (2)) may be constructed relative to
Do,o, D'o,o, D"o,o
as follows.
~ (al)...
(~m)O,
Let
w ~ (T1)...
(Tn)Oo
~e put ~XT
~def (~1) "°" ( ~ m ) ( T 1 ) ' ' "
(~n) OXO
and define 0°
D'a,T £ (~ X T ) a ,
~def 0,
0 (~)* ~ def
H
~,~
0r
49 , ~
D
•
~I
... x m
x I
... x m
D ~~, T
~ ^z
~Z~X~y~11
D"~ , T
TI
~m
T ~ ~x y x I
Tn °°" Yn "Do,o(XXI"'•XmJ(YYI"''Yn
Yl
*
~n
D'O ~ o Z X l
. D"
°'• Yn
o,o
•
zO
..x m 0
"'"
TI o°"
0 Tn ,
Yl "'" Yn
oSm
J'
°
Then
D$,T(Do,TX
) =
Lx I
=
...x m
. D$,o(Do,o(XXl...Xm)(YOT1°..oTn))
~.x I
~m °°-x m
• XXI~..X
= 0 , and
Do,o'
Y
°'I
and similarly
for
We may take pairing
D"
function
j
=
X
.
a,T
0×0
m
with inverses
o,o'D' D"o,o
J1' J2
to be given by the standard
(~ $.9
)
Do, ° ~ ~y.j(x,y), D,o,o ~ it' D"o,o ~ J2' Alternatively,
=
if we take
0 ×0
jlj(x,y) = x, j2j(x,y)=y.
~ ((0)(0)0)0 , and we define
Do,o =def ~Oy%(O)(e)o
~Y,
D'
o,o -=def ~(o)(o)o,o n o,o'
D"
o,o
def ~(o)(o)o,o
H~
o,o
where Aa,T then
~xay ~
def
(I) is satisfied
variant
of
°
y x
again;
qf-N-HA
w
,
o,o
in fact,
with the
~
we can now establish
X - operator
Lx.tx= t
of the induction
(cf. also the discussions
1.6.18.
Historical
notes ; variants
A quantifier-free to
qf-!-H~
only,
~)
extensional
version
In Kreisel
recursion,
1959, Specter
Spector's
x= ( y--= z)
,
contain
constants
for defining
recursive
functionals
1958.
axioms
it is obvious
(correspondirg
The system is sketched
and rules is given. that an intensional,
1962 , and Grzecorczyk
theory is studied. system corresponds O, S , constants
~nP
functionals
paper also contains
use
From not an
was intended.
of the quantifier-free
functionals
x , but without
in 1.8.4, i 2 % [ - 3 q ) -
in Godel
list of primitives,
1958 , however,
(2) in a
with rules
in the literatureo
was first introduced
3 in G~del
does not contain
theory of primitive
i°eo no detailed
footnote
t
1.6.16
as a primitive,
t = t' = kx.t = kx.t', rule
if
Kxy. y ,
def
(projection)
qf-WE-~
and
m.
(substitution)
sat~fying
by composition
the generalized
to ~s
1964 an extensional
If we disregard
His primitives satisfying
for
~s~
=
~ x I o o .x n : xl , and schemata
(primitive)
induction
version
the schema of bar-
rule
recursion.
(1.7.10).
Spector's
5o
In Kreisel
1959, the type structure
section 5 of Specter schemata,
1962 , Kreisel's
The first
1964 describes
R ' x y O = x,
satisfying
Ix=x 7
R'xy(Sz) = y z ( R ' x y z )
°
In Tait
also describes
1967, the full intensional
D i l l e r and Sshutte (contained in The pairing
E~
to
theory. 07 $7
and
D x y = xyy 7 07 S7 I,
and
defined ones.
in detail° with logical
I - ~w
operators
, but with
x
=y
is Vxa~
ya
as a primitive.
1971 seems to be the first paper where a neutral
qf-N-HA
theory
w ) is taken as a starting point.
operators
described
of the subject ; the first variant This v a r i a n t has the advantage al equality for the typed
in Io6o17
seem to belong to the "folklore" ~73. in 1.6.17 is e.g. found in Luckhardt 1970 ~
of being consistent
X - calculus
studies of the f u n c t i o n a l s
theory of combinators,
from p r e v i o u s l y
theory,
studied ; Tait's f o r m a l i s m corresponds
X - abstraction.
and contains
C x z y = xyz,
The second set is based on
pairing operators
as an axiom instead of h a v i n g
As noted in
of the q u a n t i f i e r - f r e e
Bxyz = x(yz)7
five schemata for defining new f u n c t i o n a l s Grzecorczyk
types.
by a schema p e r m i t t i n g
two variants
set is based on closure u n d e r composition7
I, By C 9 D, R~
Other
product
schemata are very similar to Spector's
but have to be supplemented
Grzecorczyk
includes
of
are eogo Sanchis
with a n o t i o n of
(cf. end of Io6o177 N-HA w, 19677
Barendre~t A).
in the context
Stenlund
~mke~sion-
of a typed
1971, ~ 9 7 ~ .
51
§ 7.
Induction and simultaneous
recursion.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
1.7.1.
Contents of the section.
detail concerning and extensions
The section discusses various points of
induction and simultaneous
of these systems.
recurslon in
qf-~-HA
~, HA w
The reading of the section may be post-
poned until the need arises. In 1.7.2- 1.7.7 we describe a method, constants for definition by simultaneous simple recursion. interested
due to Sch~tte,
for obtaining
recursion from the constants for
Proofs are carried out in
qf-~-~.
If the reader is
say in a discussion of modified realizability,
interpretation
(in § 3.4, 3.5 respectively),
and wishes te avoid the tedious
details of 1.7.7, he may either simply believe the result, structure extended by Cartesian product of formation simply postulate
in the description
of
or the Dialectica
N-HA ~
(cf.
or use the type 1.8.2), or
simultaneous
recursion
outright. In 1.7.8- 1.7.10 we obtain an induction lemma, interpretation
In 1.7.11 we briefly discuss iterator may replace Subsection
1.7.2- 1.7.7.
1971) how the
the treatment of simultaneous
recursion and
HA ~ .
Simultaneous
Definition.
(following Diller and Sch~tte
the recursor as a primitive.
1.7.12 discusses
the induction lemma for
1.7.2.
needed for the Dialectica
in § 5.5, taken from Spector 1962.
recursion in
q f - N - H A ~.
We put
prd ~def kx. RO(Xyz . y)x,
x=y ~def Rx(~uv. prd(u))y x>y
~def x ~ y
xZY
~def (x>y) V x = y ,
The following
x
~def y > x
x!y
In
(i)
prd(O) o o ,
(ii)
x;O
= x,
( i i i ) prd(x)/o(iv)
Sx=Sy x=x
(vi)
Sx~x
below,
but espec-
is that we wish to verify that they indeed have quantifier-free
Lemma.
(v)
7
The reason for explicitly proving all these elementary
proofs. 1.7.3.
~def Y Z x
two lemmas will be needed in part in 1 7
ially in |.~.|O. properties
/ O,
= x~y = o = I
qf
N-HA ~
prd(Sx) ° x x~Sy = prd(x=y) x/o
5~
x~0-~
(vii)
x=Sprdx
(viii)
Sy_<x "~ y < x
(i~) (x)
y<x
-~ x "--y- S(x "--Sy)
Sx'--y = 0 "~ x'-Uy = 0 .
Proof.
(i), (ii) immediate
(iii),
Contraposition
of
, Induction on
(iv) Sx~Sz
= x~z
,
y :
then
f r o m the definition. x = 0 -~ prd(x) = 0 . assume
Sx'--SO= prd(Sx--O) = prd(Sx) = x=x'--O ;
Sx~SSz
= prd(Sx~Sz)
= prd(x ~ z )
= x~Sz
apply
;
rule of induction. (v)
: By (iv) and induction on
(vi)
:
Induction on
x.
(vii) :
Induction on
x:
then
x.
O/ 0 ~ O= SprdO;
assume
z / 0 ~ z = S prd z ,
S z ~ 0 ~ S p r d ( S z ) = Sz .
(v~i) : TO show
x±Sy/
x = Sy ~ y . p r d x ,
0 V x = Sy " x = y /
so
x ~y=
~Sy/O-prd(x~y)~O,
Sprdx
so
(ix)
: y<x~
x!y~O
(x)
: By induction
1.7.4.
Lemma.
In
Proof.
I n d u c t i o n on
I
(by vi).
(byiii).
~Sy/O-x~y~O
; x~y~O on
O.
m prdx=
~ Sprd(x~y)=S(x~Sy)
.
y.
qf-N-HA
~
y < x ~ x ~ (x.ty) = ~ .
y.
(I)
O<x--
x:(x±O)=x~x=O
(1.7.5
(2)
so<x - ~
(~=so)
: x= prdx : I
Sz<x -
(x=Sz)
= Sz
, (v)),
(1.7. ~ ,
(~i~).
Assume
(3) If
SSz<x,
x~Sz
then
.....S ( x ~ S S z )
SSz<x,
.
hence
Sz<x
( 1 . 7 . 5 , (viii) ; with 1.7.5(Ix)
(x=Sz)
= Sz/ O,
o
prd(x~ (x~SSz)) (1.7. 5 ,
x=
=x~S(x~SSz)
=x~
so
x~(x=SSz)
= SSz
(vii)).
Hence
(4)
(3) - [(ss~<x) - x ~ ( x = S s ~ )
(2),
: ss~]
.
(4) give the i n d u c t i o n rule
(5)
sy<x-
(x: (x=Sy)
Hence also by i n d u c t i o n f r o m 1.7. 5 .
Simultaneous
sy.
(I),
r e c u r s i o n in
For each sequence of types of constants
=
Ri ~I~.,.~O
(5),the a s s e r t i o n of the lemma. qf-N-HA~.
~I' "''' G n
~ i = 1,...,n n
we wish to construct
(abbreviated
as
R) =
a sequence
such that for
55 sequences of variables
x ~ xl,...,x n , y m yl,...,y n,
where
x i E ~i '
Yl ~ (~I) "'" (%)(°)ai (1)
~o
= ~,
~(s~)
o~(~)~.
In order to ob%ain constants
R
as required in (I), it is sufficient to
=
establish the recursion rule : I If (2)
~, ~
are sequences of closed terms (of fitting types),
is a sequence of closed terms
~
such that
there
~0 = ~,
T(Sz) = Sz(T~). To obtain (I) from (2), we apply (2) with (with
~
and
~
having the same types) for
satisfies the equations for
~O~y = ~ , and therefore
k~
kzx~.
R~z
,
T(Sz)xz o ~(T~)z
b~=y z. T z x y
.x , t, ~
~(~y)v
respectively.
Then
=T
i.e.
,
satisfies the equations
The method for establishing
kv°~.
(I) for
R.
(2) below seems to be due to Schutte * (cf.
Hindley, Lercher and Seldin 1972, page 156) ; a stronger result (simultaneous course- o f - v a l u e s
recursion)
is established in Diller and Schutte
1971 ; this implies (2). 1.7.6.
Pairing functions for objects of equal type.
Let P ~
be a pairing function of type p xS y q O
For
P
= x~ '
Pa xcya(Sz)
(G)(~)(O)~ , satisfying
= ya .
we may take P~ ~def kx~yazC . Rx(ku~v ° . y)z .
1.7.7.
Theorem * (Sch~tte).
Proof.
We establish this by induction on the length of the sequences
The recursion rule (2) of 1.7.5 holds.
For length I the solution is given by hypothesis) to construct (5)
Rt(kuv. svu).
(2) to have been established for t1' "''' ~n+1
~, ~
~, ~.
Assume (induction of length n.
We wish
such that
/ ~i(O) = ai [ ~i(Sz) = biz(~Iz ) .-. (~n+1 z)
where Let
ai, b i
(1~i!n+1)
are constants of the appropriate
R' = kxyz . Rx(kuv. yvu)z , so that
type.
R'xyO = z , R'xy(Sz) = yz(R'xyz) .
I am indebted to R. Hindley for communicating to me a correction to page 156 of Hindley, Lercher and Seldin 1972, together with the proof given below.
We put t = X U l . . . u n . R ' a n + l ( X V . bn+lV(UlV ) . . . (UnV)) t i = XVUl...UnW. P ( u i w ) ( b i v ( u l v ) oo. ( U n V ) ( t u l . . . U n V ) ) ( w : v ) By induction t~O
hypothesis, = ~a.
1
t~(Sz)
i
* "''' t*n t1'
there are
= tiz(t~z )
'
such that,
for
1
(t~z) "'"
"
We put ~. -= ~z. t*zz,
l
tn+ 1 -: X~. t ( t ~ ) . . . Now
(tn~)~.
(3) is proved by induction.
Case a,
I< i< n . {.0 = t~O0 = ~ a , O i
I
{i(Sz)
= a.
1
I
= t~(Sz)(Sz)
= tiz(t~z ) ... (t~z)(Sz)
=
= P(t~z(Sz))(biz(t~zz ) ... (t~zz)(t(u~z)... = biz(t~zz ) ... (t~zz)(t(u~z)...
(u~z)z))1
(u]z)z)
= biz(~iz ) ... ( ~ n Z ) ( t n + l Z ) • Case b,
i = n + I.
We first establish
(4)
t~(z+w)z 1
We prove For
= ttzz
for
I < i < n,
all
w.
1
this by induction
on
w.
For
w= 0
(4) is immediate.
w > O,
t~(~ +S.)~
= t i*(s(~+w))~
=
= ti(~ +.)(t~(= + w))... ( t ~ ( ~ + w))~ = = P ( Z [ ( z + w ) z ) g ( z ~ (z+w)) = t ~ ( z + w ) z = t~zz (the exact form of the expression z ~ (z+w) = 0 , derived Now we establish, (5)
by induction
for all
on
is irrelevant on
w
from 1.7.3
(v) as basis).
bn+ I ~ ( t ~ ( z + w ) v ) . . .
(t~(z+w)v))z.
z.
t~n+10 = t(t~O)...
(t~O)O =
= R ' a n + l ( X V . bn+ 1 v ( t ~ O v ) . . . which is equal to the right hand tn+1(Sz)
here ; we must use
z, w
~n+1 z = R ' a n + l ( X V .
We use induction
~
= t(t~(Sz))...
(t~Ov))O = an+ 1
side of (5) for (t~(Sz))(Sz)
=
= R'an+1(XV.bn+ l v ( t ~ ( S z ) v ) = bn+l~(t~(Sz)~)...(t~(S~)~)*,
...
z = O.
(t~(Sz)v)(Sz)
=
•
55
where
¥ m R'an+1 (~v" bn+ I v ( t ~ ( S z ) v ) By the induction h y p o t h e s i s
for
...
(t~(Sz)v))z
z , using
.
w= I ,
¥ = ~n+1 z • Also,
by the induction
hypothesis
for
z , using
Sw
~n+1 z = V' = R ' a n + l [ X V . b n + I v ( t ~ ( z + S w ) v )
for
...
W
(t*(z+Sw)v)]z
,
hence
~n+l(SZ) = bn+lZ(t~(Sz)z ) ... (t~(Sz)z)V' = b+lz(t~(S~+,)z)... (t~(S,+-))~' (by (4)) = R,an+l[XV.bn+l v(t~(Sz+w)v)... (t~(Sz+w)v)](Sz) . This establishes
Now we can complete
(5).
~n+1(Sz)
= t(t~(Sz))...
the proof :
(t~(Sz))Sz
=
R'an+1[~V.bn+1 ~(t~(Sz)v)...
(t~(Sz)v)](Sz)
=
bn+1~(t~(S~)~) ... (t~(s~)s)~", where ~,,
- R,an+~(~v. b + 1 , ( t ~ ( S , ) v ) . . . = ~n+flz ,
tn+l(Sz)
= bn+iZ(t~zz).-.
bn+1~(~Is)...
=
and since also is irrelevant)
(t~zz)(En+Im)
(~n~)(~n+~),
~n+10 = t ( t ~ O ) . . .
(tnO)O = R'an+IgO = an+ I
(the form of
the proof is completed.
1.7o8-1.7.10.
The i n d u c t i o n
1.7.8.
Let
Lemma.
(t[(S~)v)~ =
hence
~
lemma for
qf-N-HA
w.
be a sequence of terms of
N-~W
and let
a sequence of terms defined by means of the r e c u r s i o n operator tOxv
= v,
=
where in
qf
=
x,y ~ O . N
Let
=
Q
= T(x--Sy)(tyxv) =
=
be a predicate
be
,
=
such that
(x, =v not free in
F)
HA m
rbQ(x, Then in
t (Sy) x v ~
t
such that
~_x_v)-Q(s,,v),
rbQ(o,=~).
qf-N-~®
Proof.
In order not to encumber our t y p o g r a p h y we let
Assume
z< z ,
U.
Then
x--z = S(x--Sz)
(1.7.3,
t -= t , v -=v , T = T .
(ix)) ;
=
=
56
t(x~Z)XV
= t(S(x~Sz))xv
=
= T(x~S(x~Sz))(t(x~Sz)xv)
=
= T ( x ~ (x~ ~))(t(x= s~)xv) = = Tz(t(x=Sz)xv. Since
~Q(x,
Txv) ~ Q(Sx, v) ,
it follows that
Q(~, T, (t(x~Sz)xv)) This implies 1.7.9. then in
Q(z,t(x~z)xv)
Lemma.
When
~ Q(Sz,t(x~Sz)xv)
satisfies
.
the conditions of the previous lemma,
~
qf - ~ -
r p y~ Proof.
Q
- Q(Sz, t ( x ~ S ~ ) x ~ )
- Q(y,~(x~y)x~)
Induction on
.
y.
O i x -- Q(O,~xx~)
( s i n c e C ffQ(O,v)).
A s sume
Then Sz!x
~ z < x,
hence
Sz~x--
Sz!x
" z!x
hence
Sz&x
,
(Q(z,~(x~z)x~)
-- Q ( S z , ~ ( x ~ S z ) x ~ ) )
-- Q ( z , ~ ( x ~ z ) x z ) .
hence
szix 1.7.10.
Induction lemma.
Q(s~,~), Proof.
Q(sz,~(x~ s z ) x S )
then
rp
In
qf-N-HA
Q(~,~)
(x C
By the previous lemma,
r pQ(x,~ox~),
i.e.
Note that for
v
w
n o t o c c u r r i n g f~ee i n
P ~ x~x
~ Q(x,t(x~x)xv)
r). , hence
rpQ(o,~).
consisting of a single variable,
the proof only requires
simple recursion. 1.7.11. Theorem. Schutte
1971.)
corresponding TE T )
(Replacement
of
If we replace in axioms by a constant
Ra
by the iterator
qf.N_~W J
,
J
; Diller and
the constants
R
the iterator of type
with its 7
(for each
satisfying %xyO
= x , J xy(Sz)
then a constant Proof.
The
define
Pa
satisfying
= y(Jjyz)
(x~ 7, y E (~)~) ,
the axioms for
R
becomes definable.
X- operator can be defined as before. In terms of ~ G x~ G as x y "Ja (H @y ) ; then obviously G
Pox y 0 = xa ,
P x~yC(Sz) =
y~
J~
we may
,
57
Next we define
U m kx I . Jo(xI(SO))S.
Ux+O
= x+(SO)
ux+(so)
° s(x~(So)).
Then
This function enables us to define a predecessor
function :
~o y • j(Ho,oO) U y o 0 .
prd ~
We prove by induction on
y
J(Ho,oO)Uy (SO) = y . Then it follows that prdO = J ( H o , o O ) U O 0
prd(Sy) NOW define
QT Q
= Ho, o 0 0
= 0
= J(no,cO)U(Sy)O
° U(J(no,oO)Uy)O
= J~no,oO)Uy~so)
o y.
o
as
~ kxyz[x(y(prdz))(prdz)]
(z~ O, y E (O)v, x E (m)(0)T) .
Then
%xy(S~)
= x(y~)~.
Finally we put
~
~ ~'y(')(°)'~°[j(o),(H,oX)(%y)~
]
and then
RTxTy(m)(O)'o =
RTx'y(')(°)'(Sz
x',
) =
y(Rxyz)z.
Q.e.d. Remark.
In the sequel we have usually dealt directly with
tive ! occasionally,
in applications,
R
as a primi-
there might be a slight advantage in
using the iterator as a primitive. 1.7.12.
Simultaneous
recurs%on and the induction lemma in
We note that the proof of the induction lemma (1.7.10), neous recursion is available, difficulty.
For the case
can be carried over to
v= v
(i.e.
v
qf-HA~. provided
qf-HA m
simulta-
without
consists of a single variable)
we
only need simple recursion. In the proof of closure under simultaneous in establishing
(5) in 1°7.7.
A(z,w) where
3[z,w]
by the following
the crucial
step is
(5) as
~ (~n+l z = S[z,w])
represents
Assume ~n+IZ E ~,
recursion,
Let us abbreviate
~ ~ (T)O.
the left hand side of (5) in 1.7.7. Inspection of the argument in
sequence of equalities:
I.?.7
shows that
58 = ~(bn+IZ(q(Sz)z)... = u~(bn+lZ(t~(Sz)z)... = u~(bn+IZ(t~(Sz)z)... = u~s[S~,.]
u~(~n+1(Sz))
the a s s e r t i o n
u~(~n+1(Sz))
r[ua's[z'']'z]
(I) where
r[u~,v~,z]
Le~ us put
stands f o r
0 ~
n+1
from
then ( I )
(t~(Sz)z)v;)
is equivalent
.
to
T o u , e[,,1].
intuitively :
L"
Vw<x(um({n+1(Sz))
introducing
0 °
f
=
fu%x
(2) can be expressed
f(To~),(Sx)
0
for all
=x
0
recureion,
can be expressed by
such that
- u¢s[z,O]i
. lu~(~n.l,)
- u%[~,S~]
I .
as
= 0 -- f~(S~)x=O.
po x 0 = X 0 ~ qo x
.
quantification
by primitive
lu~(~n+~Z)
fu%(Sx)
pc, q¢
= ums[Sz,w])
system bounded
a function
fu%
N o w let
= =
o
In the q u a n t i f i e r - f r e e
Then
=
,
u~(bn+IZ(t~(Sz)z)...
r[u~,v~,,]~
Tom =(~n+1 ~)
(2)
can be o b t a i n e d
= r[u¢,~n+IZ,Z]
To ~ X u % v ~
Therefore,
= ums[Sz,w]
(t~(Sz)z)s[,,.]) (t~(Sz)z)({n+lZ)) (t~(Sz)z)s[z,Sw])
,
~e T
be defined by
p(¢) X(~)T = pT(X(a)TO¢) q(¢)~
X0
= Xx¢.%x
0
Obviously p~qcx So
q¢
o
provides
= x
o
an e m b e d d i n g of the natural
numbers in type
Now put
B(v(°)~'~) ~ef f(v(°)%)~(P~ v(°)~1)) = 0 and let
T 1 ~ Xv(°)%.
P(To(v(°)%)~)(%(Sp~(~(°)~l))),
then
B(Ttv(°)%,~
) -- B(v(°)~,S~)
e .
59
Since obviously Therefore,
~(qoo)
ua(~n.10)
= uCs[O,w],
by the induction lemma 1.7.10
for v yields
extend the induction lemma for v.
B(v(°)~,O) .
B(v(°)~,z) .
Substitution
~u~(~.l~) ~u°sE~,~l.
Thus we obtain simultaneous
variables
we also have
recursion in qf -HA~ J
qf-HA
W
, and now we can
to an arbitrary
sequence of
of
~0
§ 8.
N -HA w
More about
1.8.1.
Contents of the section.
neous information of
N - H A m,
This section contains further miscella-
which may be consulted by the reader when the
n;ed arises. First the extension of the type structure by Cartesian product formation, together with the addition of pairing operators as primitives is discussed in 1.8.2 it is shown that this extension constitutes an expansion of any theory in the language of
N_~m.
Subsection 1.8.4 is devoted to the discussion of the
X-operator
as a
primitive. Subsections 1.8.2.
1.8.5 - 1.8.9 discuss reductions of the type structures.
Theorem
(Cartesian product types and pairing operators).
Let the type structure
T
be extended to
T'
by adding a clause
~,T E T' = O'XT E T'
T 3)
T I , T 2
and replacing in
~
by
~'
(1.6.~).
Furthermore, we assume the existence of constants D'
E (~)~
D"
E (~x ~)T
D'(D X y ) = x , Let
N - HA m ~ p
q f - N - H A~ m )
D"(D x y ) m y ,
N-HA m , and ~
is an expansion of
,
satisfying D(D'x)(D"x) = x
denote the extension of
is an expansion of
Do,,7 E (a)(7)~ ~
.
N. - HA m . thus . obtained. .
q f - ~N - HA'~ ~ p
Then
N - HA pm
(defined analogously to
N - HA m. ~ ~ p @
Proof.
To each type
a E T'
of types in
we assign a sequence
T,
as
follows :
(i)
O* m ( 0 ) .
Let
~*
(ii)
~ (~1 . . . . . a m ) ' T* ~ (T 1 . . . . . • (~×7)*~(~I,...,%, 71 .... , 7 ) ,
).
(iii) [(~)73" m ((~l)...(~m)T1 ..... ( a l ) . . . ( ~ m ) ~ ) . m I ..., H a.,~.) ~a*,7* m (n~*,~*'
We define a sequence
Further we define a sequence
~p.,~.,T. For
~.
=~,
m X~.
~p.,~. T . ~z(y~)
such that
such that
.
we take the sequence as defined in 1.7.5.
where
~ ~
(t 1 ....
,tm),
m (Sl,...,Sm) , is interpreted as
t i m s I & ... & t m = sm . Now we define a mapping
F
on terms and formulae of
~ pw, N-HA
as follows.
(i)
To each
x a , ~ E ~'
(el,...,~n) and
Fx 'a
~ ~ (x ~I I ..... x an n ) , where
we assign a sequence
~ a*.
If
x a, x 'a
are distinct variables,
(iii) If
Fy T m y ,
Fx a m ~ ,
sequences
Fx a
have no element in common.
Fna, T m ~a*, T* ' FZ p,a,~ E =Z0 * ,a*,~* , rR to=o, r(s)=s.
(ii)
then
of o p e r a t o r s
~ ~a* '
we take for
DDa, ~
~a*,T*
kxy. y , and for
and
the c o n c a t e n a t i o n
of the
rD,
rD"
we take
r%,~
ro,.a,~ = x ~ - - ' [ "
~a.,T.,
(iv) (v)
rtt, m (rt)(rt,). r(t= s) m (rt= rs).
(vi)
r
preserves
propositional
r(A o B) m r(A)o r(B)
operators
for
and Yk , i.e.
r(~) m Yk,
o = ~, V, &.
(vii) r(vxaA) ~ Vrx~(r(A)) , r(~xaA) ~ ~rxa(r(A)) . First note that
r
is the i d e n t i t y on formulae of
n a m i n g of variables). of d e d u c t i o n s
It remains
AW~A N - H.~p
that
=
~-HA w
(modulo re-
to be shown, by i n d u c t i o n on the length ~N - H A w ~ F ( A )
.
This turns out to be com-
pletely trivial. 1.8.3.
Remark.
The theorem also extends
obtained by adding d e f i n i t i o n without
C a r t e s i a n product
additional 1.8.4.
definition
The
schemata for functionals,
types contains
schemata
~-operator
to certain extensions
~
for
variant of the
N- IDB~,
B ~
in § ~.9)-
as a primitive notion.
Instead of h a v i n g a theory
E-HA ~
with p r i m i t i v e s kx ~
,
Z
version
kE-HA w,
ators.
kE-H~A w
is similar to the d e s c r i p t i o n
of
with
H
may c o n s i d e r an a l t e r n a t i v e The d e s c r i p t i o n
~ - ~
if only the theory
a "simultaneous"
(examples
of
we
as p r i m i t i v e
oper-
of
E - HA ~.~w , w i t h the f o l l o w i n g differences~ (a)
Ha, T ,
(b)
The t e r m - d e f i n i t i o n
kx~
Z0,a, T are added.
T m 5) (c)
the operators
are omitted from the list of c o n s t a n t s ;
If
(1.6.4)
tETm T ,
The d e f i n i n g axioms for
is extended with a c l a u s e :
then
kx~.t E Tm(a)T .
H0,a,
Ep,a, ~
are replaced by the rule of
- conversion : ~-CON
(kxa.t[xa])t '~ = tit']
If we make changes
(a),
(b),
(c) i n
(t'
E-HA w t
we
free for
x
in
o b t a i n a theory
t). hE-~
~o
(cf.2,~8). Now
E-HA W
and
hE-HA w
are e q u i v a l e n t
in the f o l l o w i n g
sense : as
w
~2
has been shown in 1.6.8, we can define a the rule of operators
I- operator in
k- conversion holds ; conversely, ~
_, E
_ _
satisfying
lng p r l m l t l
es in
E-HA
zp,~,~ h e f
~(o)(~T~y(~)%p
Hence the union of
,
~-H~ m
kE-HA w
such that
we can define
the defining equations for the correspond-
by p u t t i n g
H~,~ =def kxPy~'x~
. x~(y~) .
E_~m
and
~E-HA~
is a definitional
both ! and similarly for the quantifier-free ~qf-~-HA~
in
theories
extension of
qf-~-~
and
(the latter defined in the obvious way).
The description
of a
i- variant of
(qf -) N - H A w
or
(qf -) ~ - ~
is
not such a simple matter however.
The problem is this : if we simply in-
clude the rule
in our system (say
s = t = kx.s= Âx.t
between closed terms of
s=
( s, t
H
A,B
~),
In fact, if
S, T
are recursively enumerable,
inseparable.
recursively
inseparable
1967 , p. 94) such that
c ~ {xl ~ ¢ A }
Then
C
is recursively
(by completeness So
then equality
= ~.
~ ~An~
Let
then
be a pair of recursively
sets (e.g. as in Rogers
(~)
~),
decidable.
{j(~s~,~t ~ ) l ~ s = t }
closed terms of
For let
cannot be recursively
A,C
of
enumerable,
~
for
Zo I-
is a pair of recursively
Now the statement recursive
P.
Let
istic function of x~ A
*
x~A t
P.
reducible
to
S,T
is equivalent
*
(qf-) ~ - H A ~
restricted
to
recursively VyP(x,y) [
x~ B ~ ~ ~x E B
(by (I))) . inseparable
sets.
for some primitive
representing
the character-
Zy([ ~ t x y ~ 0)
s= t = kx.s= ~x.t ),
respectively, Hence
has therefore "ordinary"
means.
(since
ky.O ; and so the pair
via the mapping
the pair
The problem of the description
distinguishing
BcC
Then
Zy~Pxy
(Ho[ers 1967, p. 80).
enumerable,
be the closed term of
~ ~t~ (by the proposed rule
i 0 C = ~,
predicates) ~ ~~ b ~ ¢ A
of a
S,T
to be solved in a different
A discussion
I - l-
inseparable.
(qf-) ~ - H A ~ manner,
or
namely by
and equality established by
of this possibility
after the treatment of computability
is
k x . j ( ~ t ~ ] rky.0~) .
is alsor¢cursively
~- variant of
provable equality,
ArC
in Chapter II.
is better postponed till
63
1.8.5-1.8.8. 1.8.5.
Reduction to pure t.ypes.
Pure t y p e s . OE
T I)
The p u r e t y p e s
P
are defined
inductively
by
P:
We introduce an abbreviation using natural numbers to indicate pure types : (n)O -=def n+1. We now wish to construct a mapping for
a E ~P' and such that to each
F~ E ( ~ ) a , F
and
F,
P'P f = f,
~
of
T
onto
P,
such that
~ E ~T there exist mappings
definable in
N - HA w,
such that in
~ =
F E (g)~ qf-WE-HA
w
P P'f' = f' .
This is done in a number of steps. 1.8.6.
Injectiq~ in higher types.
We define mappings j
of type I
(I)
mpj , with left-inverses
into objects of type
j+l,
pm.0
~0 0 0 _ . I I~ mPo ~ ~x.y .x , Pmo = ~x .x u, and for XxJyJ.xJ(pmj_1(yJ)) , mpj ~xJ+lyj'1"xj+1(mpj -I(yj-I)) . pmj
One readily verifies that
which map the objects
as follows:
mpj E (j)j+1,
j>0
pmj E (j+1)j.
By induction on
we find that pmj(mpj(xJ))
= xj .
For we have
Pmo(mPoxO ) = Pmo( ~y o .xO~j = ( ~y o .x o )0 = x o , and for
pmj(mpj(xJ))
= pmj(~yJ.xJ(pmj_l(yJ) ) = = ~yJ-1[~vJ.xJ(pmj_1(yJ))](mpj_1(yj-1)) = ~yj-l.xJ(pmj_Impj_1(yj-1))
Now we construct type-increasing mappings m pmj E (m)j ( m ~ j ) by
j>o
=
= ~vJ'1.xJ(y j-l) = x j . m
mpj E (j)m,
with left-inverses
I mp~ ~ kxJ.x j , pm~ ~ XxJ.x j , mpjm+l ~ kx J . mPmmP~(xj) ,
(2)
pmm+1 _ . m+1 m~ m+1\ J = ~x . pmj~x ). 1.8.7.
Coding of
n - tuples for all pure types.
Let
J' J1' J2
denote the standard pairing function with inverses for
the natural numbers.
We extend these to all pure types by putting
j ( x m + l y m+l) ~ XzTM . j(xm+Izm, ym+Izm )
J
~4
•
,
m+1
•
,
m+1
31~x J2~X
.
) ~ ~z m. j1 x )
kz TM
This may be extended .2
_
32x
to
.
.2
z
m+1
z
m m
.
p - tuples, _
.p+1~ n
m+1
.
.2
putting
_
xn
. n n p+~) ~ ~(x~, ~.p (x~n ..... ~p+~))
~x~, ....
.p+1 n ~ n 31 x ~I x ,
.p+1 n ~ .p . n ~k+IX 3k 32x
Now we are able to describe
the coding of
for p - tuples
where As inverses
.
.
.
of different
pure types:
m xn(P)~ ,mPn(p> p
g(x~(~),.. ,xn(P>)~ ~ ~P<~p~(~)x~ (~> .
1~k%p
.
"
m = m~x~n(1) ..... n ( p ) } .
we have
.p,m~ m~ m j~(x m) 3k,l kx ) ~ pm 1 so that k,n(k) 1.8.8. (i)
....
Description
If
~ = O,
(ii) If
of
~, ~ ,
C~ ~ a,
a = (al)...
7' .
F f = f,
(~p)O,
m = maxlnl,...,npl,
then
m x~FV .p,m Fax~ = XY " ~ ~IJ1,nl F'xm+1
1.8.9.
Reducti.on to numerical
O ~ T =o ,
T O 2)
~1,...,an,~ T
~n
1~i~p,
we put
D~ = m+1 , and
ym)
Ft ~p,m ( ~2 e2~n2 ym) ... (F~p J1,np~p'my m ) ,
6 ~TO
~1'''''~n
qf-W.~E-H~ ~.
extension
~ (at× ... X=n)T
(the numerical ~o 2)
t~pes in
the following
To I)
and let
for
= ~Ylal ""Yp~p " xm+1 JP(~IYl .... ' ~pYp)"
Let us consider
restricting
7'f = f.
D~ i = n i
types)
be the
T ~o
6
of the type
~T0
substructure
E ~n
=
(alX'''~an)O
of 1.8.2,
D~,~,
D'D(x,y) is an expansion case which
of
T
obtained
~o
6 =nT .
qf-N-HA
~
when extended
~
Da,~,
T :
to
Note that in virtue constants
structure
D"o,~
= x,
added
D"D(x,y)
of the original
especially
interests
T
and with
~O
such that = y,
system us,
to
D(D'x,D"x) qf-N-~
qf-WE-~
= x
; and similarly ~.
for the
~5
The type structure
T ~o
may be reduced to
T ~n
by mappings
F , F~,
as
follows : (a)
~ = 0 ~ then
(b)
~ : X :t
r
• "
~ :
F x ~ = x~ ,
r:
: x[y~, . , x _ ] ~ ( r
F~ xCW : (~)
Oq = ~ ,
(p¢...×%)0~
n
:
F'x q = x
(np¢...~%,)o.
, y. Pl 1'''''Fpn yn)
xn~(r~Y~'''''r~yn)
kYl ' ' " Yn"
"
( ~ ¢ . . . . ~ % , ) , , n, : ( ~ ¢ . . . ~ m ) O .
C~ = (OOlX . . . OOnx 141x . . . X ~ m ) O ,
Then and
gm
rq X q = X[YI' . "''Yn' . . .z1'
'Zm](F~ xa(F'01YI,..., r ~ Y n ) ) (z I ,...,z m ) ~I ~m F,xO~ = kYl "'" Yn • F~(kz I ...Zm " xCm(ZplY1'''''r'0nYn' z1'''''Zm)) "
Here
k[Y1' .... yn] t[Yl, .... yn]
yl,...,yn;
i.e. if
k[y I, . ..yn] . t(Yl, . .
t
expresses simultaneous abstraction w.r.t.
is of type
.,yn) . . E (alX
v,
Yi ~ ~i
%an)V ; but
(1
abbreviates
~ylXY2... kYn.t , hence is of type (al)(~2) ... (~n)T. x(Yl,...,yn) dicates application to the arguments y1,...,y n (simultaneously).
in-
We leave it to the reader to verify that the following schemata for defining functionals of
T
imply the definition schemata of
qf-WE-HA
~n
w
~
(via the mappings described above) and vice versa. 0
is a constant of type
0
(with the usual axioms).
(ii) S
(i)
is a constant of type
I
(with the usual axioms for successor).
(~di) If
t[Yl,...,yn]
yiE¢i
(1
is a term of type , then
(c[1~... × ~ ) 0 , and for
0,
containing
X[y I ..... Yn]t[Yl,...,yn]
YI'''''Yn
free,
is a term of type
ti~ qi '
(k[yl,...,Yn]t[Y1,...,yn])(tl,.O. ,tn) = t[t I ..... tn] . (iv) There is a term ~ E ((a1~ ...Xap)0 ~ ( ( a 1 ~ . . . × ~ p ) 0 X 0 x a 1 × . . . × a p ) X 0 ~ ~I ~ . . . x ~ p ) 0 , a I
such that
ap
~ , ( x , y , O , Y 1 . . . . . Yp ) = x ( y I . . . . . Yp) cr 1 ~p ~(x'y'Sz'Yl . . . . 'Yp )= Y ( ~ [ Y l . . . . . Y p ] ' ~ ( x ' Y ' ~ ' Y l
.....
Y p ) ' Z ' Y l . . . . 'Yp)
(This reduction may then afterwards be combined with the reduction to pure types as described above.) conditions first reduces
The proof of the equivalence of the closure
(iv) to comparison with the recursion rule, which,
however, is equivalent to asserting the existence of constants
1.7.5).
R
(see
"
66
§ 9°
Extensions
1.9.1.
of arithmetic.
Introduction.
divided,
according
I° ) Extensions by adding to additional
The various types of extensions
to their language,
of arithmetic w.r.t, ~(~)
into three categories :
constants.
or
H~
for species or relations introduced by (iterated) (g.iod'S).
in 1.9.2 below. investigation, first-order
~
generalized
with
induction for
with predicate
Such extensions are briefly described
constants
inductive and discussed
With respect to the various methods for metamathematical they behave in most respects like
HA
itself,
i.e. as typical
systems.
2 °) Extensions
of arithmetic
in a language with variables and quantifiers
for species of natural numbers added. prehension
Examples:
or addition of transfinite
a certain primitive recursive well-ordering,
definitions
may be
the same language or a language obtained
one or more predicate
reflection principles,
of arithmetic
can be studied.
pretations,
as well as normalization
can be adapted to such systems, the systems under
I°).
In this context full impredicat~ecom-
Pure realizability
Variants
and pure functional
theorems for natural
deduction
but not in such a straightforward such a s ~ r e a l i z a b i l i t y
intersystems
way as for
do not readily
=
extend to these systems.
For a more detailed description,
see 1.9.3- 1.9.9
below. 3 ° ) Extensions
of arithmetic
in a language obtained by addition of function
symbols and function quantifiers "non-committal"
to
very elementary ones,
~(H~A) . Such extensions may (apart from such as
E~L
described
be grouped according to their intended interpretation (A) The function variables (i.e. completely determined
in 1.9.10 below)
into two classes:
are thought of as ranging over "lawlike" objects,
sequences
given by a "law" or prescription).
As
long as our concept of "lawlike" has not been analyzed to a degree which prevents identification
with "recursive",
spired by the idea of lawlike that lawlike
we may expect systems to be in-
sequences to be consistent with the assumption
sequences are recursive
(Church's thesis,
obtain systems which are proof-theoretically has also to incorporate
additions as under
see 1.11.7).
and functional
one
1°).
Systems for lawlike sequences behave still very much like to realizability
To
stronger than arithmetic,
interpretations;
HA
with respect
with respect to Kripke
seman~cs and natural deduction they have not yet been investigated. (B) The function variables are thought of as ranging over some kind of choice sequences
(i.eo sequences for which it is not assumed that they are a priori
completely determined by a law). see Treelstra
1968 , 1969 .
For detailed discussions
of this concept
Their essential feature is that they enforce
67
certain continuity conditions on operators defined for all choice sequences. I.e., if
~
is a type 2 operator, defined for all choice sequences of a
certain "universe", v~
where
~
v~(~=
satisfies ~x -
~x= <~0 .... ,~(x-~)>
~ = ~) , o
Continuity conditions do not increase proof-theoretic strength ; but it is also possible to postulate the schema of bar induction for choice sequences (which is simply false for the universe of recursive sequences) and which does increase proof-theoretic strength. Realizability and functional interpretations can be adapted to these systems (replacing partial-recursive-function application
{. I(o)
by continuous-
function application). For a description of the principal systems which fall under this heading, -z4 see 1 . 9 . 1 % 1 O W o 1.9.2.
Extensions of arithmetic eApressed in
~(~)
o__~r ~(~A)
extended
by relation constants° The most obvious extension is obtained by addition of a local reflection principle to
RF(m)
~:
Proof~(~, ~A~) -
( A closed)
A
or a uniform reflection principle
RFN(~)
ProofHA(X,rAy u) -- Ay
( VyAy closed)°
For a general discussion of such principles,
see Kreisel an__~dLevy 1968°
Another me~hod of extension is the addition of the schema of transfinite induction for certain arithmetically definable (in fact, primitive recursive) well-orderings of the natural numbers;
i.e. if
~
is such an ordering,
we add TI(~)
Vx[(Vy<x)iy--Ax]-
VyAy°
A third, and very interesting possibility is the addition of constants for species introduced by generalized inductive definitions (g.i.do)o Assume language
PA
to be a new (unary) predicate constant not occurring in the
~.
such that
A
Let
~
be a system with
~(~)=~,
and let
A(P,x)6~P]
is "monotone" :
~[e,P,] ~ A(P,x) ~ Vx(Px-P,~) - A(P,x) (where Then
[[P,P'] PA
schema :
is as
~,
but relative
~ P , P ' ] )°
is said to be introduced by a g°iod., if we add an axiom and a
,
6S
A(PA,X ) ~ PAx and for all
Q
in
~(PA)
The best known example is Kleene
(see Kleene
0 ~ the set of recursive
1944 , 1955, or Rogers
the definition in the form described
ordinals introduced by
1967, § 11.7, § 11.8).
(To bring
above, we have to rewrite the definition).
A simplified version is obtained by taking e.g.
A0(Q,x ) ~ (x=l) v(a(x)o&x=2 (~)°) v [x=~.5 (x)~ &Vy(~{(x)2}(Y)a ~Q({(x)2}(Y)))]. Still simpler is
here
kxy.[x](y)
functions,
is an enumerating function for all primitive recursive
ky[n](y)
representing
the
n th primitive
recursive
function in
the enumeration. We may then define, when
PI
permitting
over
quantification
has been introduced,
Ap2(Q,x ) ~ P1 x V Vy6 P l ( Q [ x ] ( y ) ) etc.
(cf. chapter VI).
definitions 1.9.3.
a new predicate
P2 '
PI :
This procedure
,
gives rise to generalized
inductive
of higher type.
Language
of
HAS ~o To the language of ~
(n_> O) , to be denoted by
we add variables for X n, yn, Z n
shall often omit the s u p e r s c r i p t ) ,
n-ary
(when irrelevant
and s e o o n d - o r d e r
relations
(species)
to the discussion we
quantifiers
V2' ~2
omit the subscript when the context makes it clear that second-order fiers are i n t e n d e d ) ,
The o n l y s e c o n d - o r d e r
(we
quanti-
terms o o n s i d e r e d a r e s p e o i e s
variables° 1.9o4.
Comprehension
A comprehension
(1) where
principles°
principle
zx n ~ l . o . X n [ A ( x l A
does not contain
is a schema of the form
. . . . . Xn) < ~ x n x l . . . X n ] , Xn
free.
We call the schema (i)
arithmetical
comprehension,
if
A
is a formula of
~
if
A
is a formula of
HAS ~o
(abbreviation :
ACA ) (ii)
predicative
comprehension,
bound species variables
(abbreviation : Pc~
)
not containing
69
(iii)
(full, or impredicative) not containing
Xn
comprehension,
if
A
is any formula of
~So,
free (abbreviation : CA )o
The system in the language of
HAS
based on the axioms,
rules and axiom
~ O
schemata of
I~
(but with respect to the extended language),
quantifier rules and axioms for second-order
quantifiers,
together with
is called
HAS
.
~ O
1.9.5.
Extensionality°
We denote by EKT
EXT
the axiom schema
Vxy[Ax & x=y-~Ay]
~Ve define
HAS
as
HAS ~
Note that in
+ CA + EXT. O
HAS,
EXT
may be replaced by the single axiom
vx I Vxy [ X l x & x = y - X ~ y ] 1.9.6. HAS
Theorem.
+ CA
If
then
First proof°
~H
~
o
is one of the systems
H+EXT
Let
°
is conservative
over
~o' H
HAS~o+ACA ,HAS o + P C A ,
Wor.t.
be a mapping of formulae of
formulae of
HAS
HA
into formulae of
~ O
HAS ~ O
A
, given by : of the form
~(A)
is obtained from
Xtl...t n
by
A
by replacing each sub-formula
~1...Xn(t1=x1&...
&tn=X n&Xxl...xn)
o
Then one readily verifies by induction on the length of deductions systems
H
of
for the
mentioned :
Second proof.
Let
¢
be a mapping of formulae of
HA~S° , which is defined as the relativization
HAS
into formulae of
to extensional
species,
i.e.
~[~nA] ~ ~n(Ext(Xn)-- ~[A]), , [ ~ A ] ~ ¢{n(~xt(Xn) &*[A]) and
¢[Ao B] ~ ¢[A] o 9[B]
,[(Qx)A]
~ (Qx)¢[A] Ext(xn)
for
for
o ~-,
Q ~ V I, ~I '
V, & ,
and where
~xt(X n)
~def Vx1°''XnYl . "Yn(Xnx1°°'Xn&X1=Yl . . . .
Then also, if all free second-order variables of
A
is defined by
&
Xn=Yn'Xny1"'°Yn ) "
are among
XI,.o.,X n ,
one proves by induction on the length of deductions ~+EXT
~ A(X I .... ,Xn) ~ ~ + E x t ( X l ) + o°. +Ext(Xn) ~ ¢[A(XI,.,.,Xn) ] .
The verification 1.9.7.
Lemma.
is quite straightforward If
H
and left to the reader.
is one of the systems
HAS ~ O
HAS
+EXT+PCA,
HAS
~ O
~
ed by restriction servative over
+EXT+CA,
and
H'
+EXT,
HAS
+EXT+ACA,
~ O
is the corresponding
system obtain-
O
of the predicate variables
H' .
to unary ones,
then
H
is con-
7o Proof.
n ~2' "°" ~o' VI'
Let
actual variables
be the list of species variables
in this case, not the syntactical
ables for variables)
with
We define a mapping
T
• (t=s)
n
arguments,
for all
(i.e. the
(= metamathematical)
vari-
n °
as follows.
~ t=so
""
o(n,i)(~ntj'''tn
) "
(~n was
T(Ao B) ~ (TA) o (TB)
for
o = ~, &, V o
• ((Qx)A)
for
Q = ~I' V~,
~ (Qx)~(A)
(QVj(n,k))T(A) This mapping transforms
for
each proof in
~
defined in
1o3o9 C.)
Q = ~2' V2" into a proof in
~' , with a con-
clusion which only differs in the naming of second-order variables. If we wish, we might also have kept the variables for one-argument in the proof unchanged by the translation, of
T
Let
(the modification m
species
by a slightly modified definition
depends on the proof under consideration).
be the maximum index
i
such that
V!
occurs in the given proof.
1
we then
v t,
put
T'((Q~k)A) The verification
VL (k,n)( n t1"''t)
~ (QVI+ =Jik,n) ~ ' ,,)
(A)
for
Q = V2' Z2 , n / J .
that the mapping has the property stated is quite straight-
forward ! we consider the only case which is not quite trivial, of the comprehension
schema for
n - argument
~v~ v~l...~n[A(x1 .... ,x n) ~ This translates
for
into
(under
i.e. instances
(n/ I) species :
~kXl...%].
T )
~E'~j(n,k) 1 Vx I . "°Xn[A(xl .... ,xn) @-~V j(n,k)(VnX1"''Xn I )]"
This follows from ~V 1
.n
I
.n
j(n,k)VZ[A(Jlz,.--,JnZ)
together with 1.9.8.
over
EXT.
Theorem.
Proof.
HAS ~ O
+ EXT + ACA
By the previous lemma, HA,
~--~Vj(n,k)Z]
where
H
is conservative
over
it suffices to prove
is the restriction of
HAS
H
HA. to be conservative
+ EXT + A C A
to unary species
variables. We further remark that we may restrict attention to instances taining
(at most) one free numerical
vxy
~x~w[A(~,y,~) ~->x~]
is a consequence
of
EXT
and
parameter,
since e.g.
of
ACA
con-
71
Let now any proof assume
~
w
in
H
of an arithmetical statement be given ; we may
to use finitely many instances of the comprehension schema, say Vy ~X I Vz[Ai(Y,Z ) ~
We define a predicate
X~z] ,
0
( Vyz Ai(Y,Z )
closed).
C(x,y,z) :
C(x,y,z) ~ (x=O&Ao(Y,Z)) V ( x = I & A I ( Y , Z ) ) Vo.o V . . . V(x=k&Ak(Y,Z)) Let
Vo, VI, v2, ...
be the numerical variables of
let
Vo, V1, V2, ...
be the (unary) species variables of
Let
m
be the maximum index
Now we define a mapping
i
such that
V. I
~,
and ~.
occurs free or bound in
w
a :
~(Ao~) ~ ( A ) o ~ ( B ) for o a((Qvn)A) ~ (Q~n)O(A) for Q
~ v, & , ~ , V 1 , ~I;
~(Vnt ) ~ C(JlVm+n+l, J2Vm+n+l ~ t) ~((Q2Vn)A)~ (QIVm+n+I)~(A) , Note that
~
where
Q ~ ~, V.
is the identity on arithmetical formulae ; a
preserves logical
inferences, axioms for equality and successor and induction (at least as far as they occur in
w )o
Now consider an instance of
ACA
occurring in
w ; it translates under
into Vy~Vn+m+ i Vz[Ai(Y,Z ) e-~ C(JlVn+m+1, J2Vm+n+1 , z)] . This is obviously derivable in that a
~(EXT)
transforms
1.9. 9.
is derivable in w
~, H~.
into a proof in
Formulation of
HASSS
with
since
Ai(Y,Z ) *-~C(i,y,z) ; also note
So, after intercalation of some steps, HA. ~- terms.
Instead of formulating second-order logic with a comprehension schema, it is sometimes more convenient to use a more general class of second-order terms. So
CA
is replaced by a rule of term formation : whenever
a formula of
HAS~o,
then
kx 1..oxnA(xl,...,xn)
A(Xl,..°,x n)
is
is a second-order term, with
the rule {Xx1...Xn.A(x I ..... x n) }(t I .... ,tn) ¢-~ A(t I .... ,tn) • PCA
is represented by a similar rule of term formation where
mitted to contain bound second-order quantifiers, etc.
A
is not per-
72
1.9.10-1.9o11.
Intuitionistic
1.9.10. D e s c r i p t i o n The system EL
EL
as d e s c r i b e d
of
analysis with v a r i a b l e s for sequences°
EL.
to be d e s c r i b e d b e l o w is a slight v a r i a n t in K r e i s e l - T r o e l s t r a
1970 , § 2.5.
For sequence v a r i a b l e s we use either case of
N - H ~ w)
~(~)
by the a d d i t i o n
and an a p p l i c a t i o n
operator
kx,
the term d e f i n i t i o n
and extending
(i)
function variables
(ii)
one-argument
(iii) if
~
as
I,
Z
I,
11 I ~
function
W
HA
I
(as in the
~, 8, Y, ....
~(E~L)
and quantifiers,
R , and a b s t r a c t i o n
operators
by
(i.e. terms for functions) ;
constants t
I,
of sequence v a r i a b l e s
of
are functers
V
are functors ;
a (numerical)
term,
then
Ap ~t
(abbreviated
~ot ) is a term ;
(iv)
if
t, t'
if
t[x]
are terms,
quantifiers,
%0 a functor,
is a term,
is n o w f o r m a l i z e d
a rule
kx.t[x]
Rt~0t'
is a t e r m ;
is a funotor.
by a d d i n g q u a n t i f i e r rules and axioms for f u n c t i o n
of
t-conversion
and d e f i n i n g axioms for REC
y
"Ap" , a r e c u r s o r
is a functor,
(v) EL
x I,
or we use g r e e k lower case letters
is obtained f r o m
of the system
k-CON
(kx.t)t'
= [x/t']t
R
f Rtq00 = t
\ Rt~0(St')
= ~j(Rt~0t', t')
and a q u a n t i f i e r - f r e e QF - AC
a x i o m of choice
Vx Zy A(x,y) -~ Z¢~ Vx A(x, ~x)
(A
quantifier-free) o
OO
EL
is e s s e n t i a l l y
definable
in
a subsystem of
N-~W
o
( kx
N-~
requires,
ice.
only as a syntactical
only deals with equality at lowest type, QF-AC
w,
intuitively
the primitives operator,
of
E~L
but since
are EL
this makes no difference°)
speaking,
%hat the u n i v e r s e
of f u n c t i o n s
CO
is closed u n d e r tension of recursive
"recursive
HA,
in"°
by i n t e r p r e t i n g
EL
is easily seen to be
all f u n c t i o n v a r i a b l e s
a
conservative
ex-
as r a n g i n g over total
functions.
Sometimes we can get by with a system of e l e m e n t a r y restricted
language,
such as
EL
in I ( r e i s e l - T r o e l s t r a
analysis with a 1970 ; EL
more
is a
~O
definitional
extension of such a system.
such systems by The system functions
H
in Howard and Kreisel
to be closed u n d e r
w e a k for the f o r m a l i z a t i o n which
we
need.
In the sequel we shall denote all
E~L o
"primitive
1966 only requires recursive
of the e l e m e n t a r y
in"o
the u n i v e r s e
of
This is somewhat too
theory of reeursive
functionals,
73
1.9.11. Some notations and conventions, ~0 = < > ,
= <~o, hence
=I . . . . , ~ ( x - 1 ) > ,
Ith(~x) = x .
~6 n ~def Vx< Ith(n) we put
(~=
(n)x) o
Furthermore
Jl ~ ~def Lx ° Jinx , J2 ~ ~def kx . J2~x, j(a,~) ~def Ix. j(~x, ~x) . (If we have not included k as a primitive,, jl ~, j2 ~, j(~, ~) can nevertheless be used in contexts like
A(JI~),
A(j2~ )
is taken as an abbreviation for the formula A(~) 8
every occurrence of
has been eliminated.)
Bt
by
jl(~t)
A
etc., where
A(JI~ )
etc.
obtained by replacing in
and repeating this process until
(~)x ~def kyo~j(x,y) .
1.9.12- 1.9.16. Formalization of elementary recursion theory in
E~Lo
1.9.12. We have to rely heavily on the development in Kleene 1969 . developments in Part I of Kleene 1969 can be carried out in QF-AC
includes all instances of Kleene's oo Kleene 1969, cf. footnote 7 in Kleene 1969).
X2oI~
E~L
The
(since
needed in Part I of
That we assume our coding of
sequences of natural numbers to be onto the natural numbers (contrary to Kleene's definition) is inessential.
(~I ~ ) ( x ) ~ y ~(S)~y
~de~ ~ ( ~
We define
rain [ ~ ( ~ . ~ z ) / O ~ ) ± l
~def ~(~ m i n z [ ~ ( ~ z ) / O ] ) °
~IB,
~(B)
by
~ y
I = y°
We may use the partially defined expressions constructed from terms and the partially defined application operations viations, similar to the use of
.!o , °(°)
p - terms (Cfo 1.5. IO ) constructed by
means of partial reeursive f~uction application° terms in this case also.
(~lS)(x)
~, ~ v
We shall speak of
p - functors.
and ~(~)
By virtue of (I), every
are partial recursive functionals of
p - term of type
0
and
{~I}(~, ~')
Io9.13. Coding of sequences and
We put
p-
respectively.
n - tupleso
~u(~1, .... %) = ~ . ~ u ( % ~
no, ~I
~, 8, x
and
such that
corresponds in a standard way
p - term in the sense of Kleene 1969, replacing
kx. {~o}(x , ~, ~')
We denote
....
~, ~ respectively, hence we can find certain numerals
to a
p-
If they are of type I (i.e. when they represent a
partial function) we may distinguish them as functors by
as systematic abbre-
..... ~x)
~I~' ,
~(~')
by
74
~i e =
~"
Furthermore we let
~zo
Ji ~x°
k~
= o,
(0 < i
k~(m.~)
be functions satisfying
o ~ i m * <Jin x> .
We abbreviate ~I(~I .... '~u ) ~def ~ 1 % ( ~ I ..... ~u )
(~I ..... ~U ) ~def ~(~u(~1 ..... ~u )) ° 1.9o14. Theorem.
(i) Let
(primitive) recursive
f
~[%,..°,~n]
..... %]
(ii) Set ~[~I ..... %] be a p-term tive) recursive f~ such that
Proof.
"'' ~")~
By K l e e n e
.....
then there is a (primi-
O~
1969 to prove
the existence
~ ~[~1 .....
%)
(i).
of a numeral
%](x)
By Kleene 1969, such that
.
1969, "34.1, *34.2 (page 69)
{~}(~,~I,.... %) ~e
of type
~[~I ..... %]"
41 we can prove
{~t(x,%
(T, U
"
~e use lemma 41 and § 4 of Kleene
Pc 67, lemma
p - functor ; then there is a
such that
m
f~l (~I ..... %) ~[~I
f$(~'"
be a
primitive
" ~ mi~yT(~
, ~, ~ly .... ,~y)
reoursive)o
put f 0=0 m f
(e.m)
= 0
It is easily verified that
if-,T(B
f
m
, x,
k lnm , . .
. ,knnm) o
satisfies our requirements.
(it) is proved similarly. 1.9o15o Theorem
(s- m - n
theorem analogue).
(i) There exists a primitive recursive function
A
of two arguments such
n
that (writing
~^n ~
for
(~A n ~1)I(~ 2 .....
^n(~,~) )
~n ) ~ ~1(~1 . . . . .
~n) .
(it) Similarly, there is a primitive recursive function
A~ n
such that
(~ ^~ ~I)(~n-~(% . . . . '~n )) ~ ~ ( % ( ~ 1 . . . . ' ~ ) ) " Proof.
(i)
By Kleene 1969, lemma 41, "34.1 there is a numeral
m
such that
75
(~I (81 ..... ~n))(x) ~ ~miny T(~, x, ~,~ly,..o,~j). me put (using 1.9.14 (i) implicitly) (,~^1~1)(o) = o (o~ ^ ~1) (~: * u) = y + I ~ (~A~I)(£*U)
~(lth(u)=y)
A
,",T(m., x , ~ ( l t h u ) ,
~l(lthu),
= 0
cases.
in all
other
n-I k 1 u .....
n-1 kn_lU )
(ii) Similarly. I o 9.16. T h e o r e m (i)
For each
(Recursion ~
there exists a
F o r each
Proof.
~
there
(i) Consider
~1(6,~1 .... Take
~ = cA
n
¢.
(~
~
such that
= ~1(~ . . . . ,~n) •
~l(~,~,'-,~n) (ii)
theorem analogue).
exists
a
~
such t h a t
~ l ( S A n 6' ~I ..... Yn ) "
''n)
~" '~I(6 A,, ~''I .....
There exists an
¢
such that
"n) "
Then
~)1(~t, . . . . 'q) =" st (¢,'I . . . . . %) ~1(¢^~
¢,~I ....
'~n) ~1(~'~I
.....
~n )°
(ii) Similarly. 1.9.17 . D e f i n i t i o n s If
t
is a
x 9 we put If
t
If
t
A°x,
p - term of type
in the p a r a m e t e r s
is a
p - functor,
0,
A°~, A1eo which is p r o v a b l y defined for all values of
of
Similarly
is to be a
AI~.t
t
0 , we take of
t
A1x.t
different
~o
Ao~.t
different
we take
in the parameters
According
A1x,
of type
AOx.t -=def kXot o
is a
reoursive
of
p-term
from
to be any
from
to be any ~,
~,
~,
q0~ primitive
such that
such that
to 1.9.14 and 1.9.15 we can always construct
primitive
such that
such
~.
reoursive
76
1.9.18.
S2stems of i n t u i t i o n i s t i c lawlike
sequence ;
For a universe
of lawlike
seem to be i n t u i t i v e l y the strongest
analysis based on the concept of a
IDBo sequences,
justified,
various forms of axioms of choice
notably
ACoo , but also
- v~[A~- ~[(~)o:~ Vx~((~)~, (~)s~)]] RDC I
ACel , and even
principle
implies
ACol , hence
ACoo ;
see Kreisel
"
and T r o e l s t r a
1970,
t h e o r e m 2.7.2. It follows from the results of G o o d m a n in fact conservative of this book ;
over
HA;
1968, and E,
that
E~L+ RDC I
is
the work of Goodman falls beyond the scope
the proofs are very long and the m e t h o d cannot be readily
fitted into the f r a m e w o r k of the rest of this book. However, pretation, Kreisel
it is not hard to establish, that
E L+RDC I
and T r o e l s t r a
the c o n s i s t e n c y
is consistent
1970 , 3.7).
(of. § 3.2,
To obtain a p r o o f - t h e o r e t i c
strengthening,
i n t r o d u c e d by a g e n e r a l i z e d
example here b e i n g the theories
IDB
is obtained by adding a constant
functions) KI.
to ~
and if
also e s t a b l i s h e s
Church's
thesis state-
involving natural
we have to add a constant for
inductive and K
d e f i n i t i o n ; the principal
ID~I . (for a u n a r y predicate
Q
of
EL , together with two axioms and a schema = kn. Sx ~ K ~
A K ( Q , ~ ) ~def Z y ( ~ = kx. S y ) V ( a O = O & V x Q ( k n . ~ ( ~ * n ) ) )
for all El, K2
1.11.7).
are reduced to statements
inter-
(Cfo 5.6.~6, and
only.
a species
ID~B
~
axiom for systems with f u n c t i o n v a r i a b l e s ;
ments i n v o l v i n g functions numbers
relative
The same i n t e r p r e t a t i o n
of Church's thesis
acts as a r e d u c i b i l i t y
by means of a r e a l i z a b i l i t y
in the language
of
we put
IDB
may be combined into
AX(X,~) -- x ~ IDDBBI
may be defined as
ID~B+ ACol o
(ID~B I
in Kreisel and T r o e l s t r a
here ;
!DB
corresponds
to
1970 is an expansion of
IDB
of K r e i s e l - T r o e l s t r a
ID~BI
as defined
1970. )
v--~- O
The axioms
E l - K3
are in fact equivalent
schema (of. K r e i s e l - T r o e l s t r a
1970 , 3.2.1):
to the f o l l o w i n g a x i o m and
77
(~)
~a
for all
(2)
Q
[Vn(~n/O--~n) ~ V n ( V y ~ ( . . #) ~ n )
-
of the language,
Ka & a n / O
~
~o]
and
~ V~(an= ~ ( n . m ) )
.
(I) is called the principle of induction over unsecured 1.9.19.
Systems of intuitionistic
sequences.
analysis based on a concept of
choice sequence. As already remarked in the introduction
to this section, universes e~fls
of
io na l
choice sequences are supposed to be such that an-~operator of type 2, defined on the whole universe
should be continuous,
i.e. satisfy
~Vithout introducing higher type objects in the language, of expressing
the eimples~ way
this continuity property is by the schema (weak con[i,uity
schema)
wc-~
V ~ X x A ( ~ . x ) -. v~x:: xz v13 ~ ~ : A ( ~ , y )
o
This principle may be conceived as being obtained by combination of the above-mentioned
continuity property for type-2 operators with the following
"selection principle"
(axiom of choice) which is itself not expressible
in
,:(~) : -*
V~A(~,x)
:~
v~A(~, ~ )
A stronger axiom of continuity
C-N
V~XxA(a,x)
-
C-N
.
is expressed as follows
~:~[::o~& v n ( ~ n / o -. V ~ n A(~, ~n "--I))
where Ko(a') ~-def Vnm(~:r~/ 0 ~ ~ a = ~(n*m)) & Vt~ ~Lx:(~'(~x) / 0 ) (C-N C-N
corresponds
to the strong continuity
.
of Howard and Kreisel
1966).
expresses that there is a modulus-of-con[i~luity functional ; W C - N
only expresses local continuity. 1.9o20o Bar induction. The schema of bar induction, 1966 , appears in various forms.
(1)
v~P~
(2)
Vnm(Pn ~ P ( n * m))
(3) (4)
Vn(Fn V ~ P n ) Vn(Pn -- On)
(5)
Vn(VyQ(n.~)
discussed
in Howard and Kreisel
-- Qn) o
Then bar induction with the monotonicity ~S
extensively
~e list some formulae first :
condition,
Bi M,
can be expressed
and bar induction with the decidability condition, BID
as
The weakest version is
(1) & (4) & (5) QO (with I' quantifier-free). B=QF It is shown, in Howard and Kreisel 1966 (Remark 4, page 337) that B I M +
can
be strengthened to
Similarly, if & is supposed to be monotone (i.e. then BID or BIQP may also be strengthened: (1) & ( 3 ) Bc (4) & (5) & km(Qnd&(n*m)) (1) &
(4) & (5)
& Vnm(Qn-.B(n*m))
tfnm(Qn
4
%Qn
+
VnQn
-+
&(n*m))
),
quantifiex free).
(for P
It should be noted that (~owardand Kreisel 1966, theorem 8 ~ ) EL
+
ACol
+ WC - N +
BIM
t
C-N
and also (~owardand Kreisel 1966, theorem 8E)
l.9.21,
Extended bar induction.
The schemaof extended bar induction is described below.
Let
R
be any
unary predicate of functions; then we put
contains all codings of finite sequences of elements of R. denote the coding of a sequence $o,u.,Bx-q by '
R"
Let us
' is a sequence of rp such that ( T ) ~ = Az.x , ( ( P ) ~ =+ By ~ for y < x , (T) =Xz.O for y > x . Y * denotes concatenation; we abbreviate <'@* as < is h , O . Now the schema of extended bar induction EBID is given by
. >'
where
79
(1) (~) (~)
~ V!~ ~ ~" (P~ v ~1~) Vl~ ~ ~ " (P~ - ~ )
(4)
~1~~ ~{* ~ ( ~ $ x )
(5)
v~
~
(v~
R ~(~%)
- ~)
.
1.9.22. Theorem.
E~ c + pc I ~ ~BI~ where
DC I
DC I
is the schema Vc~A(%~)
Proof.
-~ [?[(y)o:~&VzA((~)z,
We first show that
(a form of
DC I
(~)Sz)] °
implies the following more general
RDC I )
To see this, assume
v~
s A(~,~) o
s ~
Let
Then obviously
(using classical
logicS)
v~ ~ A*(~,~) Let
~E S ; by
DC I , there is a
?
such that
[(~)o = ~ ~ vzA*((~)z'(~)s~)] Then we prove by induction on
°
z
v~[(~)~ ~ s ~ A ( ( ~ ) ~ , ( y ) S ~ ) ] . Thus
(6) follows.
Now apply
(6), with
s - ~
~sume
&~q~,
A(~,~) ~ ~
(~)-(~),~(<>~),
Obviously,
if
R~
Hence by c l a s s i c a l
&-~q~,
logic
((~)-(~)
~S~s
as in~.~.2~).
contraposition
Z~ R(-~Q(~.~))
Thus
v~s
~(~ : ~. %)
A(~,~)
of
(5)
yields
and t h e r e f o r e
also
schema
80
By (6) there is a
(~)o
~
such that
< >~ ~ V z ( ( ~ ) ~ s ~ v~A((y)~,(~)Sz)) ,
~
hence, by induction on (7)
z
Vz(RU(7)z & Ith(y)z = z & ZgE R ((Y)Sz= (Y)z* 9) & ~ Q ( V ) z ) "
NOW (7) implies
(8)
(Y)s~ = (~)~ *<((~)Sz)Sz > ~ ((~)Sz)S~
We wish to construct
6
(~)z = ((~)Sz)Sz ' ioeo Now we see from (7) that for all
z
such that
VzmQ(~z)
implies by (3)
1.9.23. Remark.
= (~)
Z
~
R°
This is achieved by taking
Z
6 = Xx((7)sjlx)Si~(j~x
(by (8)), therefore
Vz~Q(Zz)
~
).
; but on the other hand,
8~ R*o
Vz(~P(~z))
This contradicts
(Luckhardt
(4), since
o
Attention has been draw~q to the schema
work of Luckhardt
((?)Sz)Sz E R
1973, and Scarpe!lini
EBI D
1972 A).
by recent They showed
how to construct models for the theory of bar recurslon of higher type which could be shown to be models in a theory corresponding show how
EBI D
tinuous functionals (1 9 10).
EL + EBI D .
are a model for the theory of bar-recursive
By the preceding theorem,
bar-recursive
to
can be applied to show that the so-called extensional
We con-
functionals
this also gives a modelling of the
functionals which can be shown to be a model in
E~Lc + DC I ;
but this is already explicitly in the literature,
e.g. Kreisel 1968 , pp.146-147°
1.9.24. Fan theorem.
in its simplest form may
The so-called
"fan-theorem"
be stated as follows : FAN
V~6 B Z z A ( ~ , x ) -- Zz V~E B Zy V~6 B ( ~ ' z = ~ z ~ A ( ~ , y ) )
Here "~
~6 B
is an abbreviation for
V x ( ~ x ~ I)
( ~6 B
.
may be read as:
belongs to the binary spread"). Kleene's *27.7 in Kleene and Vesley
1965, page 75 is a generalization
FAN ; it is shown there that ~
+ wc-N + BI b b F A
by first showing that EL~..+ BI D ~ FAN where FAN '
( FAN,
Vn(Rn v-.an) & V ~
corresponds
~mR(~)
to Kleene's "26a)~
-~ ~
V~E B ~ _ < z
R(~)
of
81
Here we have restricted our attentions (i.e. satisfying are equivalent
~6 B ) ;
to similar principles
Let us denote by
where
~6 S
For every
implies
if
~
S~S'
~[S'] = S , FAN
belongs
to the
and
FAN'
spreads.
~
fan $ .
~
such that
8 6 S ~ Vx(~x!ax) o
is itself a fan.
for two fans
~=
w.r.t.
FAN
the more general principle
{~I Vx(~xi=)}
Furthermore, such that
FAN*
abbreviates :
S
in the binary spread
for arbitrary finitary
fan S , we can find a function
The species
to functions
but it is not hard to show that
for
S, S'
~6 So
we can find a projection
Therefore
S , as is seen by applying
FAN*
FAN*
w.~.t.
w.r.t.
S'
S' to
v ~ s, ~ A ( ~ , y ) . The following mapping transforms a sequence of
O's
~ where
Ix
and
O, I ~2+I,
I aO+1, O, I ~I+I
stands for
every function of natural numbers into
1's , ioeo an element of the binary spread :
I, I, .o.,
I
(x
times).
It is easy to see this mapping is bi-unique, fan of the binary
spread.
Thus we may derive
1.9.25- 1.9.27~ Extensions 1.9.25. Extensions
A first e x a ~ e
FAN*
from
N-HAWo
inductive
is the theory
definitions ; IDBWo ~2
the natural numbers,
with objects of finite type over three trees of the first class, and trees of
the second class (trees of trees of the first class). ed at length in chapter VI° obtained from
T2
equivalent
is obtained by extending 0
extending
and
K
T2
to
~-HA ~
TI
IDB
as regards proof-theoretic
and the Brouwer-operations)
in the same manner as
A first example of such an extension appears in Howard reworked version in Howard In Troelstra
strength,
~-HA ~
thereby
extends
H~A.
1963 , a completely
1972.
197~ systems
~-ID~5 w,
E - IDB w,
W E - IDB~
We briefly outline these systems by first introducing N - H A~ pw .
~I , the theory
to trees of the second class.
to a theory of finite types over two basic
(the natural numbers
(a variant of)
This theory is discuss-
contains as a sub-theory
by deleting all reference
Another example,
types
FAN .
%o theqries in all finite types with sets introduced
b~,, ~eneralized
basic types:
of
and transforms a fan into a sub-
The type-structure
are described.
N - IDB w
is now extended to a structure
~K:
similar to
82
There are constants D ,T, Da,T,' D"~,T,
0, S, H
,
Furthermore, there is equality a 6 TK° I
Zp,a,T, R
and moreover constants =a
as before, pairing constants 9 I, 92' ~3' ~ '
I(p,~,T 6 TK) .
as a primitive constant for each type
The axioms and rules contain the axioms and rules of
is an "injection"-functional
from
K
into
(0)0,
916 (O)K, ~26 (K)(0)K ; the connection between
I(~1~)y
= s~
is given by:
(x,y6 0, e6 K) 0
e, f, e ~, e", el, .o., f' , f" , .o.
@36 ((0)K)K
I, 41, ~2
N - H~p,
I6 (K)(0)0o
(x,y~o)
I(92ex)y = I e ( < x > . y ) (We use
so
for
K-variables.)
is a "sup"-operator for sequences of elements of
K
and satis-
fies l(~3Y)0 = O,
l ( @ 3 y ) ( ~ * n ) = l(yz)n
(y6 (0)K, z,n6 0) .
&
~
is~constant for definition by recursion over Is0 = Su-~ ~ e x y le0 = 0
and
Z's
= xu,
(o)~, y~ ((o)~)(K)o) o
kv° Y (~2ev)xy
H,s
with axioms
-~ ~ exy = y(lv. Y~(@2ev)xy)e
(u~ o, ~ Here
K,
is assumed to be s~ntactically defined in terms of
(cf.1.6°a).
This completes the description of
N - IDB w .
E - IDB ~
is then obtained by
requiring x K = yK~r-9 Vz°(Ixz = Iyz) and also
hereditary extensionality, ice°
~(~)~ ~ y(~)~ In
WE-IDB
~
If where
P
~-~ w ~ ( ~
= y~).
extensionality is weakened to
b~-t~s,
then
bP~F[t]:F[~],
is a propositional equation between terms of type
0,
and
t,s E ~ ° I-IDB ~ 6 TK
is obtained by adding to
.~N-~D S
a constant
E
for each type
with axioms E xy=0 x=y
V E~Xy= I
~-~E x y = 0
,
thereby making equality decidable for all types. There is still an intermediate type of theory possible, where equality is neither extensional nor intensional (but also not neutral). dicate these extensions of We only add %o
N-HA w
N-~w,
N - ID~B~
respo by
Let us in-
InN.~-HA w,
In~ _ IDBWo
S3
x
I
F o r a model 1.9.26o
~-~ Vz°(xz
Theories
w i t h bar
on (essentially) B~
been first
introduced
For the most
since
the
of h i g h e r
type ; N - H A
+BR.
type are e x t e n s i o n s
structure,
constant
in S p e c t o r
1962.
w i t h a n e w schema
for type
(The most
1968 , G i r a r d
N - ~ PHA °°
of
~ ).
important
1972.
for c e r t a i n
Such s. theory further
Furthermore
has
refer-
Scarpellini
1973. )
of bar r e c u r s i o n
such
in § 2.6.
of h i g h e r
same type
convenient
tion of a type
e.g°
recursion
1968 , Kreisel
Luckhardt
theory
ICF
(the b a r - r e c u r s i o n
are H o w a r d
1972 A,
o
see
of bar r e c u r s i o n
constants
ences
= yz)
of such a theory,
Theories based
I
= y
~
formulation,
as an e x t e n s i o n
of finite
sequences
assume of
sequences
may be i d e n t i f i e d
that we w i s h
N-HA w .
of objects with
We note
of type
special
to describe that
a
the
the addi-
is an expansion,
sequences
of type
(O)a,
as follows. Let the natural
numbers
n O = n, Then
<x
n(~)~
, ooo, x _I > z(O)%
n
be ceded
= kx ~ . nT,
may be coded
= u
,
z(°)°(i+
into h i g h e r
n XT =
as
Dn
z
n T
types
as
n
as fellows:
.
with
I) = x ~
for
i
I
z(°)~i N o w let
c
of type
natural
numbers,
[o]
a,
[el(i) = u i
for
jyEc]<
see,
being
continuous
z )o
If
y[c] < Ith(c),
of type
B
then
ranging
over finite
same n o t a t i o n s
for
for
if
sequences
as for
(u £ ~) ,
(0)~ , w h e r e
[e](i) o o ~
constant
of
sequences
of
Ith(o) o
o = <Us, .oo, Ux_1> ,
i~x.
satisfies
that
(i.e.
B
defines
yz , z 6 (0)~
B yzuc
is r e d u c e d c
- B y~uc o u ( ~ v . ~ y ~ u c ( c . ~ ) ) c .
depending
;
to the c o m p u t a t i o n
such that
of continuity),
eases w i t h
y[e]
is the set of axioms
a functional,
is d e t e r m i n e d
consequence
BR
, the
lib(o) - B y~ue ° ~c
of
If the set of
~u
c I . c2 , d
i<x,
intuitively,
y[c] ~ I t h ( c )
of type
a sequence
[y[c]ilth(c) To
i > u °
and let us adopt
such as
denote
The b a r - r e e u r s i o n
~R
for
be a v a r i a b l e
objects
Let
= 0 °"
y[c]~lth(c)
for all
on a finite
the c o m p u t a t i o n of
Bcyzu(c*V)
is w e l l - f o u n d e d ,
the c o m p u t a t i o n
BR o
we must
c
think
initial
of
y
for all
for v6 ~.
(classically reduced
structure.
as
segment
B yzuc
will be e v e n t u a l l y
of our type
of
to
a
84
1.9.27. Girard's theor 7 qf functionalso This theory is introduced to be able to give a direct Dialectica interpretation of the theory of species (i.e. not via a theory with variables for functions 9 as in S~ector 1962 (Girard 1971 , Girard 1972). The type structure
(i)
$S
is defined by
0~s
(ii)
~, ~, ~', ~ ,
..o
belong to
~S
(4, ~, ~', B ~ . . . .
are called vari-
able types)
(iv)
if
~[~] 6 ~S'
then
V~.o[~],
The functional constants contain
S~.~[~] E ~ S "
S~ E
T, g p , ~ m ' R , Do,m' o,~'D' D"G,T, OO
for all6Tp 6 ~S ' and two injection functionals
Ivan, T
and
I~,T
~ satis-
fying
Finally, there are two operators
DT, ST
(net functionals with a type, but
corresponding to schemata for introducing new funetionals). Let type
%6 ~ ~
be a term, not containing free variables containing in their
free°
Then
DT~t
is a %erm of type
V~°~
with axioms
I V ~ [ @ ] , T ( D T ~ t ~[~]) = t°[ T ] Let
% 6 (o[~])T ,
~
of a variable free in ST ~ t
not
occurring
free
in
m , and not occurring in a type
t ; then
~ (~[~])~
,
with the axiom ST ~ t ( l ~ [ ~ ] , p
s°[d) = t(~[~])~s°[~]
.
Also
O(O)T x
0
~: OT,
Do,T OoOT
= 0o' XT
ST ~t(o~.~[~]) = t(oo[~])
(t6 ( ~ [ d )
p) o
For this theory also intensional and extensional versions are possible. example9 we may add the equality functionals
E
as in
I - }L~~.
For
85
§ 10.
Relations between classical and intuitionistic systems: = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = translation
into the negative fragment.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
1.10.1.
Contents of the section.
For classical
predicate
logic and arith-
metic there exist a number of mappings into the "negative" corresponding
intuitionistic
these translations
systems in the literature ; t h e definition of
can be readily extended to higher order languages.
survey is given in Luckhardt Gentzen
1933 , Kuroda
HA,
~
(i)
Hc ~ ~A ~-~ A
(ii)
He ~
A ~
(iii) For all
let
and let
All translations
197~, chapter III.
1951 ; cf. also Kleene
For definiteness, logic or
fragment of the
~ Hc
1933,
(many-sorted)
predicate
be obtained by addition of the excluded third. properties :
for all
A~ Fm(~)
H b <0A , for all
A ~ Fm(~)
there exists a
negated prime formulae by means of As remarked in Luckhardt
A
are Godel
1952, § 81.
denote intuitionistic
have the following
A E Fm(H)
References
B , constructed V, &, ~, ~
1973~ all these translations
sense that, for any two translations
~, ~'
from doubly
s~
~ ~ ~A ~--~B.
are equivalent,
satisfying
in the
(i) - (iii)
~ ~ ~A *--> ~'A . Below we primarily discuss 1.10.2.
Definition of the mapping'.
higher-order)
language,
of variables), translation")
&, V, ~, A
P' ~ ~ ~ P
(ii)
(A&B)'
(iii)
(A-- B)' ~ A'-- B' (VxA)'
(A vB), ~ ~ ( ~ A , ~
(vi)
(ZxA)'
m VxA' ,
Then we define
P ;
for variables
be any many-sorted
~
V, S
(first-or (for any sort
the mapping ~ (the "negative as fellows :
~ A.
X
of all sorts
x
of all sorts.
B')
~ ~ Vx ~ A' , for variables
Remarks.
In a system
~
with a language
HA, H~Am, [ - ~ )
clause
resulting translation given by (i) - (vi). HAS , we may use as
.
for prime formulae
(iv)
(e.g.
~
~ A'~B'
(v)
1.10.3.
Let
based on the logical primitives
by induction on the formula complexity
(i)
(i)
the variant due to Gentzen.
t= s
in
(ii) If we use
where prime formulae are decidable
(i) may be simplified
to
P' ~ P;
is then obviously logically equivalent
and
P' ~ P
the
to the one
In systems with two types of prime formulae,
P' ~ P
HA~S)
~
such as
for the prime formulae which are decidable p, m ~ ~ p
(such
for the other prime formulae.
for prime formulae,
we have
A" ~ A' ; and we always
8E
have
A" *-~A'
in intuitionistic predicate logic.
We may also let the
treatment of prime formulae depend on the context : if they appear unnegated in
A
then
we put a double negation in front, otherwise we do not change them ; A
is further defined by (ii) - (vi).
1.10.4.
Convention.
that
is defined by 1.10.2,
'
Then also
We shall give our proofs below under the assumption (i) - (vi).
W e sometimes find it convenient, however, prime formulae 1.10.5.
P
A" m A' .
P' ~ P
to assume
for decidable
(cf. our remark ~.I0.3 (ii) above).
Definitions.
(a) We define the strictly positive parts (s.p.p.) of
A , for
A
in a
given language, inductively as follows: (i)
A
(ii)
If
is a s.p.p, of B&C
or
(iii) If
B ~C
is a s.p.p, of
(iv)
VxBx,
If
A ;
BVC
SxBx
are s.p.p, of
A , then so are
A , then so is
is a s.p.p, of
A,
B, C
C
then so is
Bt
for any term
(b) A Harrop formula is a formula which does not c o n t a i n a s.p.p, with or
S
V
as a principal operator.
Alternatively,
the class of Harrop formulae
ly by the clauses (i) prime formulae belong to (ii)
A,BE A =A&B
E A,
(iv)
B ~ A
E A.
1.10.6. A
t.
=A--B
Definition.
(iii)
A
can be defined inductive-
A,
A ~ A = VxAE A,
In any language
~
(as intended in 1.10.2), a formula
is said to be negative, if it is constructed from negated prime formulae
by means of
V, &, ~, ~
.
(In systems with decidable categories of prime
formulae we shall assume that a negative formula may also contain unnegated prime formulae out of the decidable categories.) 1.10.7.
Remark.
In
~
there are Harrop formulae which are not provably
equivalent to a negative formula. SyT(x,x,y)]
(see
3 ~ 2 ) •
An example is But conversely,
m Vx[~ ~yT(x,x,y) every negative formula is
a Harrop formula. 1.10.8.
Lemma.
Let
H
be a formal system based on many-sorted intuition-
istic predicate logic, and let
A
be a Harrop formula constructed from
decidable or doubly negated prime formulae. ~A~-* Proof. (i)
Then
~A.
By induction on the complexity of
A .
The assertion holds for double negations of prime formulae and
87
decidable
prime formulae. -7 m /k *-e /k .
(ii)
A*-e
If
mmA,
B(--e t o m B ,
then
(A&B)
e-~(mmA&m-TB)
4"-e(A&B)
(1.1.8, v) (iii) If
Ax ~
m max,
then
V x A x -~ m - 7 V x A x -~ Vx m m A x - * (iv)
If
m -7 B ~ B
,
VxAx.
then
m re(A-*B) ~-~ (A-*m roB) ~-e (A-*B) 1.10.9. sorted)
Lemma.
Let
predicate
H
logic ; let
Vx -7 m A x
DNS
be a formal
For
A
system b a s e d on i n t u i t i o n i s t i c
be o b t a i n e d from
H
(many-
by a d d i n g the schema
-* ~ m V x A x
(for all sorts of v a r i a b l e s (i)
H'
(1.1.8, IV).
x).
not c o n t a i n i n g
Then
V :
H ~ -n -7 A ~--~ A, (ii)
For all
A
H'~mmA
~-eA' .
(iii) If all subformulae B e-e m -7 B )
in
H~mmA
of
H,
A
are stable
this l e m m a is used in 1.11.4.) (i),
(a) Let
mm
(ii),
(iii) can be proved s i m u l t a n e o u s l y
A.
We consider
B x e - e (Bx), .
1.10.10.
Let
e-* Vx -~-~ B x e - e V x B ' x
Lemma.
Let
H
by i n d u c t i o n on the
cases :
~-~ ( V x B x ) ,
language,
of the excluded
ZxBx.
m m Bx~--~ (Bx)'
denote m a n y - s o r t e d
for a first- or h i g h e r - o r d e r of the principle
(i)
two typical
(~xBx), <--~ m Vx m (Bx), ¢ e I V x -i B x ~ - - ~ m m
(b) (For (il) or (iii) only.) m m VxBx
stable if
~-*A' .
(Note :
of
is called
then
Proof.
complexity
(B
intuitionistic
and let
third.
Hc
predicate
logic
be obtained by a d d i t i o n
Then
HC b A ~-eA' HCbA ~ i b A '
(ii) Proof.
(i) is obvious.
(ii) f r o m left to right can be proved by i n d u c t i o n
on the length of d e r i v a t i o n s is trivial.)
in
Hc .
(The i m p l i c a t i o n
We take for example Codel's
fication. (a) B a s i s : If
~c ~ A ,
A m BVB-*
B,
A
is an axiom.
then
from right to left
system as a basis for our veri-
A' - m (m B , & m
B,) -* B'
88
which
is equivalent
lemma
1.10.8
If since
in
then
roB, & m C ' - ~ m B '
,
VxmB'x-~mB't
,
axiom
Induction
length
k + I.
completely Assume
A'
step.
of l e n g t h
rule applied
hypothesis,
Now
The
in
H
I)
obviously
A,
suppose
various
cases
to be a p p l i e d
rule
is
since
holds
PL2,
if
in
H e ~A
H e ~A
cases
PL15,
A'
H.
PL3,
by a de-
by a d e d u c t i o n
according
so
PLT,
of
to the final
PL8,
and QI
A ~ C V B 1 -~ C V B 2 ,
hence
H~
B'x-~C' .
H~-~C'-~mB'x,
~c,-c,,
so
Theorem.
follows
are
and
Let
from
Since
m C ' &-~B~ -4 m C '
by c o n t r a p o s i t i o n
to be applied
again,
E - HA ~ ,
.
mB~
mC, &~B~ -- mB~
By c o n t r a p o s i t i o n
,
f r o m our i n d u c t i o n
~ C ' &mB~ -- mC' &mB~ . to be
Q4,
so
A = XxBx -~ C .
A' -= m V x m B ' x ' 4 C so w i t h
H ~VxmB'x-~m
QI
mC'
By
'
H ~ ~C'-~VxmB'x. By lemma
1.10.8
and r e m a r k
ffb(~sx'C)' H
be one of the
HAS • P C ~ p H A S .
Then,
systems if
Hc
HA, denotes
N-HA m ,
HA m
the c o r r e s p o n d i n g
system, He ~ A +-->A '
(ii) H c ~ A Proof.
of
etc.
for any f o r m u l a
to d i s t i n g u i s h
mB~-*
hypothesis,
(i)
which
by contraposition.
and
By contraposition,
classical
this is d e r i v a b l e
Then as before,
,
H ~A'
-4 re(me' & m B ~ )
the last
1.10.11.
because
hypothesis
-4 "~C' & ~ B ~
I - HA ~ ~ ,
H
less difficult.
that,
then
in the derivation.
mC~ &mB~-~-~B~
1.1o.7
are even
Assume
_< k ,
We have
A' -= m ( m C ' &-~B~)
induction
in
trivial.
by i n d u c t i o n
Assume
;
(1.1.8,
-~ m B t ' - ' m V x m B ' x
=- m ( ~ B ' &-~-~B')
the last rule
mC' &mB~
A' ~ B ' - ~ m ( m B ' & m C ' )
A' ~ B ' t - ~ - ~ V x m B ' x .
schemata
duction
This holds
so by c c n t r a p o s i t i o n
by c o n t r a p o s i t i o n
A -= B V m B
The other (b)
,
m - u B' -" B' .
etc.
A = Bt-~ZxBx,
If
to
1.10.7.
A ~ B-4 B V C ,
m roB, - * m ( m B ' & m C ' ) If
H
and r e m a r k
We have
additional
= only
axioms
The e q u a l i t y
H~A'
.
to add to the proof of 1.10.10
a discussion
of the
and rules.
axioms
are trivial
to deal with,
as are the d e f i n i n g
axioms
for the constants. Further that
H~
we have A' ;
In the case of
to verify,
for any instance
A
of the i n d u c t i o n
schema,
this is obvious. HAS+ P(~o~HAS
finally,
we have
to check
that f o r ~ i n s t a n c e
s9
A
of
WCA,
resp.
CA , ~
A'
Let e.g. A ~ ~xnvx1"''Xn [B(x1"''Xn ) ~-~xnx1"''Xn ] " We have to show (I)
~
m v x n m v x 1 . . . x n [B,(x1...Xn)
We note that in
H
for some
e--~m mxnx1...Xn ] .
yn
VXl...Xn[B'(x1...Xn)
e-->ynxl...Xn].
Hence also n
vx1...xn ~ ~ [B,(~l...x n) ,~y Xl...xn], and thus using
~ m (D I&D2) *---~m m Dl & m mDz, aria
~ ( D I-D2) e - ~ ( ~ D
I-~D2)
VXl...Xn[m mB'(x I and with 1.10.8,
. . .
(1.1.8, v, iv), Xn)<--em m y n x I
. . .
Xn]
1.10.7
Vx1...Xn[B,(x I ... Xn) ¢--~ m mYnx I ... Xn] . Hence (I) follows,
by
~IYn D ( Y ) ~ V Y
1.10.12.
Corollar 2.
Then
is conservative
Hc
1.1o.13.
Let
~ over
nmD(Y) .
be one of the systems indicated in 1.10.11. H
Some further information
w.r.t, negative formulae. is given in Kreisel
1962C,where
also
lemma 1.10.9 (ii) is stated (Theorem I, Corollary). The concept of a Harrop formula (ice° a formula without strictly positive occurrences of V, Z) appears first in Harrop
196Oo
9o
§ 11.
General discussion of various
schemata and proof-theoretic
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
closure conditions.
1olloio
Introduction°
Certain schemata and rules will appear frequently in
the statements of metamathematical results in the sequel.
In this section
we briefly discuss the principal ones. Let us define a rule as a set of
(n+1) - tuples of formulae ; an element of
this set is an instance of the rule. of a rule,
F1,...,F n
If
are called the premisses,
Fn+ I
A rule is said to be a derived rule for a system (Fo,...,Fn>
~,
is an instance
the conclusion°
if for each instance
of the rule,
~ + r h F o ..... ~+r~rn_ I ~ + r ~ F n o A rule is said to be an admissible rule for a system from null assumptions") if for any instance
H
(or:
"derivable
of the rule
PFo ..... ~ffFn_1 ~ F n . Admissible and derivable rules are instances of proof-theoretic closure conditions of a rather crude kind : they only involve the set of provable theorems.
By a study of normalization for systems of natural deduction,
one
can obtain more delicate proof-theoretic closure conditions involving the deductions themselves. For reference in the discussion below, we now briefly recapitulate the intended interpretation of the intuitionistic logical constants. I°)
A proof of
A&B
consists of a proof of
A
and a proof of
B ;
2° )
A proof of
AVB
consists of a proof of
A
or
B ;
3°)
A proof of
VxA
consists of a construction
proof
c
of the fact that
yields a proof of
A proof of
5° )
A proof of
is in the domain of the variable
A d , together with a proof
~xA
consists of a
Ac , and a proof that
proof that
of
which, applied to a
~'
x,
of this property
H°
4°)
of
Hc
d
a proof of
~
c H
Intuitively,
A--B of
A
c
c
in the domain of
belongs to the domain of
consists of a construction into a proof
nc
of
B
E
x
and a proof x °
which transforms auy
(together with a proof
E'
satisfies this condition). the predicate
HA(C )
("c
proves
A ")
is assumed to be de-
cidable. For a more detailed discussion of this interpretation, Troelstra 1969~ @i.
see Kreisel
1965,
91
1.11.2.
Disjunction
and explicit definability property.
Nearly all intuitionistic satisfy the so-called
ED
ff
~xAx
formal systems discussed in this monograph
"explicit definability
= ~n
(~An)
(~xAx
(x
a numerical variable).
In virtue of
implies the disjunction
property
ffA~ffA
or
fib
~(~x=0--A) & (x/O~B)) <---*A VB,
(AvB
closed).
With respect to other sorts of variables, of
ED
~ gxAx x
= ~t(~At)
(~xAx
is now any sort of variable,
elements of the range of the variable ED, ED', DP (constructive DP,
we often e n c o u ~ e r
a generalization
of the form
ED' where
(ED)
closed)
ED
~P
property"
closed) and
t
x )
ranges over terms (definable
of the same sort as
x.
have often been presented as criteria for the "constructivity" character)
of a formal
system.
Of course,
if we consider e.g.
there is a property of the class of informal proofs which parallels
DP ; a proof of establishing
A vB
DP
should contain either a proof of
fies a similar closure condition° as we stated
(Similar,
DP , the formal proof of
existence follows from the existence be contained as "sub-proof"
A
B ;
but not the same condition :
of a formal proof of
in the proof of
B
whose
A V B , need not
A V B o For a closure condition the condition
on informal
we must establish more J)
Also,
ED, ED', DP
structive character"
are neither
of
HA
(so
sufficient nor necessary
of the system studied,
ly the intended interpretation, HA"
or a proof of
or the formal proof of
on formal proofs which more closely resembles proofs,
A
means that the set of formal proofs of the system satis-
HA' U HA"
since there are divergent extensions
is inconsistent)
and on the other hand, there are systems intuitionistically constants~ systems,
acceptable,
HA+N,
which both possess
~, H A ~
H A + CT
DP.
HA,,
ED, DP ;
, which are obviously
on the intended interpretation
but which do not possess
one may take
for the "con-
i.e. they do not enforce unique-
of the logical
As an example of the diverging
+ IP
(cf. 3.7.4
(i), 3.7.4 (ii),
3.2. ZT). We present an example of an intuitionistieally does not satisfy Let
ED
(from Troelstra
~
justified
system which
; the example is due to Kreisel)°
Proof ~ ProofHA , and Ax ~def Pr°°f(x'rO = I~) V V y m P r o o f ( y , r O = 11) .
Since
}~
is intuitionistically
consistent
(on the intended interpretation),
9~
Vy~Proof(y,~O
for any n u m e r a l
~
is i n t u i t i o n i s t i c a l l y
= 17)
note that because
VyTProof(y,~O
of [.
TmAx .
Further
~ ~Proof(~,rO=17)
Therefore
~ ~-~Vy~roef(y,~O
for any numeral
true, hence also
= I~) , we must have
:
~)
~ o
Also
W~Ax Now assume
~
[~y Proof(y,'-o = I~) v V y ~ P r o o f ( y , %
~ ~ x A x -~ A ~ ,
then it would follows
o I~)] °
that
[ ~ " P r o o f ( y , ~ O = l ~) V V y - T P r o o f ( y , r 0 = l ~)] -~ V y ~ P r o o f ( y , % = l ~) ,
~ y P r o o f ( y , ~ O = I D -- V y ~ P r o o f ( y J 0 = t ~) ,
and thus
~Vy~Proof(y,~O=1 ~) T h e r e f o r e HA+ ~txAx
t h i s c o n t r a d i c t s G o d e l ' s second i n c o m p l e t e n e s s theorem, does not satisfy ED, since m + ~ A x ~ A ~ , but~ot Io11.3.
Schema
D:
Vx(AVBx)
-~ ( A V V x B x )
,
x
~+~Ax~A~.
not free in
Ao
This schema has a t t r a c t e d a t t e n t i o n because i n t u i t i o n i s t i c logic w i t h this schema added, models w i t h constant In G o r n e m a n n schema D
domain
is s e m a n t i c a l l y
characterized
(see e.g. G ~ r n e m a n n
the d i s j u n c t i o n property.
predicate
by K r i p k e -
1971).
1971 it is also n o t e d that i n t u i t i o n i s t i c
possesses
hence
However,
predicate
logic +
this p r o p e r t y
is
lost as soon as we go to arithmetic. Actually, complexity A m ZyBy, ~Bx
HA + D = ~ HA c , as is easily seen ; by i n d u c t i o n on the logical
of
A
then
V Zy By ; if
hypothesis, Vx~Bx If
one v e r i f i e s (Vx~Bx Bx,
V ~By
ZyBy
therefore
D
note that
restricted
Let for example A ~ ~Vx Cx, H~+DC
~ A V VxBx;
which is impossible.
(1)
HA,
~Bx
and if
-TBx V Zy By °
V ZyBy
for all
7Bx,
then
By i n d u c t i o n
x , and thus
Vy(By v - T B y ) -~ Vy(By V-7 V x B x ) <--~
spoil the d i s j u n c t i o n property.
Let
Dc
to closed instances°
Vx Cx
Bx -= Cx
be a
Then hence,
~+~O~vxc~ in
hence also
E.g. if
V -7VxBx o
Even closed instance~ o~ D indicate
Vx(A V ~ A ) .
o
A -= V y B y ,
6--> V x B x
of
V ~yBy)<--9 Vx(-~Bx V ~ y B y ) ;
then
Bx V ~ B x ,
the pmov~6~lity
rossersentence Vx(A V B x )
if
~+D
or ~ +
° Dc
for
holds in would
H~ + D c , snd let F~ ; by
satisfy
Dc
DP,
bWCx
At the same time it follows
that
D°
is not derivable
not even the rule for sentences :
t-W:(AvBx)
=
kAVVxHx.
This contrasts with the ease for i n t u i t i o n i s t i c
predicate
logic, where cut
93
elimination for a calculus of sequents or normalization for natural deduction readily yields tained from by
(I).
(In a closed normal deduction,
V - introduction from
V - i n t r o d u c t i o n ; hence Since
D
A
Vx(A VBx)
A V B a , and this in turn from
or
VxBx
must be obA
or
Ba
can be derived.)
is obviously invalid for our realizability interpretations,
and
the rule is not admissible even for arithmetic, we shall not spend further attention 1.11.4.
on it. The schema
Vx m m A
~ mmVxA
.
(DNS = ~ouble negation _Shift).
To this schema attaches considerable technical interest,
since, as we
have seen in 1.10.9, in theories based on intuitionistic logic + DNS , the negative translation satisfies
mmA
Also it permits us to derive, Assume DNS;
VxmVy~A'(x,y) hence
, then
mm~zVxA'(x,zx)
<--gA'
for all
A o
in intuitionistic analysis, Vx ~ m S y A ' ( x , y )
, i.e.
,
so
m VzmVxA'(x,zx)
(AC)'
from
m~Vx~yA(x,y) o
AC: by
In other words, the
'- translation interprets classical analysis, formulated with sequence variables and the axiom of choice, in the corresponding intuitionistic theory + DNS . This fact constitutes the starting point of Specter 1962. Io11.5o
Narkevts schema and rule.
Marker's schema in its most general form can be stated as
M
Vx[AV~A]
& ~xA--
ZxA.
A simpler and weaker form is MpR
m~XxA c MpR
Let us use for A of
x
~ SxA for
(A
primitive recursive)o
restricted to closed instances.
MpR
ranging over natural numbers,
which can be tested for each :xA
(i.e.
mm~KxA )
x
~
expresses : if we have a property
(i.e.
Vx~ V~A))
for all
x ,
such that
that
A .
and an indirect proof
then this amounts to a direct p r o o f o f
other words, Marker's principle enables us to assert, A
Intuitively,
m~xA
:xA o
In
for a computer testing
guarantees that the computer will find an
x
This is obviously an enlargement of the concept of "con-
structive". o
MpR
is not derivable in
~
(Kreisel 1958A).
Consider e.g. the follow-
ing instance
(I) where
~Vx~Bx Vx~Bx
~ ZxBx
is a
~ossersentence~r
Naking use of the closure of
~
HA °
under
(of. e,~. 3.1,7)
94
it would follow that if
Bx
m~
(I),
is primitive recursive, hence
In the first case HA ~ 9~ ~ B x
o
,
~ B ~ or
~ ~B{
in the second case
° HA ~ ~ V x ~ B x ,
B o t h cases conflict with the assumption that
rossersente,]¢e HA,
HA L ~ V x ~ B x
for
V x ~ Bx
ioeo is a
HA o
and many other intuitlonistio formal
systems, have been shown to be
closed under the rule ]V]R
bVx(Av~A),
~ =~ZxA
= ~ZxA,
and a f o r t i o r i under ~RpR
~ ~ x A
= ~ A c MRpR
and its specialization Using lish
as follows.
For all
assuming Hence
A
primitive recursive,
to closed formulae°
w - consistency and classical metamathematics freely, we can estab-
ZR~R
closed.
for
[,
Assume
~A[
~- consistency of
~n(~
~A~) ;
/ ~ZxAx
or
~ ~A~
H~,
, A
~xA
~ n ( ~ ~ A n ) ; then,
we obtain a conflict with
arguing classically,
For m o r e details on
primitive recursive,
Suppose
~ n ( ~ A [ ) ; hence
~ ~Vx~A ~xAx
° o
}~i9, see § ~.8, § 5.4.
Technical interest of Zarkov~s schema also derives from the fact that it is validated by Godelts Dialeotica interpretation, and that by a result of Godel (Kreisel 1962 , § 3) completeness Worot. intuitionistio validity (conceived as the analogue of classical set-theoretic validity) implies the validity of Markovls schema in intnitionistio arithmetic. In recursion theory, there is a rather special application of Zarkov~s schema in the proof of Post's theorem (a set N\X
are both roeo)°
X~N
is recursive if
X
and
The application involved corresponds to a schema of
predicate logic of the following form : V~y(A V ~ A )
& V~y(B W B )
& Vx~(~A~yv~Bxy)
& V~Xz(A(x,y)
&B(x,z))
- W(~A~yV~Axy).
Or with function variables: ~ Z y Z z ( a y = b z ) & Vx ~ ( Z y ( a y =
x) V Z y ( b y = x ) )
This schema is presumably weaker than
-- Vx(~y(ay: x) V m ~ y ( a y = x ) )
o
MR , but the precise logical relation-
ships are unknown. I°11.6.
Independence-of-premiss schematao
An "independence-of-premiss schema" is a schema of the form (x in
A)
not free
95
(I)
(A--~B) -- ~(A--B)
where
A
in general must satisfy additional
restrictions
~,
either syn-
tactical or logical. The principal
instances of independence-of-premiss
schemata which we shall
encounter are
(so
A
must be of the form
Ie°
~C
here),
Vx(A V--A) & ( V x A - - ~ y B )
and the weaker
-- ~ y ( V x A ~ B )
and the still weaker IPpR
(VxA--[yB)
-- ~y(VxA--B)
On the intended interpretation A~xB
,
the " x
for which
(and not only on
A
~,
primitive recursive).
of the logical constants,
B"
may depend essentially
schema (I) expresses,
that for
the
x
does not depend on the proof of
indicate a priori an
x
which should satisfy
ence-of-premiss
B
schemata do affect the intended
of the logical constants:
they restrict
if
(2)
~A-
(A
admissible
~B
independence-of-premiss A certain technical
1.11.7.
So independinterpretation
of the form
A~xB
.
show more or less the same as the con-
of intuitionistic
that the system discuss-
implication
enforced by the
schemata. interest
by modified realizability
see e.g.
holds.
the type of mappings from proofs to
sistency of the schema relative to the same system:
metic !
satisfying the at all : we can
rules of the form
restriction)
ed permits the interpretation
restricted
A
= b~(A-B)
under an additional
IPpR
A
A A
(constructive)
proofs which can be used to establish implications The corresponding
if we assert on the proof of
being true).
An independence-of-premiss restriction
(A
of
IP
is in the fact that it is validated c (i.e. (see § 3.4). Not even IPpR
interpretations
to closed formulae)
is derivable
in intuitionistic
arith-
3o1.11)o
Church's thesis and ruleo
In a formal
system with function
symbols,
(the intuitionistic
version of)
Church's thesis can be expressed as cT
v~ ~x Vy ~z [Txyz & ~x = uz].
Combined with a choice principle
, we obtain a version which can be co expressed in the language of arithmetic : CT o
Vx ~ A ( x , y )
AC
-- ~z Vx ~u(Tzxu & A(x,Uu)) o
96
The conceptual interest of
is in their bearing on the question: o do the concepts of "humanly computable function" and "mechanically computable
function" coincide ?
CT
and
CT
(here "humanly computable"
should mean "computable by
an idealized mathematician in the intuitionistic of these matters,
see Kreisel
ly in the fact that
CT
sense":
1970, and especially Kreisel
for a discussion 1972)! and second-
implies the incompleteness of intuitionistic predi-
cate logic (sketched in Kreisel 1970, Technical Note I : for a more detailed exposition see van Dalen~). Church's thesis turns out to be consistent with all intuitienistic formal systems not involving the concept of choice sequence, containing the fan theorem or bar induction The underivability of
CT
o "Charch's rule" takes the forms:
where CR
~6 GR
abbreviates
is obvious,
~xVy~u(Txyu
since
CT
& ~=Uu)
o
is
false in
HA c .
and
o
A weaker
version of Church's rules takes the form
WCR
If
~Vx~vAxy,
such that WCR
then there is a recursive function
Vn b A(~, ~ n ) .
is closely connected with
tion,
ED
is equivalent to
0bviously~ f
and especially not
(cf. § 3.2).
WCR
implies
ED : for systems with a recursive axiomatiza-
WCR
ED o
(Kreisel 1972 ) . Conversely,
if
ED
This is seen as follows. holds, we may construct
as fn = min m Proof(jlm, CA(~, J~2m) ~)
which makes
f
recursive, in view of the recursiveness of "Proof"°
As remarked in Kreisel 1972, this result tells us where not to look for a conflict with Church's thesis ; all the usual systems satisfying be expected to yield a refutation.
ED
cannot
Chapter II MODELS AND COMPUTABILITY
2.~.I.
Definition over the tFpe structure~
repeatedly with definitions structures, if
~, T
type structures,
i.e. type
obtained from certain basic types by closure under the condition :
are types,
structure being T
In the sequel we shall meet
over applicative
then so is
T.
(~)~,
our principal
example of such a
In the discussion below, we restrict our attention to
for simplicity. An applicative
t'E ~,
t,t'E M
subset
M'cM
(i.e.
~
set of terms then
M
tt'E M .
is e set of terms such that if A basis for an applicative
such that the closure of
is the smallest applicative
applicative
s~s
M'
set
tE (a)T , M
is a
under application yields
set containing
M' ).
M
Examples of
:
(a) The set of closed terms
CTM
of
N
HA w
with the constants of
~-HA w
as a basis. (b) The applicative and type
0
(c) The applicative the type if
x~
set generated by the basis consisting of the constants
variables of
0
~-HA~
(CTMo).
set generated by the basis consisting of the constants,
variables
is the type
and a single fixed type
~
Further examples of applicative ~
with
(i)
T
l(~)~n
Po(t~ .... ,tn)
(ii)
if
~(~)~(t~, ....,tn)
(i)'
an applicative Po(t)
if
(ii)' P(~)T(t)
set
of an M
t~ ..... the 0
of the
(a), (b), (c). n - ary relation over the type
of terms takes the following form : and
A(t~ ..... tn)
(A
being any given
~t~ ~ ~ ... v t ~ M(p~(t~ .... ,t~) ~ ~(t¢~,...,~oq~).
(for an example
see
2.2.5)
takes the following form
set) :
t6 O, if
if
Ao(t )
A(~)T(t)
and
te
and
~t' ~ ~(P t' = P
Such a definition may be viewed as a superposition over a type structure, applicatively
i((~)~) = l(~) +l(T) + ~.
t~,...,t n E M-
A slight generalization (M
i(0) = 0,
in examples
for an applicative
predicate);
(CTMo(x~),
sets are now provided by restriction
In its simplest form, a definition structure
variable
variable).
Let us define the type level as follows:
types to types
~
from type
where each type 0
~
tt') . of a sequence of definitions
is not only viewed as obtained
but also acts as a "ground type"
for more complex types w.r.t,
the property
A
.
(or "basic type")
98
A definition according to (i), (ii) or (i)', (ii)' is a definition over the type structure of metamathematical properties of metamathematical objects (i.c. terms).
Of course, similar definitions are possible within the system
itself, i.e. we may define properties within the system according to a schema :
(i)"
Pc(X ....
%ef
(ii)
..... x =)~) mdef VY~'''Ynk ckY1'''''Yn) " P~(xlY1''''XnYn))"
and such a schema similarly permits generalizations and variants(cf. Z 5 15 for an example). 2.1.2.
Establishin~ properties for applicative sets of terms.
We consider three types of definitions of a property an applicative set (A)
Q
Q for the terms of
M:
is defined as a unary predicate over the type structure, according to
clauses'(i) and (ii) in 2.1.1. (B)
Qt mde f P(t,...,t) , where
P
is an
n - ary predicate defined over the
type structure according to clauses (i), (ii) in 2.1.1, and holds for (C)
Q
A(t,...,t)
t~ M.
is defined inductively over the type structure according to (i)' and
(ii)' in 2.1.1. In each of these cases, establishing
V t6 M[Qt]
ing
M,
V tE M'[Qt]
Lemma.
If
for a basis
tl,...,t n
M'
of
may be reduced to establish-
because of the following lemma :
are terms such that
Q(ti)
for
1~i~n
defined by a definition of type (A), (B), (C), then for any repeated application from Proof.
tl,...,t n,
Qt
, and
Q
is
obtained by
holds.
Quite straightforward ; we have to show that if
t IE (~)~ , t 26 a , then
t
Q(tl), Q(t2)
and
Q(tlt2) .
Similar lemmas reduce the establishment of properties
Q
defined via defini-
tions of type (i)", (ii)", or one of the many variants, to the establishment of
Q
2.1.3.
for a basis of the set of applicative terms considered. Definabilit 2 aspects.
Suppose
Q
to be an
(A), (B), (C) in 2.1.2. of
Q
(so
Q*
themselves) expect
Q*
n - ary predicate defined by a definition of type If we wish to consider an arithmetical version
Q*
is a predicate of godelnumbers of terms, not of the terms
then if there is no bound on the type level, we cannot in general to be arithmetically definable, since
~*
should satisfy e.g.
for a definition of type (A) : Q*(m)
4--~ S~ (type (m)= r ~ u & Q ~ ( m ) ) ,
Q~)T(m) ~ (where
ra ~
Vn (Q~n ~ Q~(app(m,n)))
denotes the code number of type
~ , app(n,m)
the arithmetical
99
representation of the application operation),
therefore with increasing type
level the logical complexity of the arithmetical formula indefinitely ; for an actual counterexample So for an arithmetiz~8 version,
~(m)
increases
see 2.3.11.
the applicative
set of terms considered
will have a bound on the type level. As we shall see, in our application~,the definability of the arithmetized predicates usually ensures formalizability of the proof of numbers 2.1.4.
n
of terms in the applicative
Sets of terms closed under
@*n for all godel-
set of terms considered.
~ - abstraction.
If we consider sets of terms not only closed under application but also under
~ - abstraction,
the reduction effected by the lemma in 2.1.2 might not
be sufficient since the effect of closure under
~ - abstraction might force
us to consider a very "large" basis in the sense of 2.1.1. Let us call a
k - set of terms any set closed under application and
abstraction w.r.t, variables of the set. which yields
M
A
k- basis for
by closure under application and
~-
is a subset
~- abstraction.
The appropriate trick for establishing a property (A), (B), (C) in 2.1.2 for a
M
Q
defined according to
X- set is then to prove a stronger property
Q*
("Q - u n d e r - substitution"): Q*(t)
holds if
Qt I
holds for any
tI
obtained by substituting for some
(not necessarily all) occurrences of variables in holds
(and possibly renaming bound variables in
free in
t'
becoming bound after substitution),
t t
terms
t'
for which
so as to avoid variables
(see e.g. 2 . 2 . 2 7 - 2.2.51).
Q
100
2.2.1.
In Godel 1958 , the concept of a "berechenbare Funktion"
(= computable
function) is regarded as a primitive concept, and it is considered evident that each primitive recursive functional definable in Godel's theory (i.e. a functional represented by a term of
~_~w)
is "computable".
In Talt 1965 this concept is made the subject of a formal analysis, and it is shown that each constant term of type implies that each term of type version of
~_~w)
O
0
"reduces to" a numeral, which
can be formally proved (in a suitable
to be equal to a certain numeral.
As a by-product,
Tait's analysis yields more : all terms can be brought into a standard form ( "normal form"). This is exploited in Tait ' 1967 , to show that the closed terms of yield a model for
N_~W
I - H ~-~ A ~ , if equality between terms is interpreted in the
model as : reducing to the same normal form.
Tait 1967 also introduces the
inductively defined formal computabil~[N predicates
("Comp"), a device which
has been extensively used since. In the present section we discuss computability predicates and use them to prove normalization and strong normalization theorems for the terms of
~_~w
and extensions. The main novelty is the simplification of the treatment and strengthening of the strong normalization theorem in
2 2 ~9.
We first discuss computability for the terms of with classes of terms with In 2.2.35,
~-operators
instead of
~-~; H, Z
then we deal as primitives.
the various proofs of normalization and normal form theorems
occurring in the literature are discussed and compared. Additional material on computability is given in the next section. We feel the notions of computability and strong computability have a certain intrinsic interest, because of their intuitive simplicity ! hence the rather extensive discussion,
with description of different approaches,
below.
It should be noted, however,
that for all applications of computability given
in the sequel of this chapter, in proofs of results which do not require the notion of computability for their formulation, to standard computability of terms of type 2.2.2.
O ;
we may restrict our attention the remainder is a luxury.
Definition of reduction and standard reduction for terms of
We say that a term is a contraction of
t t ),
contracts to a term
t'
(t
contr,
t' ,
~-~. or
if one of the following clauses is satisfied :
(a)
t ~ Htlt 2 , t' ~ t I
(b)
t ~ Ztlt2t3,
(c) (d)
t ~ Rtlt20, t' ~ t I t Rtlt2(St3) , t' ~ t2(Rtlt2t3)t3.
t' ~ tlts(t2t3)
t'
101
Ne adopt the terminology of C u r r 2 - F e 2 s
195~
and call
t
in
the clauses (a) - (d) a redex (and similarly for other types of contraction introduced in the sequel). If t
t'
is obtained from
(i.e. a subterm of
t' ~I t
t
t
contracting a single subterm (occurrence) of
is replaced by its contraction)
then we write
t ~I t' "
or
A sequence (finite or not)
to, tl, t2, ...
is said to be a reduction sequence o f
to
with
ti+1 ~I t.1
(starting from
for all
i
t o ).
A term which does not admit any contractions, is said to be in normal form. A finite reduction sequence ending in a term in normal form is said to terminate . We say that of
t
t ~ tI
ending with
t'
(t
reduces to
t' )
if there is a reduction sequence
(a reduction sequence from
t
to
t' ).
A reduction sequence is said to be strict, if the contractions (a) - (d) are applied only in case
tl, t2, t 3
are normal.
Attention to strict reduc-
tion sequences implies prescribing a certain (partial) order for the contractions.
The order in which contractions have to be executed can be made
completely deterministic by introducing the concept of the leftmost minimal redex
(lmr).
The
lmr
of
t
is a subterm of
t
when
t
is not normal,
t
(t
assumed to be
of
t
otherwise undefined. We define the
lmr
by induction on the complexity of
non-normal). (i)
If
t ~ ~tl...t n , ~
tl,...,ti_ ~ of
a constant or variable,
are normal,
ti
not, then the
and lmr
is the
lmr
t.. l
(ii) If
t
is a redex and (i) does not apply,
A standard reduction sequence such that We write 2.2.3.
ti+ I t ~w t'
then
to, tl, t2, ...
is obtained from
ti
imr(t)
is
t
itself.
is a reduction sequence
by contraction of the
lmr
if there is a standard reduction sequence from
of t
ti . to
t' .
Comparison of standard and strict reduction.
Intuitively we feel that there is standard and strict reduction :
little essential difference between
strict reduction corresponds to the natural
idea of contracting ("computing")
starting "from the inside"
(i.e. starting
with redexes not containing other redexes) ; standard reductions make in a convenient but arbitrary way (since not directly related to the partial ordering of the tree of subterms)
the procedure completely deterministic.
We show Proposition.
If
t
strictly reduces to
standard reduction sequence from
t
to
t' , t' t'
normal,
then there is a
102
Proof.
By induction on the length of strict reduction sequences.
Suppose (1) the assertion to hold if than
k
t
strictly reduces to
t'
in less
steps ; we now prove the assertion for strict reduction sequences
of length
k
sequence•
So assume (2) also the assertion to have been proved for all
by a sub-induction on the complexity of the first term of the
strict reduction sequences of length
k
starting with a term of complexity
< i. Let
tl, ..., t k
be a strict reduction sequence from
let the complexity of ing
tI
be
i.
Then either
t2
tI
to
tk , and
is obtained by contract-
t I , and then the assertion readily follows from induction hypothesis
(I).
If
t2
is not obtained from contracting
tI ,
tl, ..., t k
starts
with an initial segment tl = ~s~1)... s(ml) q~sl2).., s(2)m' "''' ~ 4 n) "'" s(n)m where
~
normal.
is a constant or variable of N - H A w, and s 1(n), -.., sm(n) are ~I) ~n)~ s. , ..., s (1
Then the sequences
duction sequences after omission of repetitions. (i)
all sequences
s! I), .... s! n) 1
repetitions.
have length
Then either
after omission of
1
Then by induction hypothesis (1), there are standard
reduction sequences from
s! I)
to
1
~S~ n) ... Sin) -= tn
to
tk
reduction sequence froml t I
s!n)
(I < i < m)
l
and from
~
which may be combined into a standard to
t~ ; or
(ii) there is a sequence
s.- ), ..., s (n/ of length k, 1j~i, for 1 1 < j < m , and ~s ~n)
without repetitions s (I) = s (n) s(mn) __ tk J Then the assertion follows by the sub-induction hypothesis (2).
and
(Cf. Tait 1967 , II on page 203.) Corollary.
All terminating strict reduction sequences starting from a given
term terminate in the same term. In the sequel we shall establish a much stronger result (see 2.2.23). 2.2.4.
Alternative definition of may also be defined as a relation between terms, inductively generated
by the following closure conditions: t ~ t, = t~t",
t ~ t' = tt" ~ t't", Htt'~t,
t ~ t' = t"t ~t"t',
Ztt't"~tt"(t't"),
Rtt'0~t,
t ~ t'
and
t I ~t"
=
Rtt'(St")~t'(Rtt't")t" .
The equivalence of this definition with the one given in 2.2.2 is intuitively obvious, and in fact formally provable in
H~,
e.g. by an appeal to the
theory of § 1.4. 2.2.5. ability.
Definition of computabilit2~ strict computability1 standard computWe define a predicate
Comp = U {Comp~ I ~ E T},
defining
Comp~
to3
by induction over the type structure : (i)
ComPo(t ) mde f t E 0
(ii)
Comp(a)v(t ) mde f V t'(Comp~(t')
and
t
reduces to normal form ; ~ Compr(tt'))
and
t
reduces to
normal form. Similarly we define
Comp', Comp~
to" in the definition of
Comp a
and
Comp", Comp~ , replacing "reduces
by "strictly reduces to" and "reduces to
... by a standard reduction sequence" respectively. 2.2.6.
Theorem.
All terms
t
of
~-~.~
satisfy
Comp"(t) , and hence
have a terminating standard reduction sequence. Proof. term
We note that if t
Comp"(tl) , ..., Comp"(tn) , then
constructed by repeated application from
lemma in 2.1.2.)
Comp"(t)
tl, ..., tn.
for any
(Cf. the
This is an immediate consequence of the definition.
Hence it is sufficient to prove
Comp"(~)
for
~
a constant or variable
of cur theory. (i)
Comp"(O)
is immediate°
(ii)
Comp"(t o) = Comp"($t °)
(iii) Comp"(x °) (iv)
Let
is also immediate, hence
~ = (QI) ... (am)O.
If
Compq1"(tl)' "''' ComP~m(tm) , there are
terminating standard reduction sequences for readily combined into a terminating x~tl...t m. (v)
If
Hence
CompS(t3) , then
We put (I)
t3
v(t3)= k°
There is a uniquely determined for any
and
If
Comp"(Ra)
Comp"(t2)
k.
Let
and
by proving
Comp"(t3)
and
~ ~ (~1) °o. (Cm)O°
Comp"(ti) , 1 ~ i ~ m + 3 ,
v(t3) = O .
There are standard reduction sequences from 1~i~m+3.
k
t.
= Comp"(Ratlt2t3) )
by induction w.r.t, Let
t~.
We now establish
Vtl V t2 Vt 3 (Comp"(tq)
(a) Bas~s.
which are
has a standard reduction sequence terminating
t!~ m %kt~ , t~) ~ St
v(t3) = k
tl, ..., t m,
standard reduction sequence for
Comp"(x a) .
in a term in normal form such that
Comp"(S) .
is immediate.
t~ m O,
ti
to
t!1, t~l normal,
there is a standard reduction sequence from
Rtl...tm+ ~
to Rt~.. "t'm+3 ; Rt~t~t~.. "t'm+3 ~I t~ta''" t'm+3 , and since, according to our hypotheses and the remark at the beginning of the proof, Comp"(t~t~t~...t~+3) If
t~ ~ St
for any
(b) ~nduct~o~_~tep. If from
, also
to
t~1,
Rt~t~t~...t'm+3
is already in normal form.
Assume (I) to be established for
Comp"(ti) , 1 ~ i ~ m + 3 ti
Comp"(Rtl...tm+3) .
t , then
t'i
n
, ~(t3) = k , there are standard reduction sequences normal
(I < i < m+3) , t~ m 6kt~
Then there is a standard reduction sequence from
for a suitable
Rtl...tm+ 3
to
t~.
104
Rt~. . ti+3; also Rt~t~(skt~)t~. t'm+3 ~I t2(Rtlt2(~ ' , , k-1 t3))(sk-lt~)t4""t~5; ,, . . . and this term has a terminating standard reduction sequence by our hypotheses and the remark at the beginning of the proof. Cases (vi) and (vii), where it is to be shown that
Comp"(U)
and
Comp"(Z) , are left to the reader. 2.2.7. (i)
Remarks. Comp'(t)
from 2.2.6.
and
Comp(t)
However,
for all terms
Comp'(t)
and
t
Comp(t)
of
N-H~ w
follow directly
can also be proved directly
along the same lines as in 2.2.6. (ii)
In the definition of
the condition
"and
t
Comp',
Comp"
we might actually have left out
reduces to normal form" in clause (ii).
To see this, note that one easily proves by induction that 0 ~ (defined by 0 o O, 0 (~)T = H T0 ~ ) is normal and satisfies Comp~ (with the weakened definition of
Comp").
Comp~(t0~1...oam) quence of of
t
.
Now if
Comp"(t) , ~
~ ~ (~I)... (am)0 , then
Hence there is a terminating
tO GI ... 0 am , from which a terminating
can be extracted.
Similarly for
(iii) If we are interested slight simplification
standard reduction sestandard reduction
sequence
Comp'
in the computability
of closed terms only, a
might have been achieved by omitting clauses
(iii)
and (iv) in the proof of 2.2.6. 2.2.8. then
Lemma. t
If
t~ 0 ,
t
a closed term of
We proceed by induction on the complexity
closed and normal.
Then
t
has (possibly)
0, S, Stl, R, Rtlt2...tn, tl, t2, ...
have type
normal).
tmO,
Stl,
t.
Suppose
t
is
one of the forms
H, ~tl, Z, Ztl, Ztlt 2
we are done.
Rtlt2...t n If
(n h 3 ) .
t ~ St I ,
0)
Finally,
if
t ~ Rtlt2t3...t n ,
numeral,
so
t
by induction hypothesis
then (since tI
tI
is a numeral,
is closed, normal, therefore
then by induction hypothesis
t3
so is
t.
is a
cannot be in normal form.
Theorem.
term of type Proof.
of
But the only forms of this list which could
of type
2.2. 9 .
form,
0 , are O,
If
in normal
is a numeral.
Proof.
(with
N-HA w
0
(On assumption
of consistency
of
reduces to a uniquely determined
N-HAW.)
Each closed
numeral.
Theorem 2.2.6 and lemma 2.2.8 imply that each closed term of type
zero reduces to a numeral. consistency
of
N_~W
The uniqueness
since if
t~,
of the numeral follows from the t~m,
~m,
it would follow that
~o5
2.2.10. type
Theorem.
0
over
Proof.
N-HA ~
is conservative w.r.t, closed prime formulae of
~,~obtained
by omitting induction from
~-~.
By refinement of the argument in 2.2.9, noting that
~ t I= t 2 . t°~,
For let
s°~m,
~_~w
hence
~ t o = s°
~ ~ t ° = ~,
Then we can find
~ ~ s° = m .
t1~t 2 ~, ~
implies
such that
By consistency of
~-~,
hence ~ b to = so
=~, 2.2.11.
Remark.
In
2.5.6
it will be shown how to prove uniqueness of
normal form for all types (i.e. every terminating reduction sequence of terminates in the same term) by means of a model for 2.2.12- 2.2.19 . 2.2.12.
N - H A ~.
Strong cqmputabilitz.
We shall now refine the preceding discussion, by proving a stronger
theorem : each reduction sequence starting from a term
t
shall call such a theorem a strong normalization theorem.
terminates. A term
t
We is
said to be strongly normalizable, if all reduction sequences starting from t
do terminate.
In order to prove this theorem, we have to modify our
definition of computability to a definition of strong computability. 2.2.15. SC
Definition.
Strong computability for terms of type
, is defined for all
(i)
SC (t)
iff
t6 0
a6 ~
~ , denoted by
as follows :
and every reduction sequence starting from
t
O
terminates. (ii)
SC(~)T
We put
iff
SC ~def
2.2.14.
Lemma.
terms such that Proof.
U
~t'(SCa(t' ) = SC (tt')).
{SO Let
Ia e
~1.
x e (el) ... (an)0,
and let
ti e ~ i ' 1 ~ i % n
ti
is strongly normalizable ; then
If
SC(t a) , then
be
SCo(xtl...tn) "
Obvious.
2.2.15.
Lemma.
ta
is strongly normalizable ; also
SC(x°). Proof.
We establish the assertion of the lemma by induction over the type
structure. Assume for al subtypes
O
of
(~)T
Let SC(t(a)T) . Then SC(t(~)~x~) . tion sequence starting from t. We from t(~)TX~, say tx, t~ 1), t~ 2) ,
as for
t (k)
is defined ; if
t (p+I) 0
t (p+2) '
0
..
the assertion of the lemma to hold.
Let
t, t (1), t (2), ...
be any reduc-
now define a reduction sequence starting ...
t(k) E t(k)x as long o breaks off at t (p) , we take
as follows:
t, t (~) , t (2), ...
a standard reduction sequence starting from
t(P)x.
~"
By induction hypothesis and
SC(tx) , tx, t (I), t~ 2)
so has an ir~ial segment of the form
tx, t~1)x . . . . i tik)x
terminates, and such that
lOd
t, t (I)
t (2)
Now assume Then
t,
t( "k"~
is a terminating
and let
~ ~(~1)"'(%/0'
t.
are strongly
normalizable
SCo(X(~)'t~ I ...tm) , and thus 2.2.16.
Remark.
inductively
(where
SC~i(ti/, ;
sequence for
l
t
SC(t).
hence by lemma 2 . 2 . t 4 ,
SC(a),(x(~)" ).
Instead of using
SCp(O p)
reduction
SC (x p) , we might also have established
0 ° ~ O,
PO ~)~" " = ~o,TO
, as in 2.2.7)
to-
gether with the induction hypothesis. 2.2.17.
Definition.
, where sequences
T
A reduction tree of a term
is a non-empty
such that
n*~E
to the elements of (a)
~<>=t
(b)
If
T,
n * E T
for
t~ ..... t~
I < i < n , and
ordering is prescribed
Definition
definite,
t
tl,...,t n,
t' , then
~(n. ) = t!.
contractions
for all terms.
is the number of elements in
(or classically,
Let
Konig's lemma)
The
T. SCo(t ) ; then
the reduction
hence there are only finitely many terms in normal
such that St*
from
we may assume that in a uniform way an
(to be used in the proof of 2.2.19).
is finite,
not of the form
is a complete list of terms(without
among the possible tree
by 2.2.13 and the fan theorem
level)
finite
a function which assigns terms
which are obtained by a single contraction
length of a reduction
Remark.
~
such that
To make the description
form,
set of natural numbers representing
T = n E T , and
q~n= t' , and
repetitions)
tree of
consists of a pair
o
n E T,
2.2.18.
t
t >t i
for any
t*.
Then
In giving this definition, to the fan theorem,
for
I < i
Let
t i = sP(i)t~ ,
we have made an appeal
(on the meta-
by assuming that a finitely branching
all its branches finite is itself finite. theorem in the proof of 2.2.19
t~
~(t) = max Ip(i) I 1 ~ i ~ n l .
The implicit appeal
(via this definition of
~)
tree with
to the fan
can be avoided
in two different ways : (i)
by giving a proof of the uniqueness
on the strong normalization define t'
v(t)
normal,
as the t>t'
p
of normal form which does not depend
theorem itself (2.2.23),
such that
t v~ ~Pt"
t"
(ii) by strengthening
2.2.19 .
SCo(t )
to:the reduction tree of
trees, which is notationally Theorem.
St ''~
; or
This requires in the proof of 2.2.19 manipulation reduction
and which enables us to
not of the form
For all terms
strongly normalizable.
t
of
t
is finite.
and recombination
of
SC(t) , and hence
t
awkward. N - HA w,
is
lo7
Proof. (I)
We first note that If
SC(ti),
from (II)
If
I < i
tl,...,t n ,
(III) If
, then
(ii)
tea
= (~I)... (an)O , then
If
SC(t) * (~tIE SC I)... (VtnE SCan )
is strongly normalizable). SC(~)
for
~
a constant or variable
of
(cf. lemma in 2.1.2).
SC(O)
is immediate.
SCo(t) , then
from
application
SC(t') , as readily follows from the defini-
n By (I), it now suffices to prove (i)
formed by repeated
SC.
( ttl.., t
our theory
t
$C(t) .
SC(t) , t ~ t '
tion of
then for each
St
SCo(St) , since any reduction
must necessarily
t, tl, t2, ... (iii) SC(H ,~)
be of the form
is a reduction
holds.
For let
sequence.
tl, ..., tn
H~, t I ... tn E 0 , and suppose arbitrary
reduction
sequence
sequence
starting
St, Stl, St2, ..., Hence
where
SCI(S ) .
be terms such that
SC(ti)
(1~i~n)
starting from
H
.
Now consider an
Tt I ... t n.
This will be of the form H t (k) t (k) ~a,Ttl "'" tn' H~,~ t(I)I "'" t(1) n ' "''' ~,T I "'" n ' t(k)t (k) t (k) t~k+1)t~ k+1) t (k+1) I 3 "'" n ' "''' "'" n ' "'" (Such a
k
must occur, otherwise
ti' t(1)i ' t!2)i , ... infinite reduction Now
after omission
sequence,
contradicting
tlt3..°tn~t~k)t~k)°°°t~k)
(ii) SC~t(k)t(k) I ~ " ..t(k)~ n - ° o°o t (k+1) n (iv)
,
..o
one of the sequences
would,
of repetitions, SC(t~)
, and by (I)
become an
and 2.2.15.)
SC(tlt3°..tn)
, hence by
t(k)t(k) (k+1)(k+1) 1 } "'" t(k) n ' ~I ~3 "'" sequence starting from t (k)t(k) I 3 ° t(k) n
Now
is a reduction
o.
,
and therefore terminates. SC(Z0,~,~) is proved similarly.
(v)
For
(vi)
Now we have to show that R
SC(x~) ,
see 2 . 2 . 1 5 . Ra
is strongly computable.
E (al) ... (~n)O , we have to show that
tl, ..., t n
such that
SCa (ti) ,
SCo(Rtl...tn)
I < i
Now if for
We apply a sub-induction
~.r.t. ~(t3) . (vi) a. If
v(t3) = O , a reduction
one of the following
forms
sequence ( .
~
starting from
indicating
Rt I ... t n
has
the sequence fo,fl,f2,..°):
l l
(I) i,
with
tl O) ... t (°) ~ tl... tn n (2) Rt I °.. tn, Htl I) ... t (I) RtQ 2) t (2) Rt~ p-l) t (p-I) n ' "'" n ' "''' "'" n ' Rt(P)t(P)o tiP) t (p) t(P)t(P) t~ p), t (p+2) t(P+3) I 2 ''° n ' I 4 "'" ' ' ....
108
In case (1)
~
obviously
nates since by (I) , (ll)
terminates, in the second case goct(P)t(P) ~ I 4 "'" t nkp) ) ' a n d
~
termi-
t(P)t(P) t(P) t(P +2) t(P+3) I % "'" n ' ' ' "'" is a reduction
sequence starting from
t(P)t (p) t! p) I 4 .... (vi) b. Assume SC(Rtlt2t 3 ... tn) to have been proved for all tl,t2,.-.,t n such that SC(ti) , l ~ i ~ n , and ~ ( t 3 ) ~ k . Now consider Rtlt2t 3 ... t n with SC(ti) , I < i < n , ~(t3) =k+1 . reduction sequence starting from Rtlt2t3... t n. ~ following (I)
Let ~ be any has one of the
forms:
t(i)> where Rt i t ~ Rtl °) t (O) n l n n (Actually, as follows from 2.2.23, this case cannot occur.) •
.°
•
. . .
° . .
*
(2) Rt I ... tn, Rtl i) ... t (1) Rt~ p-l) t (p-i), n ' "''' ''" n Rt(P)t(P)o t~ p) t~ p) t(P)t (p) t(P) t(P +Z) t(P+3), I 2 "'" ' I 4 ''" n ' ' This case may be dealt with as under (vi) a. case is also excluded,
(Actually,
....
this
by 2.2.23.)
(3) Rt i ... t n, RtQ I) ... t~ I) '
"'''
RtlP-I)
"'"
t (p-i) n
'
RtI(P)t2(P)#st~ o/~t~p) ... t~ p), t~ p)(Rtlp)t~ p)tO)tOt~p) ... t~ p), t(P+2),
t(P +3) .....
In case (I), it is obvious that to
(vi)a;
~
must terminate ; (2) is referred
in case (3) we use (I), (II), and our induction hypo-
~(to)ik ) .
thesis (since
Our next task will be to prove uniqueness
of normal form.
We first show
how to do this by a method due to J.B. Rosser (Rosser 1935 ) . 2.2.20.
Definition
"bounded" (a)
(for use in 2.2.21).
reducibility
t ~* t'
and
(~*)
We define inductively
a notion of
by
t I >* t~ ~ tt I ~* tt' .
(b) iHtt' >* t, ztt't" ~* tt"(t't") , Rtt'O ~* t, Rtt'(St") _>* t'(Rtt't")t" , t ~* t . In other words, if
t ~* t' , there is a derivation
sequence of assertions
t o _>* t o', t1>* t~, t2~* t~ ..... t n _>* t'n where tn = t, each t I -~* t'l either holds by (b), or is obtained from (j,k < i) by rule (a).
t'n--t' , such that t.~ -~* t'j , tk>* t~
2.2.21
t "' such that
t t >~* t"'
Lemma. ~
If
t" >* t'" ~
t>*
t' , t>* t"
then there is a
•
Proof.
Let
t~* t' , t~* t" ; we may assume
t~* t"
be established
by derivation
t' ~ t" .
sequences of length
Let n, m
t~* t' , respectively;
~o9
we apply i n d u c t i o n w.r.t, (i)
t ~ * t'
(ii)
tmHtlt2, Take
holds
n+m.
since
t' E t .
t' m t I .
Then
Then take
t" ~ H t ~ t ~ ,
t'" ~ t " . t1~*t~,
t 2 ~ * t~.
t"' ~ t~ .
(iii) t ~ Z t l t 2 t 3 , for
t' ~ tlt3(t2t3) . Then
i = I, 2, 3
(iv)
t~tlt2o,
(v)
Let
Take
"
t,~tl~
Similarly,
if
derivation
1
3'
2
thsn t"~Rt~t~O.
tmRtlt2(St3)
t ~ tlt 2 ,
t ''EZt~t~t~ ,
t'" ~ t,t'ft't'~
,
3"
t.l_ >* t'i
"
Take t ' " ~ t ~ .
t' ~ t 2 ( R t l t 2 t 3 ) t 3 .
t' m t ~ t ~ , and let the final assertion
sequence
hold by a p p l i c a t i o n
t ~ * t'
of (a) in 2.2.20 to
in the t1~*t ~ ,
t2 f t ~ If
t>_*t"
(li),
holds by (b), we may deal with this case as u n d e r
(iii),
(iv).
Hence assume
t"
tlt ,, ', 2 , and the d e r i v a t i o n
with an a p p l i c a t i o n
of rule
t~*
t I''',
2.2.22.
t~*
t I t2
Lemma.
t I''',
t~*
sequence of
(&) in 2.2.20,
Then, by i n d u c t i o n h y p o t h e s i s
tlt 2 _
(i),
t~*
ends
t 1 ~ * t~, t 2 ~ * t~.
t '" I , t 2'" such that
there are
t~' ,
so
t _>* t"
t 2''',
hence
t'~'I~2_ >* tl...... t2 '
•
If
t > t' , t > t"
there is a
t"'
such that
t' > t'"
t" > t"' . _>
Proof.
is the t r a n s i t i v e
is a sequence 0 < i
to, ... , t n
Let us write
terms.
So
such that t > * t'
to-= t ,
.
Also
t >* t' , t >* t"
Proof by i n d u c t i o n o n
n + m < k ; let n o w
t>t'_ ,
there is a
t'"
Assume
n+m=
k ;
let
n>
t I --1 ~*
t 2 "
t' --m >' t 2 ,
2.2.23.
(fig.
I)
such that
t
t" --n >* t 2 .
Theorem.
of terms of
t' >* t"' to hold for
I
Construct t such that t --n-1 >* t o ' t o _>~ t' o >~ t I , t" >* t I ; and t2 (induction hypothesis) t ~ , t o-in --n-1
Then
N-H~A m
The normal
is u n i q u e l y
Proof.
Let
normal ;
then by 2.2.22,
t ~ t' , t ~ t"
such that
t' > t'"
is normal,
and
~9.i
form determined. t'
and
there is a
t" > t'"
Then
t
t"
t'"
X-
I
of
•
such that
e.g.
for
consisting
the ~ s s e r t i o n
We can find
t' --m >* t 2 ,
then there
t n - t' , t i _>* ti+ I
t~t~<==~ Sn(t~t')
n+m.
t --m > * t" ,
t >* t'
_>* , i.e. if
if there is such a sequence
t~*t'c==~t~t'
Now we show : If t" ~* t"' .
closure of
to
t"'
t' -= t"' = t" .
t~X t g ~
110
2.2.24.
Remarks.
The essential
method of "counting" subterm occurrences >I
refers
idea of the p r e c e d i n g method is a clever
contractions :
simultaneous
count for a "single"
contractions
step (expressed by
to a single c o n t r a c t i o n u n d e r all circumstances.
n o t i n g that the method d e s c r i b e d are "quantifier-free",
is very " e l e m e n t a r y " ;
and explicit d e f i n i t i o n s
of
of disjoint ~ * ) whereas It is also w o r t h
the methods used
>*, >
O
are
ZI
in
character. B e l o w we describe less elementary,
an a l t e r n a t i v e
method
but m a t h e m a t i c a l l y
(2.2.25 - 26), w h i c h is logically
slightly
simpler.
On the other hand,
the p r e c e d i n g method does not require a strong n o r m a l i z a t i o n proved first, and the
and also applies
~ - calculus.
as a primitive, whereas
to the type-free
In the case of the theories based on the
the second method is even simpler
the first method becomes
(HRO) ;
applicable 2.2.25. 9
Proof.
If
t" _>_ t*
v e r s i o n of t ~I t" ,
tI
N-HA w .
then there is a
t*
such that
•
tl, t 2
in
t
respectively, are disjoint°
ing both
to apply.
This method is only easily
For the proof we must d i s t i n g u i s h
(a) The redexes t"
t ~I t' ,
~ - operator
of normal form is via the e m b e d d i n g
3.5.5 - 2 5 6
to the combinatorial
Lemma.
t' q> t*
see
logic
(less cases to check)
somewhat more complicated
A third method to o b t a i n u n i q u e n e s s in a model
theorem to be
systems of c o m b i n a t o r y
and
t2
in
t !
two cases.
which are contracted Then obviously
in r e d u c i n g
t*
is obtained
the order in which
t
to
t',
by contract-
the contractions
are
executed is irrelevant. (b) Let again t', t" where
tl, t 2
respectively, tI
be the redexes in and assume now
is a subterm of
Now we have to d i s t i n g u i s h For example,
let
tI
o c c u r as a subterm of t6[x ]
identical
by
is o b v i o u s l y
various
t5
.
t4(Rtst4t6[t2])t6[t2 ] , where
t~
to obtain
tI
(the case
similar).
to the form of
Rt3t4(St5) , and let
(so that we may write
t4(Rt3t4t6[t2])t6[t2]
t4(Rt3t4t6[t~])t6[t~]
completely
cases a c c o r d i n g
be a redex of the form
be obtained by c o n t r a c t i n g
with
replacing
obtained t*
t'
w h i c h are c o n t r a c t e d
to be a subterm of
contains only a single o c c u r r e n c e
N o w let
t' ,
t2
t
t2
of
x).
tI
to
t5
as
tI . t2
t6[t2] , where
t 4 ( R t 3 t 4 t 5 ) t 5 , w h i c h is
If we then apply two c o n t r a c t i o n s by
t4(Rt3t4t6[t~])t6[t2]
is the c o n t r a c t i o n of
t2,
we have
t*.
is also obtained by first r e p l a c i n g
t I ~ Rtst4(St6[t2] )
is replaced by
this redex. The other subcases are very s~milar.
t2
by
t~
Rt3t4(St6[t~])
in
t" ,
to
, and then
so that
, and then c o n t r a c t i n g
III
2.2.26. Hence
Second proof of 2.2.23.
the reduction
theorem,
or Konig's
reduction of
t
tree of
The assertion all
t'
with
Let
is obvious
t*
If
reduce
t**~ tI ,
; hence
2.2.27-2.2.34. Instead with
t*
appeal
the number on
~(t)
Assume
SC(t) .
to the fan
of nodes
in the
that the normal
term
the assertion
Let
sequences t l E t'm
so that
form
starting
from
by induction
t I ~ t* ,
t**.
Then,
to hold for
t, tl, t 2 .... , t I t ,
and ending
hypothesis.
t~ ~ t*
since
and
(by 2.2.25) ;
~(tl) < k,
8(t~) < k ,
t I ~t'm.
Computability
of dealing
~-abstraction
2.2.27.
denote
~(t) = k.
t I ~ t' , then
to a normal
t**mt~
(intuitive
for ~(k)= I.
and let
t I ~ t ~ , we can find a
let
~(t)
be two reduction
form.
be any term ; by 2.2.19
We prove by induction
~(t') < k, m
If
lemma).
t
is finite
determined.
t, t~, t~, ..., t' in normal
t
t.
is uniquely
Let
tree of
for theories
with the terms of
based on
k-conversion.
N - H A w , we define
a set of terms
as a primitive.
Definition
of
Tm' .
Tm T = U ITm~ I ~
TI,
where
the
Tm~
are
defined by Tm'
(i).
0 ETmo,
S E Tmlo)o,
Ra E Tm#
with
m = (m)((~)(O)a)(O)~
,
~ E T. Tm'
(ii).
If
x~
Tm'
(iii).
If
t E Tm# ,
If
t E Tm I~)T ' t' E Tm'~ ,
Tm t (iv).
is a variable
2.2.28.
Contractions.
"[x~]"
in the notation
free in
t o
then
"contr"
R ts 0
contr,
(d)
R t s (St')
(h)
(kx~.t[x~])t '
x ~ E Tm~.
then
tt' E Tm'T " a relation
between
refers only to the occurrences
relation
terms of
xa
is given by
t ;
contr,
bound in
~ , then
is in general
"t[x~]"
The contraction
(c)
of type
kx~.t E Tmla)m .
s(Rtst')t' ;
contr
tit']
if no variable
free in
tv
becomes
t[t'] ; a
(j)
kxa.tx ~
(k)
kxa.t[x ~]
2.2.29.
contr,
t
for
contr.
t
not containing
x
free ;
~y~.t[ya] .
~' > I "
Reductions
We define (strict,
standard)
reductions
t'
is obtained
from
t
by an
t'
is obtained
from
t
by replacing
according
to contraction
We write
t >I t'
if
rule t'
similar
~ - reduction
is obtained
(resp.
a redex in
(k) (resp.
(h),
from
to before.
t
We say that
B- , ~- reduction) by its contractum
(j)). t
by some
~ - reduction
if
112
followed by a contraction t > t'
according to (h),
if there is a sequence
--
tl, ..., t n
I I ti+~ for A term t is normal if t ~ t' -
(j),
(c), or (d).
such that
implies that
t'
We write
t ~ t l , tn
~
t !
is obtained from
,
by
t
reductions.
2.2.30.
Computabilit~
Predicates
and strong computability
"Comp"
In order to prove
and
SC(t)
k- based theories.
may be defined as before
for each
over a stronger property, tion
"SC"
for
(2.2.5,
to be called strong computability
under substitu-
(SC*) , which is defined as follows :
Defimt~on. t (notation
is said to be stron~l 2 computable under substitution SC*(t) ) if for each substitution
of appropriate of variables)
of strongly computable
types (renaming bound variables if necessary for some occurrences
term is strongly computable SC*(t)
of variables
free in
t , the resulting
(cf. 2.1.4).
2.2.31.
Theorem.
The proof is very similar to the argument in 2.2.19o
for each
t ~ Tm'
SC*(O)
is immediate.
(ii)
SC*(S)
is proved as in 2.2.19,(ii).
(iii)
Let
~ = (ml)...
terms
to avoid clashes
Proof. (i)
2.2.13).
t , we in fact have to use induction
(Cn)O.
(a) for all t I .... ,tn,
Then
SC*(x ~)
means:
t . E~. , t. 6 SC 1
I
(1
I
--
,
--
SC(x~t I ... tn) , and (b) for all
t,t I .... ,tn,
t. EG. , t. E SC 1
1
(1
,
1
t E ~ , t ~ SC , SC(tt I ... tn) .
(b) is obvious ~ (a) is proved *s in 2.2,19 ( 4 ,
(iv) sc*(~), (v)
Let
compare 2.2.19,
SC*(~[xq]) ,
obtained f r o m
(vi)a, (vi)b.
~[x ~] E (TI)... (~n)O,
kxq.~[x ~]
for some occurrences
and let
by substitution
of free variables,
kxa.t[x ~]
be
of strongly computable
and possibly
terms
some renaming of
bound variables. We wish to show that for arbitrary
t
E q , tiE ~i ' t. E SC O
0
SC((Xx~.t[xq])totl...
A reduction
sequence
of the form: ,.
~here
~
. (k)~
:~,.
t*(k)[= ]
and where
tn) .
starting from
(k°°t[=°])too°.tn (k)
. (k)
for
1
(~X~ot[X~])tetl °oo tn
, (~.t(~[~])t(ol),o.t~ ,.,(k)~.(k)~,.(k)
is obt*ined I rom
t~k~[~ :]
is either
I) .... . (k)
by ~ome
;(k+2)
+(k+])
~-~eductions,
113
(t.(k)[t(k)l~t(k)
t(k) ~ ~(k+1) {(k+2) ~ ( k + 5 ) n (t*(k)[ ~k)] sequence starting from t )t k) ... t (k) ; n quence is of the following form: (kxa-t[x~])to.--t n . . . . .
~(k+2)
or the reduction se-
(kx~.t(k)[x~)t; k) ... t~ k), (s)t~ k) ... tl k),
~(k+5) 9
9
"'"
where
t(k)t~°j~ is of the form
where
(s)t [k)
t(k)
o
from
is a reduction
"''z\ n
stkk I t ~ ... tkk] . o
n
~(k+2)
sx°, ~(k+}),
s not containing .
x
o
free, and
is a reduction sequence starting
"" The second form of reduction sequence is in fact a
special case of the first form, so that we may further restrict our attention to the first form. We see that the reduction sequence terminates, since SC*(t*(k)[x~]) , hence also
SC(t*(k)[t~k)])
since
SC*[x[x~])
SC(to)
implies
implies
SC(t~k)). 2.2.32.
Lemma.
t' ~ t*
If
t >I t' , t >I t" , then there is a
t*
such that
t" ~ t*
Proof.
Similar to 2 2 2 5 .
2.2.53.
Theorem.
Each reduction sequence starting from
in a term in normal form, which is uniquely determined
t6 Tm'
terminates
up to renaming bound
variables. Proof.
Immediate by combining theorem 2.2.31 with the previous lemma.
2.2.54.
Remark.
An alternative method for proving 2.2.32 is obtained by
adaptation of Rosser's method (cf. 2.2.20- 2.2.25) to the
k- calculus (as
in ~artin-LSf 1971 C, Barendre6t 1971 ; Hindley, Lercher and Seldin 1972 ; the adaptation is due to W.W. Tait and P. Martin-Lof). 2.2.35. HA w
Discussion and comparison of ~roofs of computabilit~ for terms of
in the literature. Tait 1965 proves normalization by means of quantifier-free
for a system of infinite terms~ into which the be embedded. HA c
This procedure is very similar to the consistency proof for
obtained by embedding
H~Ac
into classical arithmetic with the
Hanatani 1966 shows by cut- elimination for sequent each closed term system
¢o - induction,
~- terms (as in 2.2.27) can
t
of type
O
we can find a numeral
~H obtained by omission of induction from
w - rule.
calculi that to n
such that in the
N - HA ~), H ~ t = ~ o
(His
set of primitives is that of Speotor 1962). Tait 1967 introduces the computability predicate attention to "strict reductionS" (2.2.2).
"Comp" , but restricts
By means of the computability
114
predicate Tait gives a proof of normalization for closed terms of (the proof applies to open terms as well, as we have seen). his attention to strict reductions, resulting normal form.
~-H~ ~
By restricting
it is easy to show the uniqueness of the
(Tait 1967 is an elaboration of Tait 1963.)
The papers Sanchis 1967, Diller 1968, Dragalin
1968 apply b a r - i n d u c t i o n
arguments to suitably defined partial orderings. Diller 1968 shows computability for closed terms of type specified order of computation,
0 , relative to a
by means of bar induction ; it is shown that
the argument may be replaced by a transfinite induction up till the first - critical number, assigning ordinals to terms. Sanchis 1967 uses an argument very similar to Diller's bar induction, obtain a strong normalization theorem for the terms of to a weaker set of contraction rules;
to
~ - HA w , but relative
instead of (d) in 2.2.2 he only
permits (d')
Rtlt2(n+1)
contr, t2(Rtlt2n)n
(disregarding the fact that his
R
is in fact cur
kxyz. R x(Xvu.yuv)z) .
It is not obvious how to generalize his notion of successor of a term so as to be applicable to our stronger reductions. The proof of Stenlund
1971 is based on Sanchis'
proof, but Stenlund
restricts his attention to reduction sequences where a definite order is prescribed.
For yet another presentation,
see Hindle2,
Lercher and Seldin
1972. Hinata 1967 discusses the
X- version of
N - H A w , and assigns to each
term a tree, expressing the construction of the term from simpler terms. Let us call such trees construction trees.
The construction tree of a term
t
(which is not uniquely determined) has a term associated with each node ;
t
is associated to the end node.
ment (of ordinals
< ¢o )
Hinata then describes an ordinal assign-
to construction tree§, and a process for trans-
forming trees into other trees such that the term associated to the end node of the transformed tree is a reduction of the term associated to the end a
node of the original tree. formation process,
then
If~%ree
T'
is obtained from
T
by this trans-
may be assigned an ordinal which is lower
than the ordinal assigned to arithmetic extended with
T'
T.
The proof can be given in quantifier-free
¢ -induction. o
Hinata's contractions are almost the same as those described in 2.2.28. There are two differences: but instead uses
0tlt2t 3
recursively by • (0)
= t 2 ,
~(Sz)
= t3(~)z.
he does not introduce the recU~sor for
~(tl) , if
~
as a constant,
has been defined primitive
115
Then
(kx.t[x])(t')
is contracted
to
t[t'] ,
t 2... (t2(t2(Ptlt2t3)t})(St3))(S2t3)... form
binatorial
Hinata's
version.
n > O,
result implies a corresponding
contracts t}
to
not of the
Howard
R
by means of assignment
contr, n--~,
tractions
RG~
N - H A ~° contains
operator
k
recursor
R
and constants
contr. R n
Howard first considers
R~ts
by a restricted
respectively,
Next,
contr,
"restricted
opposed to strict reductions, t ~ t'
then
a non-unique
assignment
some ordinal
8
assigned
(h), R2
s
constants
for each m
~s
up to
co
N-HA ~
all numerals, S
n~ 0
contr,
with with con-
tn(Rnts)
reductlon S" , where only contractions •
from the "outside",
reduction,
and
~, 8
as
from the "inside">.
are ordinals assigned
to
~< ~.
~ to
of arbitrary reductions, to terms,
such that if
is any ordinal assigned t'
such that
to
Howard describes t ~I t'
recursive arithmetic
8< ~.
extended
theorem by
by quantifier-free
(for any primitive recursive well-ordering
satisfying certain adequacy conditions).
is an
t , then there is
This method yields a direct proof of a strong normalization means of primitive
of
successor
which involve contractions
of ordinals
and
the
to terms.
(i.e. contractions
to allow the consideration
arbitrary reduction,
¢o
k-version
as primitive
with contraction
of closed subterms are permitted
t, t ~
theorem for a
of ordinals less than
Howard's version of
permits
, as we have seen in q.7 5.
1970 proves a normalization
the abstraction
result for the com-
His rule of definition by recursion
definition of the recursor
If
if
St'.
Obviously,
S~
ptlt2(snts)
(snt3)
of order type
induction ¢o
§ 3.
More about computability° = = = = = = = = = = = = = = = = = = = = = = = =
2.3 • I "
in
A ~ + IE o ° -I - H ~'~
~-H~ ~
the following
Computability
Let us add to IE °
~Ets
= I
[ and
if
t,s
N-~
to
of contraction
~-~+lE
(e)
Eft
contro 0 ,
(f)
Ets
contro
2.5°2.
are closed terms in normal form
t~ s.
Now we extend the notions from
set of axioms
I
Theorem°
if
and (strict-,
standard-)reduction
o , by adding to the contraction
t,s
are distinct
All terms of
rules
closed terms in normal form.
~-I~ w+lE o
are standard computable,
hence
reduce to a term in normal form by a standard reduction. Proof°
~e only have to add to the proof in 2.2.6 an argument
showing
E
to be standard computable° (viii)
Assume
t,t'
to be terms reducing to normal forms
tl,t ~
respective-
ly by a standard reduction. Ett' If
reduces by a standard reduction tI
Etlt ~
or
t~
is not closed,
standard reduces to
standard reduces to 2.3.3.
Corollary ,.
to
t l~t~,
Etlt~ o Etlt ~
0 ; and if
tlt ~
is normal.
If
are closed,
t I ~ ~,
tl ~ tl ' Etlt~
Io In
I-HAW+
IE
~
every closed term of type
0
reduces
O
to a numeral. 2.3.4.
Remarks.
(i) The consistency
normal form is uniquely determined, - H ~ w + l E O ~ t =t" , t, t I, t" t ~ ~ t" . version) (ii)
The uniqueness described
~_~w+ t', t"
Corollary°
In or
!E
that the
IE ° ~ t =t' , in normal form must imply
I- ~w+
theorem (2~2.19)
(HR0-
and the proof
IE
~-~+
IE O , for closed terms
t,s
~t~So IE I .
are implied by the following O
requires
(i) and 2°3°2 we have
The equalit$ axioms axioms
o
of normal form is also insured by the model
The proofs of the strong normalization
~t=s
The
i.e.
closed~
in 2.2.23 may be extended also to
2.3.6.
~_}~W+lE
in 2.5.5.
As a corollary to remark 2.5.5.
of
stronger
set of axioms
!E I :
117
Let
~1(a)
tl,t 2
be two distinct terms with the same type taken from
the set of terms consisting of HaS,~9x1'
Z~10,~11,~12x2'
R~18x7x8 , for all IEI (b)
IE1(c )
~I' "''' ~18 E ~T"
- ~y~R~,y,
x~x'
- Zxy%~x'y'
If
~{~'
, then
Z¢3,a4,¢5,
S, R¢6 , E 7 ,
R~13x3' E~14x 4' Z~I5,~I6,¢ITX5X6 '
xJx, v y ~ y Vy%y'
~ I,¢2,
Ep,~,Txy%Ep,~,
Then
Tx'Y '
t I% t 2
is an axiom.
iS an axiom.
It follows from results in 2.5.9 and 2.5.10 that I-HA~ The
HRO-version
I-HAW+IE
O ~ I-~-~+
lEt "
described in 2.5.5 is also a model for
The proof of computability in
~_~w+
IE 1
~ - H A W + IE I . is also easily given, extending
the contractions in the obvious way. It has sometimes been argued that the rule IE is "unnatural", since it o seems to point to a confusion between "use" and "mention". This impression is mistaken,
and probably due to too much exclusive contemplation of the
term model. A glance at the the doubts,
HR0- version given in 2.5.5 may do something to dispel
since there not every object is denoted by a term ; on the other
hand, the even stronger set
IE I
shows that
IE °
is the consequence of
axioms who do not at all have the look of a syntactic criterion smuggled into the semantics. 2!.3.7.
Standard com~u.t.ab.il.ity of terms in languages with cartesian ~ro.d.u.t t2~e.
Let us consider
~ - p HA w , the conservative
extension of
N_WT~
defined
inio8.2. We add %o our contractions (g)
D'(Dtt') contr, t,
D"(Dtt') contr, t' , D(D't)(D"t)
and our concept of standard reducibility is correspondingly
contr, t
enlarged°
We extend the notion of standard computability by a clause (iii)
Comp~"XT (t)
iff
Comp~(D't)
and
Comp$(D"t) , and
T
is
normalizable by a standard reduction. Note that the existence of a terminating
standard reduction sequence for a
term implies the existence of a terminating standard reduction sequence for all its subterms. standard reduction"
Hence we may drop the condition "t in clauses (ii), (iii).
is normalizable by a
(This is seen by establishing
118
simultaneously
by induction w.r.t,
c
that (a)
O~E Comp~
and
(b )
* Comps(t) = t has a terminating standard reduction; here 0 ° =-0, 0(¢) • =- H T0 ~ , 0 ~)
with " t
2.2.7
is normalizable
by a standard reduction"
dropped.
Cf.
(il).)
Further we note that the following lemma holds for standard reducibility
The proof is entirely straightforward, using t"
by induction on the type of
t~, t, * D,t~, D1t , ~ D"t~' D"t' , and
has a terminating
standard reduction
sequence.
We extend the proof that every term of
N-HA w
is standard computable
N - ~ HA P ~ ; this requires consideration
the terms of
t,t' ,
t~v t ' = (tt"~' t't") , if
of some additional
to
cases
in the proof of 2.2.6.
Case ( i X )
(extension
Let
A
(2)
t~k~E A,
of case
(ii)).
be the following inductively
Comp"(ti) ,
(D"t)tl...t
n
e A,
l
defined class :
= (D't)tl...tnE
A,
D't, D"t C a.
Let
A denote the subset of A containing terms of type 0 only. We o readily see that d c C o m p " = Comp"(x ~) By induction over A we prove that O--
if
t E A,
then
t
has a terminating
taining a term of the form
Dt't".
standard reduction sequence not cona x this is obviously true.
For
Suppose the assertion to have been established ly also true for
also for Therefore
or
D"t.
t E A,
If it holds for
ttl...t n or (D't)t, .... tn, A c Comp" , so Comp"(x ~) .
or
t,
then it is obvious-
D't
(D"t)t 1...tn,
or
D"t E A,
t 1...tnE
then
Comp" .
c--
Ca_so_ ( x ) .
Let
~ormal ( l e n a ) .
s,tE Comp" .
~ow
~Comp"(D' (Ds't'))
using
D't
for
and
D,(Dst)_~, s,,
C_ase (x~].
Comp"(D')
have to show
Then
s ~' s',
t _~v t',
Oemp"(Dst) ,~(Oomp"(D,(Dst)) Comp" (D" (Ds 't' ) )) ~IComp" (s ')
D"(Dst)_~, t,, is established
Comp"(D't),
t_~, t,,
the
and
s',t'
Oomp"(D"(Dst)))*
and
Comp" (t ')),
lem~a.
as follows. Comp"(t~),
must h~ve a cartesian product as type,
s',t' E Comp",
and
Let t'
Comp"(t')
Comp"(t) ,
normal.
then we
Since
* Comp"(D't)
t'
and
Comp" (D" t) . Comp"(D") 2.3.8.
is established
Computability
similarly.
relative assignment
The concept is taken from Tait 1967.
of funct%pns. In the remainder
we restrict our attention to standard reduction
sequences.
of this section, For simplicity,
t19
we restrict
attention
to a single, assigned
to
x n contr, 2.3.9°
Theorem.
quence
~
additional
(i.e.
r e l a t i v e an a s s i g n m e n t of a f u n c t i o n I . If ~ is the f u n c t i o n
terms
the n u m e r a l
of
N-HA
to a term in
As compared
guished
am
All
relative
Proof.
to c o m p u t a b i l i t y
fixed type I variable, say x I x , we add to our c o n t r a c t i o n s
to o r d i n a r y
case to c o n s i d e r :
f r o m other
type
I
x
w
: representing
possess
~-normal standard
I
a standard
reduction
se-
form.
reductions,
there
must be d i s c u s s e d
variables.
a~).
W e have
is only a single
separately
to show that
and distin-
xlt
possesses O
a standard
reduction
tion r e l a t i v e tn
in
~.
form.
reduction
Remark
extension
of
If
tn
i.e.
possessing
is not a numeral,
The result
only,
t
o be a standard
for xlt ; o is a standard
~
(i). ~
for any
sequence
x lt o, ..., x!t n = x I ~ , 2.3.10.
~
to, t I , ..., t n
Let
~-normal
standard
relative
V x ( ~ x = 8x) ,
However,
of type
as a free variable,
0 ,
containing
manner
in
normal
form,
in
~-H~
~-H~
w
is a finite
contr.
~.
I
~ n = ~n
=
implies:
(il).
The p r e c e d i n g
simplicity
,
standard
1
t
where
we u s e d
forms
is a term
represents
definable
for if we reduce
sequence
a type
in this t
to
~-
finitely
k.
and
~ - normal
was m o t i v a t e d
etc.,
denote,
t
extended
are ~ u a l ,
to the a s s i g n m e n t
1 variables ;
our r e s t r i c t i o n
concern
this
of to
of n o t a t i o n a l special
case.
of computability.
by the d e v i c e
of g~delnumbeein~
we are led to c o n s i d e r
on c o m p u t a b i l i t y
of
by c o n s i d e r a t i o n s
all our a p p l i c a t i o n s
Arithmetization
forms
m a y he r e a d i l y
set or to all type
since
reduction
"SRED"
For if
kxl.t
on the
8 - normal
Then o b v i o u s l y
discussion
If we a r i t h m e t i z e
otur results Let
~-
I variable
only,
2.3.11- 2.3.13. 2.3ollo
the
to a finite type
continuous;
depends
I
which
a single
and
2 functionals
reduction
n = maxIni I I < i < k l + I.
functions
~,
is a
of
1x n. Let
~-
we can say more.
N o w the type
are o b v i o u s l y
there
many instances
xI
w.
relative
xlto , °.., X l t n
obviously
the
of all terms will be equal.
2 functional
reduction
reduc-
if t is a numeral, say ~, then n r e d u c t i o n sequence r e l a t i v e ~.
of the r e d u c t i o n
if
a standard
can be f o r m a l i z e d
as before,
the concepts
the q u e s t i o n in
an a r i t h m e t i c a l
HA
or
of
of h o w m u c h of
N _ ~w.
Z oI _ predicate,
such that
120
SRED(~tl, rs ~) quence from
intuitively t
to
s"
expresses : "there is a standard reduction
(so
"SP~D"
provability predicate for a fragment of provability
predicate
t1=t2, t2=t3,
Now inspection icate
(i) and (ii)
the proof of
tl,t2,...,t n
representing
a
successively).
such that
uses essentially
version of
term) would be provable in enumeration
expressing
that
Now let
such that
-
HA ~.~w , then,
n
(where
f(x,y)
It would follow from the assumption that
since
principles,
Term(n)
is the
is the gSdelnumber
to,tl,t2,t3,t4,..,
of closed terms of type I ! let
tive recursive function
N
only arithmetic&l
Vn(Term(n) ~ Comp"(n))
~.
pred-
(the arithmetized version of)
(cf. 2.2.5) would be provable in
primitive recursive predicate
recursive
sequence
..., tn_1=tn
~t Comp"(t)
the arithmetized
is a restricted
of the proof of 2.2.6 shows that if an arithmetical
"Comp" " were definable
clauses
N - Hv~ A ~ ; "SRED"
in the sense that it concerns proofs in a certain
standard form only : a reduction proof of
se-
has the character of a restricted
of a
be a primitive
f(x,y)
be the primi-
= rS(txg)U. Comp"
was arithmetically
definable
that
hence especially
S R E D ( f ( x , x ) , r ~ ~) o
W V x Z~z
Therefore we can find a provably total recursive by a closed term
t
function
~,
represented
of type I (cf. 5.4.29)
ff Vx SRED(f(x), ~ ~ )
.
Since on the other hand it will follow from 2.3.13 below that ~SF~D(f(x),
rY~)
we obtain a contradiction Hence our assumption false. Comp~ of
HA
~-~ t ~ + I = tx
(take
that
t et
Comp"
)o x was arithmetically
definable must be
The intuitive reason for this is rather clear : if we arithmetize for
~
of increasing
representing
Comp~
level,
the logical complexity of the formulae
increases
indefinitely.
At the same time it is clear that if we restrict ourselves tive set of all terms constructed ~n ,
this restricted predicate
As a consequence, ble in arithmetic,
from constants and variables Comp n
is arithmetically
for each given closed term or even more generally,
if
t
of type level
definable.
its computability t
free, and does not contain other free variables, ~ VxI...x n ~f
to the applica-
contains then
SRED(~t(~I .... ,~n)~ f ~l) .
is prova-
Xl,...,x n 6 0
121
Still more generally, level for
SRED(~, n, m)
ity relative to 2o3.12.
computability
(w.r.t. a fixed type
p ~ s:~D(~, ~t(x I '
-~ where
if we arithmetize
~-reductions
expresses in
,-
;1
~
="
N-HA w
with bounded type
I variable 1
r
n)
x I ) we obtain
~)
'
arithmetically
standard reducibil-
~.
Standard godelnumbering.
tions about the (standard) and certain extensions. (and variable)
c o
We find it convenient
godelnumbering
to make some assump-
to be used for terms of
Let a code number
~
N-HA ~
be assigned to each constant
Then the code number of the terms is defined inductive-
ly as follows : (i)
If
t
(ii)
If
t ~ tlt 2 , then
2.3.13. type
is a variable
Theorem.
0
Let
or constant, rtu=
t be any term of
variables and (possibly)
(I)
~-~®[-
Proof
I variable
constructed from constants,
I variable
0 variables
x I ; let
occurring free in
xl,...,x n t.
Let
N - HA w . Then
t(~,x I . . . . . xn) = y*--> S~D(~, q ( x l , ~ I . . . . , ~ n ) ' , ~ ~) o
(W.A. Howard *)o
We define for each type
is not the godelnumber
false.
°
N-HA ~
the type
of
a E T,
ing : the term with g~delnumber x
r t u _= (~) o
rt1~ * < r t f >
be a list containing all the type be another type
then
a binary relation x
VALu(x°,y u)
has the functional
of a closed term of type
o,
y~
express-
as value.
~T&L~(x°,y ~)
If is
The definition is as follows.
(i)
VALo(x°'Y°)
(ii)
v ~ ( = ) ~ (x°'y(~)~)
=def S~::(~, x, ~:~) ~_
O
dsf v~ u (VAL=(~,u)-. V~L (~.<~>, y u ) ) .
Note that
(~)
VAL(~)~(x,y) &VAL(x',y') ~ VALT(x. <x'>, yy,)°
We also need ~m~j_~o
(a)
VALT(x,y T) ~ ~Lx'[SRED~,x,×') &Normal(x')]
(b) VAL(~O~ , 0~) which is readily established is the arithmetization We now establish variables
x,
and
simultaneously,
by induction on
T .
(~Normal"
of ". ° ° is in normal form"°)
VAL(<~ >, c)
for constants c, VAL(r~ I, x) for numerical I I variable to if x is the fixed type
VAL(rx In, ~)
which the function ~ is assigned. ~--~ ...... ) n a letter dated Nay 18, 1972.
122
(a) (b)
~-~ Assume
b Vx(VALo(rXl'X)) VALo(Y,Z ) , i . e .
is immediate. SRED(~,y,r~I) . Then
i.e° VALo(~X1~., ~ ) , so VALI(~xI%~). (c) VA~o(~1,yI) -- VA~o(~S% <~I>, ~yl) is obvious SRED,
(d)
hence
~ ~ ( ~ 1 ) ° ° . (~m)0,
for
I< i < m °
VAL ( x l , Y l ) ,
by the properties of
VALT(x2,Y2) ,
VAL=i(xi+2,Yi+2)
VALo(x I. <x3,°..,Xm+2> , yly3. o.Ym+2) , hence
since SK~D(~,x I. <x3,x4,o.o,xm+2> , yly3ya°..Ym+2) , also SRED(~,~ ~,~ ~ . <Xl,X 2 ,°..,Xm+2> , H ,TylY2Y3...Ym+2)° Here we have also used that VAL (x2,Y2) implies ~x(SRED(~,Xz,× ) & N o r ~ (x)) , by the lemma. v ~ ( ~ z , ~ u , Z ~,~,~) is proved similarly.
v A ~ ( ' ~ j , H~).
~or notational s i m p l i c i t y ,
establish, by induction on
(3)
,
VALI(rS~,S) o
Let
Then by our assumptions
(e) (f)
SRED(~,rx11* / ~ )
Zet
~ ~ (~)0~ we f i r s t
y
fWoYJlYl(VAL~(%,Yo ) ~VAL(~)(0)~(xl,Y ~) VAL (rR~. <Xo,Xl,r~>, RYoYlY))
Ba~i£o Let VAL~(Xo,Yo) , VAL(~)(O)~(xl,Yl) . We wish to show VALo(rRu* <Xo~Xl,rOn, x3 >, RYoY I0y3) for all xS,y 3 such that By our hypothesis, VALo(X o* <x3> , yoY3) o By the properties of SHED ,
VALT(x3,Y3).
SRED(~,rRl.<Xo,Xl,rO 7, x3>, YoY3) i.e. SRED(~,rH~.<xo,xl,r0 ~, x3> , RYoY10Y3). Induction step° Assume (3) ; we wish to establish (3) with Sy instead of y. With the induction hypothesis, the argument is as straightforward as before. Now we are ready to show (4)
I VAL~(%,Yo) ~ VAL(~)(O)~(xl,Y I) & VALo(x2,Y 2) & VALT(x3,Y 5) VALo(rRT* <Xo,Xl,X2,X3>, RYoYlY2Y 3) -
Note that VALo(X2,Y2) implies SHED(~, x2, Also, by (3) applied to Y2 for y SHED(~,rR ~.<xo,xl ,r~2 7, x3>, RYoNIN2N 3) and therefore by the properties of
SHED
SRED(~,rHl*<Xo,Xl,X2,X3>, RYoYlY2Y 3) °
This establishes (4), and hence VAL(rH~ ,R ) . By (2) and Ca)- (f), i t i s o b v i o u s t h a t (1) h o l d s . Remark° Since standard reduction sequences correspond to a (very restricted) class of proofs, (I) may be viewed as a (weak) uniform reflection principle°
~25
§ 4.
N[odels based on partial recursive
function application : HR0, HE0.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
2.4.1- 2.4.5. 2.4.1.
General remarks on models of
Nodels ~ normal~
A model for variables,
~-HA ~
extensional
models.
is given by specifying domains for the range of the
and interpreting
the constants,
models are models w.r°t, many-sorted icate logic with equality.
tensional
~-~
Notation.
If o
(of type (N(o)T X N No
(So
N - HA ~ ,
as
cN
-N-HA ~
, let then
denote a binary operation
) representing
x
be its interpretation
(~)~
in
y
N.
(of type
o)
If
c
N
denote the
(~)~
to type
o)
is a constant of
We abbreviate
Ap(x,y)
also
xy.
Let
N, M
be models of
~, ~.
N
is an embedding if is a submodel of
O,T
striction of
ApN
constants
of
c
Definition
For any model
N
~
maps
, ~(C N ) = c M
if
to
N(o)T × N o , for all
N°
into
I~
for all
for all constants
c
of
Noc_~ ~
for all
o 6 T,
and
APN
~,T6 =T , and
is the reO N = cM
for all
N - H A m. (of the extensional of
(ii)
~or
x,y~ ~(o)~ : x ~ y
is interpreted
equivalence
relation
~).
N-H~ A w , we define by induction over the type structure :
x,y6 N o :
E - H A w.
if
N,
For
=
M,
%0 is a bi-unique homomorphismo
(i)
model of
into
~(Ap~(x,y)) = A p ~ ( ~ , ~ )
and
2°4.4.
embedding°
N-~mo
q0 is a homomorl~hism from
If
application
interprets
Submodel I homomorphism~
N
i.e. the
0 ) receive their standard interpretation.
APN'T
2.4.3.
~
1.6.12).
w - models,
is any model of
A~IA(~)~, y=)
let
(Cfo
ex-
in the model, and let )N
N
of
The model is called
in the model as the (definable)
studied in this chapter are
(objects of type
objects of type
in
the model may be called normal°
equality between the elements of the model
natural numbers 2.4.2°
since the interpretation
is interpreted by the identity in the
if equality is interpreted
All models of
Hence our
logic, not always w.r.t, pred-
systems studied.
If the equality relation ofN-HA m
extensional,
includin~ equality.
predicate
This is only natural,
equality varies in the different
domains of the model,
N - H A w.
: x~y
by
~
=def x = y
~def ~ ~ No(~ ~yz). in the model,
N
is extensional,
and is then a
124
2.4.5.
Theorem
(Zucker 1971, § 8).
there is a standard procedure E-[~ w p~
which is almost a submodel = Ap M
CM~ = CM
l(~(a)T X ~ )
( I
for all constants
If
M
is a model of
for constructing of
M,
i.e.
~E
except equality,
model
~E
of
satisfies
expressing restriction)
c,
N~- HA ~°~.~ , then
an extensional
and
~
~ ~
,
whereas for equality
x =M y ~ x =M~ y . Proof.
~e define binary relations
I~(~,y)
If we put
then
ME
on
of application
in
to
and interpret
of
For example, x6 ~,
for
2.4.6.
N ~ - - ~ I{A , w , cM
consider
The classical
coincides with
E-HA w
~
for
~
as
R' ) .
N,
, by a straight-
z6 ~0 , it follows that
We have to show that R~xyz6 ~ .
set-theoretical
model of
We prove this
~-HA~.
applications),
model of
the full set-theo-
S . Z F - set theory,
in
ZF
w,
and identify the natural numbers with
we may define the set-theoretical
in an obvious way ; objects of type
all set-theoretical
mappings from the set of objects of type
of objects of type
T ; the objects of type
2.4.7.
as in
N E , which is straightforward. ~
(abbreviated
a standard set, say the ordinal of
as the
ME
sake, we mention here the most obvious classical
If we consider say
S
~E
in
V~e also have to verify that for
belongs to
(for which we have no interesting
retical model
for
constants
z.
For completeness E-~
(R)~
y6~a)(O)~,
by induction on
~, as follows:
~°
}~E I'(~)T
forward induction on the type structure. c~
, by induction on
and we define application M,
becomes a submodel of
When restricted
constants
M
=-def x = y
x) =def l~(x,x) ,
restriction
I'
0
model
(~)T
are then
a
into a set
are the elements of
w.
Models based on partial r ecursive function application.
The models for
~-HA ~
described in the remainder of this section are
based on partial recursive function application (denoted by K l e e n e - b r a c k e t s : Hereditarily
between natural numbers
I.l.). Our basic models are
Recursive Operations
and the Hereditarily
HR0, HE0,
the
Effective Operations.
Later on, in section 6 we shall describe analogous models based on continuous function application and Kl___eene 1969; 2.4.8.
we use
Description
of
"I"
(written as instead of
I~}[8]
in K1eene and Vesley 1965,
{. I[.] )o
fIR0.
We first define, for each
6 T , a set of natural numbers
V a , as follows.
125
Vo(x)
--def 0 = 0
V(~)T(X ) -=defVy6 V a Zz 6 VT({x}(y)--z) , or equivalently
Vy ~ v ~u(~(x,y,u) avT(uu)) Now the hereditarily
recursive
(x,~)
with
Since we may assume
num~
hereditarily
x6 V.
recursive
operations
of type ~
a
consist of all pairs
to be represented
by a natural
operation~ may be supposed to be represented
by natural numbers. Application is partial recursive
function application :
(x, (~)7)(y, ~) = ({x}(y), 7) o HRO
becomes a model of
IS], [E]
-I
HA w , if we can f i n d
such that, if we abbreviate
{tol(tl,...,t
numbers
[H], [Z], [R],
{... {{{to}(tl)I(t2)}...I(tn)
by
n) ,
{[n] i(~,y) {[z]}(~,y,=) -{{~}(~)}({y}(~)) {Is] }(~) " s~ {[R] }(x,y,O) {[~] i(x,y,S~) - {y }({[R] }(~,y,z), ~) {[E] } ( x , y )
- ~glx-Yl
Such numbers are constructed
.
as follows.
We put
[hi = ~. I s ] = ~. [~]
To construct
sx ~(l=-yl)
= ~ .
a number
formalization
JR] , we may either use the fact that in Kleene' s
of recursion theory (Kleene 1969, § 1.1) definition by primi-
tive recursion is included,
and combine this with the fact that the rule of
definition by recursion permits us to construct a uniform recursor Io7.5),
sion theorem, noting that there exists a partial recursive 9(u,x,y, z)
function
such that
*(u,x,y, o) ~" x ,(u,~,y,S~) ~ {y}({~}(~,y,~), ~), hence by the reoursion
~([R],x,y,z) Then
(cf.
or, if we do not wish to use this fact, we may appeal to the recur-
H~,T' Zp,~,T' (In],
theorem we find a number
" {[Rlt(x,y,z) o
S, E~9 Ra
(~)(7),a)
are represented by
[HI
such that
126
([z], ((~)(~)~)((p)=)(p)T) (Is], (o)o) (Is], (=)(=)o) ([R], (~)((~)(0)~)(o)~). Thus
HR0
Ea, R~
is a model of
([t],s)
2.4,9.
O
t~ t
~_~w in
Remark on terminolo~z.
specifying
(0,0) , and
by
of
represents
if we interprete application,
HA ~
as indicated above,
To each closed term that
I
=
H, Z~ S,
by identity.
we can thus find a number
HR0,
HRO
such
and
beeomes a model if
[~], [Z], [S], [R], [El .
[t]
~_~e
only by
However, to avoid ponderous circum-
loeutions, or the introduction of special designations for the variants, we shall talk somewhat loosely about
HRO
wish to refer to a specific choice of speak of a "version of
of type
is an extension of ~
and if we
HRO, HRO~-I~ e
in which it is asserted that the objects
coincide with the hereditarily recursive operations of type
The language of 6'
~-}~e"
HRO"
2.4.10. The formal theories HRO
as "a model for
[H], [El, [S], [El, [R] , we shall
HR~0 is obtained by adding constants
6 ((~)T)(~)O
for
all
~,T6 T
to
the language
axiomatized by addition of the following axioms to o
= x
I-HA e .
HRO
is
i_~e:
o
Glo
~ x o
G2o
~ x a = ~ y ~ ,__) x ~ = y a
G4.
~TX(~)TY~ = U(~$,TX(~)TY~)
GSo
vx
e v ~J(%
G1 - 4 e x p r e s s
of
~.
@ 6 (~)0,
that
(T, U
as in 1.3.9 A)
y = x) o all
objects
of finite
type are hereditarily
recursive
operations ; G5 expresses that every hereditarily recursive operation is an object of finite type. HRO by
is a model for
(A~.x, (=)0)
and
2.4.11. Description of We define, for each valence relation
I~
HRO-
is obtained by deleting G5 from
H~RO. ~' ~,T
HRO.
To see this, we only have to interpret by
(A~ay
o
min Z T(x,y,z)
'
((~)~)(~)0)
@~
o
HEOo a E T,
a
set of natural numbers
as follows :
Wo(X) ~aef X = X
, Io(X,y ) ~def (x: y) o
W
and an equi-
~27
W(S)T(x) ~def V y 6 W s Zu(T(x,y,u) & W T ( U u ) & & Vy6 W~ Vy' 6 W Vuu'(I~(y,y') & T ( x , y , u ) &
T(x,y,,~,)- I(~u,~u,)). z(~)~(x,y) ~def W(=)~(x) &W(=)~(y) ~ Vz~% Vuu'(T(x,~,u) ~(y,~,~')-- l (~u,~u')). T
Now the hereditarily effective operations pairs
(x,~)
with
x 6W
If we interpret application,
=0' 0, S, H, Z, R
we have obtained a model for ity between objects of type Remark (i)°
An extension
not exist.
With
HE0
(HE0)
of type
~
consist of all
°
~-H~AWo
Iv
as in the case of
HR0 ,
corresponds to extensional equal-
= ° HEO
or
as a model,
HEO- , analogous to ~I
HRO
or
HR~0
does
should then assign a g~delnumber to
each object of type I, such that Vxly1[VzO(xz=yz) -- ~Ix= ~ly] and this would make equality between objects of type I recursively decidable, which is well known to be false. (ii).
For each closed term
Note that the
It]
2.4.12. Theorem. for
t E ~ , as in the case of
assigned for HR0 E
and
HRO, HEO
HE0
HR0
are the same.
are distinct°
In fact, if we write
Proof.
W 0 = W~,
W1 =
W~,
~
for
~ > 3 •
wn
W~ = W0,
W~ = W I
W 2 = W~,
W 3 • W~,
W n ~ W~,
is obvious.
sI~)o(X'Y) her x~ V I ~ y~ V I ~ ~nv(I~(u,v)--{x l(u) " {yl(v)) valent to
x6W~
I(1)o(X,Y) Hence, since Let
x6W~
z 6 W~,
Since
which is equl-
& y 6 W~ & Vuv(I1(u,v ) - {x}(u) ={y}(v)) , which is in turn
@-9 I~(x,x) <--->I2(x,x ) <---)x6W2 , it follows that
W2=W~.
then by definition
z6 V 3
z ~
W'
HRO~ , we have
vuv(z2(u,v)- {z }(u) -{z }(v)).
W 2 c V 2 , it follows that
W3° We note that
Now we construct
is defined for all
Ax.x6 V 2 - W 2 . x0
{x01(x ) ~ 0 {X0}(X )
{z}(w)
such that for
x / Az.z ,
undefined for
x :
Azoz
.
w 6 W 2 , hence
128
Then
x 0 6 W 3 , but
xO{W~
,
since this would imply
ix O}(As.z) i s undefined, This c o n s t r u c t i o n may be g e n e r a l i z e d ; l e t u O / u I , {u O}(x) = {u 1}(x) : 0 f o r a l l nZl , Ax.X~Wn+ 1 f o r n z l , since as before,
x 0 E~n+2-W'n+2 '
N o w we construct
{
but not on
for all Therefore,
Xo{Vn+2 "
such that
x I e W'n - W n
for
n> 3 ,
Are there m a t h e m a t i c a l l y
HRO E , but not in
2.4.13. D e f i n i t i o n
HEO ,
of
or in
since
xI
is defined on
in the future,
terms c o n t a i n i n g
type 0
w i t h infinite
complement.
(m [t]HEO )
( f o r terms
it is simplest
variables
onto the type 0
interesting
Let
functionals
HEO , but not in
[t]HR0 , [t]HE0
For a p p l i c a t i o n s
[t]HRO
x , then, since In(Uo,Ul) {Ax.x}(Uo) / {Axox}(Ul) .
Vn_ I ,
~n-1 "
Open prqb!emo
variables
c ~. Uo, u 1 be two numbe~s~suoh t h a t
Ix11(x)~o for x/x o Ix11(xo) undefined.
Then o b v i o u s l y
in
xI
x O e V 3 , whereas
t
of
F
denote
attention
infinite
set
to V
some I - I m a p p i n g of h i g h e r type
not in
V.
We then define
by i n d u c t i o n on the c o m p l e x i t y o
E-~m)o
if we restrict
from a fixed recursive
variables
w h i c h occur
HR0 E ?
of
t,
as f o l l o w s :
a
[o]H~ o ~ o, [~°]H~ o ~ ~ , Ix ]~Ro ~ rx~(~/°) [s]H~ o ~ I s ] , [~]~o ~ [~]' [ n ] ~ o ~ In], [z]~R o ~ [ z ] , (where
IS],
[R], [ ~ ] ,
[Z],
[E]
are chosen as in 2.4.8)
[tt,]~ o~ {[t]~o}[t,]~ [St']~ o ~ S[t']~ o 2.4.14.
Theorem
variables). free.
from
=less
(Provable f a i t h f u l n e s s
Let
t
be any type
0
t~S~
of HRO , u n i f o r m l y
in type 0
term c o n t a i n i n g only type O
variables
Then
~-~m Proof.
o,
and
b [t]HEO ~ [t]HRO ~ t .
For closed 0, S , and
t ,
[t]HEO
is r e p r e s e n t e d by a p s e u d o - t e r m
constructed
I.l .
Let us assume a g ~ d e l n u m b e r i n g
for such p s e u d o - t e r m s
follows :
to" = j(o,o) rSt ~ = j(l,rt I) rlt}(t')~ = j(2, j ( r t ~ , r t ' 7 ) ) .
to be given,
e.g. as
129
gnpt(x)
is the primitive recursive predicate which holds iff
x
is the
godelnumber of a pseudo-term of the dsscribed kind. A binary predicate
A(x,y)
A(x,y)
e~n be explicitly defined (Ch. I, § 4) such that
~--~[x : j ( O , O ) & y : 0] v
V [JlX= I & ~z(A(J2x,z ) & y = S z ) ] V
v [j~= ~ & m~w(~(j~j~,u) &A(~2j~,v ) ~ m u w & ~ = y ) ] . A(x,y) - gnpt(x)°
Obviously Compl(x)
is the primitive recursive function such that if
Compl(x)
is the complexity of the term represented by
One readily proves by induction on V~'[A(x,y) Let
t
and
{.}.
.
be a pseudo-term constructed from ~
for
x.
Compl(x)
& A(x,y') -- y = y , ]
Let us write
gnpt(x) , then
0, S , type 0
variables
Xl,...,x n
t(x I .... ,~n) o
Then (I)
HA ~ V y ( t ' y ~-*A(r{~,y)) .
Proof by induction on the complexity of
t.
Vy(O ~y~--~A(j(O,O),y))
is obvious.
Vy(x i~y~--)A(rxi~,y))
is readily proved by induction on
Let
Vy(t' ~ y ~ - @ A ( ~ , ~ , y ) )
t mSt',
and assume
A(rS~'~,Sy) ~--)A(r{'~,y) , hence Finally, let Assume
X.. l
.
Vy(St' ~ y +-gA(rS{'~,y))
o
t ~ {t'}(t") .
V y ( t ' ' y e-~A(r~'~,y)),
Vy(t"'y
.
Now
~--~ ~ u ~ [ A ( ~ ' ~ , u ) ~ A(~"~,v) ~ ~ ~ ~=y] ~ ~w = y] *-~ I t ' l ( t " ) =" y .
A(~{{ ' l(~"f,y) ~u~[t'
~'u a~ t" " v & ~ u ~
Thus (I) is proved. Now we define by induction on the type structure pseudo-term with godelnumber y ~") :
(i)
Into(X O ,yO ) ~ A(x,y)
(ii)
Int(~)T(x,y ) ~ Vx'y'[Int
x
is the
o
Int~(x ,y )
HE0-interpretation
(x',y') ~ Int (j(2,j(x,x')),yy')] .
Now we prove, entirely parallel to 2.3.13, that for terms structed by application from constants and type 0
variable
t(x1,...,Xn)
p
{
is an abbreviation for
t(~1,...,~n).
Combining (I) and (2), we find for type 0
terms
con-
Xl,...,x n , that
~ - ~® ff Int([~]~mO,t)
(2) where
("the
of the functional
t(xl,...,xn) of ~ - ~ :
A([t]~0,t) ~-~.~
A(r[~]HEO,t) ~
[t]}~0 ~ t,
which together yield the assertion of the theorem. 2.4.15.
Corollary.
All closed type I
recursive functions Proof. that
Let
t
of
be a closed type I
H~ ~ [t] 6 V I
terms of
N-HA w
represent pro~ably
HA. term;
(cf. end of 2.4.8).
then there is a numeral Also
~ - ~
It]
[tx]HRO ~ tx,
such where
[tx]HR o ~ {[t]}(x). 2.4.16. Generalization. based on with
In
HAc
we can easily define a version of
A - partial recursive functions
IxlA(y)
taking the place of
2.4.17 . Historical
note.
HR0
instead of recursive functions,
Ixl(y) °
HEO , for pure types, is described
in Kreisel
1959, p. 117. A
(form of)
formulated
HR0
first appears in Kreisel
in the theory of combinator~
p. 191 , lines -10 to -2. use of it in Troelstra 2.4.18.
Troelstra rediscovered
variant of
HR0
satisfying
N-HA w
with the
is most easily formulated by introducing we require reflexivity,
symmetry,
is defined syntactically: t~t'
~ t"t~t"t',
t .>t'
~ kx.t . > Xx.t,, .
are distinct
is similar to
and made extensive
Rtt'0 . >t,
conversion.
k- operator as a primitive,
(y
eq~,~litW as follows:
and monotonicityo
t~t'~,Jt'~t"
~ t~t",
not free in
t~t'
t ) ,
Rtt'(St") >t'(Rt~t")t",
closed terms in normal form, t = t'
if
t
E~tt~ I
reduces to
if t'
Reduction ~ tt"~t't",
(Xx.t)t'~[xlt']t,
E t t , >0_ t
if
t,t'
is closed, normal.
Otherwise
X~_~w
I-HAWo K~-HA w ,
We consider the introduce
t~t,
~-
intensional
transitivity
kx.t~ kY[xlY]t
Finally we add a schema
A model for
HRO
A variant
in Tait 1968 ,
1971.
Sketch of a variant of
An intensional
1958 B, lecture 60.
is briefly indicated
similar to
XK- ~
HR0
calculus,
~ 8 - conversion as
~-
can be obtained as follows° with an additional
co~stant
~,
conversion and in addition a rule of
and 8-
conversion : ~tt v cony [I,K]
if
t,t'
~tt
if
t
conv
I
are distinct closed terms in normal form
is closed, normal,
where i mdef kx.x, U mdef kxy.x~ lit't2] mdef Xz'ztlt2" For such a system the Church - Rosser theorem of uniqueness of normal form is provable
(Cfo C u r r y - F e y s
1958 , § 3D.6, chapter 4)~
131
We put
0 -= I , Obviously,
n + I = [_~,K],
Sn > n + I o
can find a term
R
_S --- ~ [ x , ~ : ] o
H.Po Barendregt has shown (Barendregt C) that we
such that
_~x~o_ = x
_~y (s~) = y (~_~yz) z R
is in normal form, and when
Rtl, Rtlt 2 We now define our
H R 0 - analogue
~(t t E V 0 =def teV
The objects of type
~
(rI~,0)
as follows°
closed,
are now pairs
grE ~ - ~,(~)(~)0)
successor,
X-HR0
normal,
of
R
are in normal form, then
-=_~)
T =def t
V a o Obviously,
tl,t 2
have a normal form.
Vt' e V ~ t " e
(x,a)~ x
E
,
tt'=t")
a (godelnumber of a) term
is going to represent
represents
V
0,
~frS~_,I)
represents
(rR~,(a)((~)(0)~)(0)~)
represents
o
Another possibility for constructing a add to the language of the
X-calculus
satisfying the reduction rules Ett' cony SO Rtt'0 cony t,
if
t,t'
four additional constants
Ett conv O
are distinct,
Rtt'(St")cony
H R 0 - analogue is the following :
if
t
O,S, E,R,
is closed, normal, and
closed terms in formal form,
t'(Rtt't")t".
Abbreviate
SO
as
I,
Sn
as
n+1. Extend now the C h u r c h - R e s s e r
theorem to this extended
k-calculus,
and thsa
proceed as before. 2.4.19. Pairing in
HR0, HE0 .
It is easy to extend
HR0, HE0
to models for
~-~A:,
E - H~Ap ~
by adding
to the definition
V x~(X) ~def v~(jlx)
~
v~(J2x)
and s i m i l a r l y ~X~
( x ) ~def % ( J l X )
& ~ ( j 2 x)
and
~ x ~ (x'y) ~def i~(Jlx'jlY) ~ I~(J2x,J2Y), ~d representing ~, D', D" by (A~y. j(x,Y),(~)(~)~X~), ( ~ . j l x , ( ~ X~)~), ( ~ . j 2 x, (aX~)~) respectively. The models so extended we shall usually also denote by
HR0, HE0o
~32
§ 5.
Term models of
2o5.1. N_ - H~A
N - H ~ m.
Definitions. , and
~-~®.
CTNF
We put
CTM
Let
CTM
the set of closed terms of type
C~M=U{CT~I~C~I,
b e c o m e s a model of
CTM~ , we interpret equality
=
[-~,
application
over
becomes a model of
CTNF
, application
of
f o r m such that
is_> t' , =~
CTNF ) , and the constants A g a i n we denote
in
in n o r m a l form in
if we let the v a r i a b l e s t
to
s
x
O, S, ~, Z, R CTN.
t, s
is i n t e r p r e t e d O, S, H, Z, R
ts ,
as themselves.
if we let the v a r i a b l e s
a s s i g n s to
range over
as j u x t a p o s i t i o n
and
this model also by N - H~A ~ ,
Ap
¢
¢
CTNF=UICTNF~I~}o
as e q u a l i t y of normal form,
Let us denote, for simplicity, CTNF
be the set of closed terms of type
the term
x
t'
range
in normal
as proper e q u a l i t y
(equality in
are i n t e r p r e t e d by themselves.
this model by "CTT~F ''
Note that for the p r o o f that
CT~I, CTNF
are models of
to make use of the fact that every term of
N-I~ ~
N
-
possesses
, we have
~
a u n i q u e normal
form. 2.5.2.
Definitions.
-HA w
Let
and the normal
.I - H~A ~ , also denoted by
CTM, CTNF °
2°5.3°
Some p r o p e r t i e s
(i)
In
of
CTEhris primitive in
be the closed terms of type
HA
¢
of
CT~F, :O{C~F~I~C~}.
models
recursive
CTNF~
closed terms of type
c~M, = U { C T ~ I ~ } , into models of
CTM~,
CTNF,
recursive,
=~
CTNF I ,
of
respectively.
CT~', CT~'
CTMt,
CTM, CTM',
~-H~ w
a
can be made
similar to the
CTNF' . is recursive,
(for standard godelnumberings)o
but not p r o v a b l y
The second a s s e r t i o n is
e s t a b l i s h e d by a w e l l - k n o w ~ type of diagonal argument : let h denote the th x closed type I term ; the e n u m e r a t i o n may be supposed to be primitive
x
recursive
in
f u n c t i o n of that
x o x .
f(x,y) = 0
n o r m a l form, Then
The
I
Suppose if
t
x,y
f(x,y)
of)
h~
is again a primitive
is a p r o v a b l y r e c u r s i v e
are g ~ d e l n u m b e r s
In
~)
is a provably recursive
(Cfo 3o~-~9) ;
CTNF,=
provably recursive for
function
such
of closed terms with the same
say
t ~h~o ° Now
f u n c t i o n of
is p r i m i t i v e recursive,
x , denoted
h ~ o Y o = 0~--~
I ~ f ( r h g o ( ~ O ) ~ , r O ~ ) = I < . - ~ h 9 0 9 o ~ 0 ; contradiction. (ii)
recursive
elsewhere.
I ~ f(rh~,
by a term
(godelnumber
Similarly for
CTM' o
a p p l i c a t i o n is recursive,
but not
(for the standard g ~ d e l n u m b e r i n g s )
in
HA.
Similarly
is e s t a b l i s h e d
in an even
CTNF'o In this case the non-provable
recursiveness
133
more straightforward
way, utilizing
(iii) The domains of the variables (in contrast to (iv)
QF-AC
Proof.
Let
co n
provably recursive in
HA.
Hence
CTM
CTN', C T ~ '
of a recursive
CT~F'. function which is not
V x ~ y T ( ~ , x, y)
holds,
and
x , but not provably recursive in
would require the existence
co Vx T(~, x, tlx) ; but since all functions in HRO , it follows
2.5.4°
CTM, CTNF, C T N ~
be the g~delnumber
for
h x + Io x CTM, CTNF, CTM', CTNF' are recursive
HR0~ ). does not hold in
is a recursive function of QF - AC
the diagonal function in
of a
t I ~ CTM
minyT(n,x,y) HA. such that
t I E CTM are interpreted by provably recursive that m VxT(~,x,tlx) . Similarly for CTNF,
.
Lemmao
there exists
For standard g~delnumberings
of partial recursive
two-place primitive recursive function
a
~
functions
such that (cf°
Rogers 1967, § 7.2, in proof of theorem IV).
Vxy~(l~t(z) ~ I~(x,y) t(~)) v==,yy,(x / x, vy / y, ~ ~(~,y) / ~(~, , y , ) ) . 2.5.5.
Theorem.
model
CTNF'
Proof.
Let
version of c
There exists a version of the model
can be embedded in
HR0
<x,y,z> ~ W3(x,y,z) HR0
by re-defining
(is isomorphic
(1.3.9
(C)).
the numbers
[c]
HRO , such that the to a submodel of
HRO)o
We define the required representing
the constants
(cfo 2.4.8) as follows. Let
[0] = 0 , and let
be any numeral
l~l(x,y,o)
~
~l(x,y,sz)
~ ty/(t~/(x,y,~),~)
{ t l ( t ° . . . . . tn)
where
~
such that
x
is an abbreviation
{...
{fttl(to)
l(tl)}
...}(tn)
.
We put
[s] = ~(Ax.x+1, ) , [n] = ~ ( ~ . ~ ( ~ . x , <x,~,2>), < 1 , ~ , 1 > ) ,
[z p,o,~] = ~(~:.~(~.~(,~.~(ll~l(~) i ( { y l ( ~ ) ) J ~ " ) ,
<:~,y,5>), <x,x,4>), <3,3,3>) ,
[R] = ~ ( ~ : . ~ ( ~ . ~ ( ~ .
[s] If
=
~(x,y,~),
sglx-yl),
to,t I E CTNF,
<~,y,8>), <x,x,7>), <~,~,~>).
<9,9,9>)-
then each has one of the forms of the following list
(s,t E CTNF) : N, Hs, Z, Zs, Zst, R, Rs, Rst, S, Ss, O, E, Es. a) If e.g.
to, t I if
correspond to different forms in the list,
t =- Zp,~,Tst,t I -= Rs't' , then
then
[to] ~ [tl] ;
134
[t o] = ~ ( ~ s . ~ ( { l [ s ] l ( ~ ) l ( l [ t ] l ( ~ ) ) , ~ ~),<[s],[t],5>) [t~] : ~ ( ~ . { ~ j ( [ s , ] , [ t , ] , ~ ) , <[s,],[t,],~>). Now
[to] / [tl]
x I , yl
,
since
~(x, )
/ ~(x',
)
for a l l
z ~' x , y , z
b) If
to~t I
also
[to]/
and
[tl]
to,t I
,
correspond
to the same form on the list, then
or type ( t o ) # type ( t l )
induction on the sum of the complexities
°
The p r o o f now proceeds by
of
t
and
t I . For example,
O
t o ~ Z p,a,~st,
let
t.l ~ Zp,,a,,T,s't' ,
then
Ito]
is as under
(a),
[t~] = ~ ( ~ . ~ ( l l [ s , ] l ( r ) l ( i [ t , ] l ( ~ ) ) , ~ , ~ ) , ). #o] ~ [ t ~ ] wo=l~ impL~ Is] = I s , I , I t ] = [ t , ] so then either type (s) / type ( s ' ) , or "type (t) / type ( t ' ) , by (~) and induction hypothesis. If
t y p e (s) ~ type ( s ' ) ,
that
~ ~ ~'
or
e / ~'
If
type (~) / type (t') , it follows that
it
follows
p / ~'
or
~ / o'
If
~ / ~' ,
If
p'
or
then o b v i o u s l y
t y p e ( t o ) / type ( t l ) ~&~Z~-~£~"
,
[to]
since
~ [tl]
.
type (to)
~ /
m (~)~,
I n the p r e c e d i n g p r o o f ,
or
m { m' ,
type (tl)
[~,~],
[Zp,~,v],
[R~],
The new definitions
but the uniformity
definition
(case (b) in
in the types is gone.
are :
[~a,~] = ~ ( ~ . ~ ( ~ 7 . ~ ( x , j ( ~ , ~ ) ) , [~p,o,~] = ~ ( ~ . ~ ( ~ ( { { x t ( ~ ) l ( f y } ( ~ ) ) ,
<x,=,2>), <1,1,1>) <"~,~,~>), <~,y,5>), <x,x,4>),
[~] [~]
= ~(~.~(~.~(A~.~({~(x,y,~),~), <x,y,8>), <x,x,7>), = ~(~.~(~.~(sgl=-yl,~), <x,x,~o>), <9,9,9>).
2.5.6.
Alternative
Since for
proof of the uniqueness
t,t' E CTNF' , t ~ t'
2°5.7. then
N - H A w)
t = t'
~ - HA ~ ~ t = t'
then
[t] = [t'] 9 it follows from
I - H A m + IE °
(and hence each
possesses a unique normal form.
Corollary to 2.5.5~ ~-HA~
<3,3,3>)
<6,6,6>)
of normal form.
implies
2°3°2 and 2.5.5 that each closed term of closed term of
.
we have k e p t the assignment ms
[E ] , we need less verification
the preceding proof is simpler),
then
m (£')v'
uniform in the types as possible ; if we use a slightly different of
w / T' .
iff
2 5.6. o
t,t'
t = t'
If
t,t'
are closed terms of
reduce to the same normal form;
can be proved in
qf-~-
H~Aw
without
N - I~Aw ,
~
hence, if the use
of induction. This may be rephrased as a conservative conservative
over
Similarly for
qf-~-HA I-HAm+
W IE
extension result : N - U A ~
without induction, o
is
for closed prime formulae.
~35
2.5.8.
Theorem.
2.5.5), CTNFt
HR0
For suitable versions of
is a model for
is also a model of
Proof°
~-HAm+
HR0
(i.e. the ones defined in
IE I ; hence, as a corollary of 2.5.5,
~I-HAm+~ IE I .
Similar to the argument in 2.5.5, we can show that
IE I
is satis-
fied. Remark.
CTNF'
closed terms
is a minimal model of t,t'
N-HAm+
IE
~
closed terms, 2.5.9.
w.r.t,
have the same interpretation
duce to the same normal form,i.e, Each model of
~-~m
if
N-~m
equality,
in the model iff they re-
~ t = t'
must be minimal w.r.t,
equality
as will be obvious.
Examples of versions of
HRO
where distinct normal terms are re~HROo
The first example is suggested by the necessity of referring to
the definition of closed terms ~',
then
between
O
resented by the same element in the version of (i)
i.e. two
[Z0,a,T]
t6 (p)(~)w,
[t]= [t'],
[Z,=,ts]
= (p)T,
in the proof of 2°2.5° t' E (0)(~')v,
[Six Is']
= [Zp,~,,t
s6 (p)a,
(under the assignment
's'] ,
a
in
If we can find normal s' 6 (p)a,
such that
described in 2.4.8),
type (~p,~ ,mrs) = type (Z ~ , ~ , , t
's')
=
Zp,m,Tts ~ Zp,a,,Tt's'
Take P = ~I' V = al, ~ = (a2)~l' ~' = (=5)ai ' ~2 / ~3 ; t ~ n l,a, t' ~ ~ 1,a I ; s ~ H I,~2, s' ~ He1,~ 3 ; then all our requirements are met°
(ii)
R(SO)(E(Hn')8)
m, T, ~ ,
for appropriate is extensionally equal to the successor function S °
m' )
(where
Now we modify our description before denote a numeral,
let
@(x,y) ~ Am.
,,(x,y) where
of
denote
[HI
ha,T, n ,,T ,
in 2.5.5 as follows.
Let
~
ks
satisfying
{~}(~,y,o) ~, Then
H, ~,
I~}(~,y,s~) ~ ly}({~}(~,y,~),~) {~}(x,y,z) ;
= sgl~-x I . ly-~l
~
is primitive recursive.
• *(~,y)+ (~=(ly-~l
+ l~-xl))
We put
. [8],
~ ~ [Z(~n')S].
~ o w we p u t
[R]
= ~Ay.
~'(x,y)
o
It
is
then
{[R] }(x,y,O)
~x
{[R]}(x,y,Sz)
~ {y}(I[R]}(x,y,z),z)
(this is proved by distinguishing
oases : y = ~
obvious
that
& x = I , or
x% I V y/~).
Also
{[R]}(0,~)
= [Sic
From the preceding examples it is obvious ~-H~i ~ ~ t / t'
whenever
t,t'
that we cannot assert
are closed terms with different normal forms.
13g
Remark.
The second example is based on another idea than the one used in
the first example. Zp,~,Ttt'
and
The first example is b a s e d on the " t y p e - a m b i g u i t y " :
Zp,a,
Tt't"
are of the same type for
The second example picks more or less a r b i t r a r i l y
~ '
two closed terms of
type I , w i t h different normal forms, but r e p r e s e n t i n g functions, 2.5.10.
snd identifies
IEo
is w e a k e r than
IE I , for example, y/y
~
Nore precisely, valid,
Ett
= 0
(0)(0)0
numeral
if
t,s
(i.e. the schema
are d i s t i n c t
IE
o
) does not
.
we can find versions
of
HR0
for w h i c h
IE
is o b v i o u s l y
o
(I) fails.
To see this, we argue as follows. type
E~ts = I
it does not f o l l o w that
Rxy~Rxy'
but for w h i c h
equal
IE I .
closed terms in normal form,
(I)
extensionally
them in the model.
W e w i s h to show that the a x i o m schema
imply
.
not c o n t a i n i n g
Take any closed term of
R° ,
say e.g.
n0
and let
,0'
~_~w ~
of
be any
such
The f u n c t i o n range of
~
of lemma 2.5.4 may be chosen such that
~
is not in the
~.
N o w we define our v e r s i o n of one e x c e p t i o n :
we define
HRO
[Ro]
as in the first p r o o f of 2.5.5, but w i t h as
• (~o ~(ar. ~(Az. {~t(x,y,z))<x,y,8>), <x,x,7>), <6,6,6>), where
~
is g i v e n by
(2a)
~(u, <x,y,8>) = ~(u, <x,y,8>)
(2h)
~(u, <~,~,8>) = ~(u, <x,[nO,O],8>).
W e note that
~
Let us indicate assignment
if
the n u m b e r assigned
to a closed term
as [t]' , and by the new one as
in normal form, w h i c h is not a numeral,
~, s
range,
ing
themselves
are in normal
are in the range of
Therefore
we ~ever 1E O
have
holds.
to u s e
But,
clause
obviously,
t.
by the original It] = [t]' , as we
For a closed term
t
o
S, E, Et So the c o r r e s p o n d i n g n u m b e r s
since
an easy i n d u c t i o n on the c o m p l e x i t y
[to]
of
form.
~, 4;
t
[t] ; then
is of one of the forms
H, Ht, Z, Zt, Zts, R, Rt, Rts,
[to] , [to]'
~.
is also outsiSa the range of
can show by an i n d u c t i o n on the c o m p l e x i t y
where
y/
of
(2b),
n t
o
hence
was chosen outside yields
[t o ] = [to],.
(I) is false :
that
that in evaluat-
137
I[R o] t(~,~) ~ fERo] ](x,EnO,O]) whereas ~ ! [nO,O]° 2.5ollo Remark.
Presupposing the theory of ccmbinators,
(Tait 1968 , p. 191 , lines -10 to -2) of of achieving the result of 2.5.5. definition of fying
~6-
HRO
is more flexible.
ccnversion,
}FRO
However,
Tait's version
is a slightly more direct way
for our purposes the present
Similarly,
the
HRO - variant satis-
described in 2.4.18, contains a
X- term model
isomorphically embedded. 2.5.12. Remark on the properties of g~delnumberings used. The construction in 2.5.5 made essential use of the lemma 2.5.4 on standard godelnumberings.
One might wonder to what extent the results
depend on the g~delnumbering chosen.
An answer is provided by Rogers 1958.
The "fully effective" numberings there are precisely the numberings which can be brought into recursive one-to-one correspondence with a standard godelnumbering.
Therefore any fully effective numbering satisfies 2.5.4
and yields the result in 2.5.5. 2.5o13. Historical note. Term models for
~-HA ~
first appeared in Tait 1963, Appendix B, which
is a preliminary draft of Tait 1967. A detailed comparison between term models and 1971 , Appendix I.
HRO
is made in Kreisel
~3s
§ 6.
Models based on continuous function application : ICF, ECF . = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
2o6.1.
Contents of the section.
In the present
section, we study models of
but based on continuous tinuous functionals
function application
similar to
HR0, HEO,
The hereditarily
con-
make their appearance
in Kreisel
1959 and Kleene
1959~(as countable funotionals) ; the intensional
continuous
funotionals
ICF
are introduced
(ECF)
~-H~A w instead.
in Kreisel
2.6.2- 2.6.10 describe continuity
1962 (page 154).
ECF, ICF
and uniform continuity
2.6o11~
and discuss the existence of moduli of
in these models°
2.6o12 extend the faithfulness
theorem from
2o6.15- 2.6.21 are devoted to the recursive the equivalence between and
~0
ECF(~)
(= ECF
HEO
to
density theorem for
relativized
ECF ; ECF
and
to recursive functions)
o
2.6.22 discusses the models elements of
ECF,
ICF
ECF r, ICF r , obtained by taking the recursive
relative
to a universe
of functions
satisfying bar-
induction. 2.6.23 describes variants application
ICF*
of
ECF,
ICF
is defined in a more uniform way than for
2°6.25 describes ECF*,
ECF*,
the interpretation
respectively, ECF,
where
ICF.
of pairing operators
in
ECF,
ICF,
ICF*.
2.6.26 describes
the analogues
IC~, IC~F-
to
HR~0, HR~0- imtroduced
in
2.4.10. Directions
for use.
realizability
For most applications
in connection with modified
and the Dialeotiea interpretation
(§ 3.4, § 5.5), it suffices
to study 2.6.2- 2.6°I0 ; a few results in § 3.5 (obtained with the help of the Dialectica
interpretation)
§ 7, in studying derivable 2.6.2.
Below we shall assume
closed under "recursive
in",
are variables ranging over to
V
require
instances
for
2°6.20.
2°6o11- 2.6.12 are used in
of the rule of extensionality°
to be a universe is
(in short,
~
We introduce
a
of functions
model
of
I V~
for each
for
~ ~ 0
for
~,T / 0
for
T / 0
HR0 , as follows :
x 6 VoI = x = x I
6 V(O)O
=- ~ = ~
¢~
Vt
(~,)o
v~ c
v I ~(~(~)~x)
~E
V1 (a)T
v~ ~
vI ~
,
e v I(41~), T
-= Vx Z~ E V I
of type I ,
~)°
~, s6 T
analogous
139
The objects
of type
a
functiona!s
relative
to the universe
I (x,O) , x E V 0
pairs
Equality
is defined
~shor~; ~ ( ~ )
(the intensional
~=0
as
(x,O) : (y,O) -=def x = y ,
, and of
(~,~)
The interpretation
continuous
ICF )
consist I ~ 6 Vq if
with
(~,a)
of application
= (8,a) depends
of the a/O
.
=-def on the
We put : =
(~,1)(x,O)
Further
([c],~)
(~x,O)
(~,(O)o)(x,O)
= (oI~.x,~)
for
~/0
(~,(a)o)(8,~)
= (~<8), o)
for
~/o
(%(~)~)(~,°)
= (~18 , ~)
for
~,,~/o
we have to show how the constants
Let us write
[0] I
(in short :
represents
(a) (b) (c)
.
may be interpreted.
[c] ) for the function
the constant
[0] -= 0 IS] - ~ . s , : [no,o] = a l £ y . x , I n , o ] -= AI~AIy.~,
(d)
(e)
ICF(~)
~/ , in short :
if
a/ 0 °
V x ( ~ x = ~x)h/for type.
of the model
cE ~
or number
such that
in our model.
[no,~]_= ^ 1 ^ % . x , [ n , ~ ] ~ AI=,A18.~
for
=,T/o.
[zp,~,o] =- al~AI~A%(=I~)(~I~)
(p/o)
,
[Zp,o,~] - Al~alSA°y(~ly)(Xy.8(,l))
(p,~/o),
[~o,~,o] -= at ~al ~a°~(~I x~" ~) ( ~ ) '
C o n s t r u c t i o n of
[R~].
s_E~_2_as 2
t Then,
by the recursion
° (~,
~, y)
=
~=o °
8(~ (~, 8, x ~ ( m ' - ~ ) ) , if
yO >
theorem
I ~o
if
analogue
I 9 16 , we find
0
such that
x~.~'--~))
if
~/o,
if
~0>0,
we may take
AIxAI~A°yo 6 o ( ~.z.x, 8, Xz.y)
[Ro]Subcase
8
yo=o
~(~o(~, ~, x ~ ( ~ - ~ ) ) , and therefore
x~.,~o'-~)
O.
~ / 0.
We can find an
,
~1 ( ~ ~, ~, Y)
=
~,
if
e ~0
such that = 0
{ 81(8l(~, 8, X z . y O _ t ) ' kz.'~O "-- 1)
140
and then by Io9o16
61
a
such that
~I(611(~,~,X~.~0=~),
X~.~=~)
~f
~0~0,
and then we may take
[~] Note
~ ~^~Y-
further that
theory
%1 ICF
~, ~, X~.y) .
is actually a model of
obtained by adding to
2.6.3.
Theorem.
A model
ICF
~(~mcXY)
= ~(~mcXY)
~
This is provable in EL. I Let ~E V 2 ; we define ~[~]n = o
if
Then put
).
possesses a modulus-of-continuity functional,
Proof.
~mc e (2)(I)0
such that
xy = xz°
~[~]
Vm±n
re[sin = l t h ( m ) + 1
(i.e. the
N - H A ~ : xl = y l ~ V z ( x l z ~ = y l z
i.e. in the model there is an object MC
Int-HA m
such that
(==0)
if
mnn,
~0,
Vm'<m
(~n' = 0 ) .
A I ~ ~[~]
[~mc] =
We have to show
(I)
~(<~](~))
if
~[~](~)
= x,
~(~x)
2.6.4.
Theorem.
object
~uc E (2)0
Muc
= ~(<~](~))
If ~[
/ O,
-
~(~y)
satisfies
~(~)=~(~) = 0
for
E L + FAN,
. y < X ;
hence
(I)
then we have in
i s immediate.
ICF(t/)
(" uc " for uniform continuity) such that
vz 2 vx I ~ B vy 1 ~ B ( ~ ( % c Z ) = ~ ( % 0 z ) ~ 2== ~y)
where, as before, Proof.
x i 6 £ abbreviates Vu°(xu~1) . I m E V 2, then V ~ x (m(~x) / 0 ) , hence by
Let
~AR
a~ v~ ~ B ~ ! ~ ( ~ ( ~ = ) / 0 ) We define ~uc(~) It
remains
to be shown that
for suitable Let
= min z muciS r e p r e s e n t e d
by ([muc],(2)O)
in
[q0uc] , as follows
B z = {n I lth(n) = z & Vi< z((n)i ~ I ) I , and put
[~uc](m) =~ minz[Z_< Ith(m ) & Vn~ Bz ~n'Zn((m)n , / 0)] + i if there is such a L 0 [qOuo]
otherwise.
is obviously recursive.
z,
ICF,
an
141
2.6.5.
The extensional model
ECF
of the hereditarily continuous functionals.
Now we desoribe~ relative to a universe of functions ~ the model
ECF(~)
(in short : ECF)
which is similar to
satisfying
HEO , but based on
continuous function application° We introduce simultaneously domains I and equivalence relations IG for all ~ E T ~ as follows. ~V0I
x6
W a1
I 10(x,y ) =- ( x = y )
---x = x ,
wl0)0 ~
EL
=
w~ = v~ ~ ~ Sx(=(~)-x) (~)o v~,(z~(~,~,) - ~(~)=~(~,))
=/o
(~)o -.
~
( o ) ~ - Vx z ~ 1 I(O)T(~' ~)
The objects of type
~
of the model
functionals relative to the universe = 0 ~ and
ECF([[) ~
(the extensional continuous I (x,0) , x 6 ~ 0 if
) are the pairs
(~,o) , ~6 ~I if o / 0 . t~ I ~ is interpreted as I~ ; application and the other
Equality at type
constants are interpreted as in
EC; 2.6.6~
is (provably in Theorem.
S~)
ICF(~) o
a model for
(Kreisel 1962 , lemma 7)
there is a "fan-functional"
~uc 6 (2)0
[-~. If ~[
in
satisfies
ECF
E~L+FAN,
(provably in
EL)
then such
that Vz2Vx I 6 B Vy I 6 B ( ~ ( % o z) = ~ ( % e z) ~ zx= zy) where, as before Proof.
1 x ~ B
abbreviates
Vu°(xu~ I) °
In the proof of 2.6°4 we must replace
by ( B z
[~uc ]
by
[~uc ]'
defined
as in 2.6.4) [~uc]'(m) = ~ m i n ~ [ y _ < z & Vn~ B Vn'~ 6 B~ Vm'(m'_
& m'~n' lwhere L 0
z =
& lth(m') =y ~ (m)[n] = (m)[n']/O)]
[~uo](m) ~ 1, i f
[~uC](m) ~0
otherwise
and where
(m)[n] =
(m)y i f y = minx[X_
+ 1,
142
Obviously, i f
~ ~ ~,
[%o]'m![%e
Also
~e have t o
show
Ira.
[~uo ]'
I I2(~,~'
Suppose
~([%e]'(~)/O).
) ,
E ~
[~ue]'m' = ~2+ 1. ~y the d e f i n i t i o n of readily of
seen t h a t
[~uo]'(m)
[~uc]'(m') 2.6.7. ~mc " Proof°
Theorem.
I~(~,~')
~ [~uc]'(~) such t h a t
= [~uc]'(~')
[~uc]'m
changed i f
max([~uc](m):l,
we r e p l a c e
[~uc](m'):l)
= [~uc]'(m')
,
~ow i t is in
the d e f i n i t i o n
and s i m i l a r l y
for
.
does not contain a m o d u l u s - o f - c o n t i n u i t y
1962, after remark
o
= x 1+ 1 ,
[~uc]m{O, [~uc]m'/O.
also is not
[~uc]'(m) ECF
(Kreisel We
[~uo]''
by
Thus
i.e.
~E m, ~ ' E m'
[~uc]((m )
z
.
,
and l e t
functional
I0).
shall show, more particularly,
that in the model there is no
such that
Vy< qrx2(~y= O) ~ x 2 ~ = x 2 ( X z o O )
.
In order to show this, we consider f u n c t i o n a l s
40' ~m,1
of type 2
such
that
4°
= k~.O ;
~m,1 ~ = {I if
VYSmo(~=o)
~o~= 0 where
m o = ~4 °
for g i v e n
~o
Now we choose r e p r e s e n t a t i v e s (i)
~ n = I o
o (ii)
n = 0
~o' ~I
of
~o' ¢m,I
as follows.
if
n = (~-~-~.O)z. < S u > . n '
for suitable
Z~mo,
or
n = (~-~y-~.O)(mo+1) . n'
for suitable
n'
~I n : 2
if
n : (kT~O)(mo+1) . < S m + y > . n '
for suitable
~I n = 1
if
n = (-~-~)(mo+l)
for
Note that
is defined for all
for all
N o w assume
n < lthv,
~
y~%0 lthn
from
and n'
~,
and in fact equal to
O;
~(mo+2 ) .
to be r e p r e s e n t e d ; if we choose
v
by
~;
then we can find a
sufficiently
large,
> m +I . o
m = maxI(n)mo+11 n
y, n'
uim,
suitable
We put
It f o l l o w s
some
9.
is always d e t e r m i n e d
such that
* * n '
in all other eases.
~o(~)
~o~(mo+1 ) = I
u, n'
in all other cases
~I n = ~o n
~I(~)
& ~(mo+1) >m
in all other cases
~IE v
& lthn>me+l}
for this choice of
m.
.
~on= I
V,
~ EV O for some
~43
For let
n < Ithv.
If
sln = 2
is excluded,
ing our choice of
Ith(n)
< m +I ,
--
since then
m.
Hence
~In = ~o n ; if
o
(n)mo+ I
either
n = (k--~.O)(mo+1). < u > * n ~ , u ~ m .
%n
Ith(n)
would be less than = Son,
> m +I , o
m , contraS~ct-
or we are in the case where
But in this case
Son = ~I n = I °
Thus we have
~(So)
= ~(~1 ) =
On the other hand,
= it follows
mo+1.
for
I x-smol
~.(m+I)(I =
that
~l(kX.O)
~(~) = s~(~.o)
= O,
)
~1(~) = 1 , whereas
becomes false, ~i~oe (b~
V y ( ~ ( ~ 1 ) ( ~ y = O)
~(~) = mo+~ ) ~ < q % ) ( ~ =
O)
holds. 2.6.8. where
We s h a l l ~
is the u n i v e r s e
is closed under mathematics 2.6.9.
now p a y some s p e c i a l
Br
(Kleene & for :
~
primitive
recursive
(I)
vs~ ~
(2)
Vz ~ S E B r V x S z
to
ICF(@)
recursive functions.
in" can n o w be established
b e l o w can be e s t a b l i s h e d
Theorem
Let us use
of (total)
"recursive
attention
Vesley
in
Rx
ECF(@) ,
The fact that HA,
and the meta-
~.
1965 , lemma 9°8 in § 9.3).
is recursive
predicate
in
and
and
Vx(~z~1)
. Then we can find a
such that
~(~) ~R(~x)
and therefore
(3)
~
Proof. R
vs~ B r ~ x i z R ( ~ ) .
Briefly,
we define
has infinite branches, ~o(X,y)
R
so that the tree of u n s e c u r e d
but no infinite
~ T(J2x , x, y) & V z ~ y
Note that
We put
~(i,x,y) ~ ~s~(i)(x,y) and define
Then for any
s
with
Vx(arx~ I)
recursive branches°
~ T ( J l X , x, z)
sequences Wor°t. W e put
144
Ws~(~)(u,y).
R~ ~ Zu<x~<~=u
(4) N o w let
~6 @.
Then there are
n = J(no,nl)
such that
~tyT(no,U,y) ~-¢ ~ y T ( J l n , u , y )
#a,u/ 0 ~
(5)
n, no, nl,
L~ u : o ~-~ ~yT(nl,u,y) ~
~T(J2n,u,y).
~ 2 1. net ~ / 0 . Then ~yT(Jln, n, y) & ~ F T ( J 2 ~ , n , y ) , ~y~1(n,y), so ~y~s~(~)(n,y). Ca~e 2. Let a m = 0 ; then similarly TYWsg(a~q)(n,y) . We
now take
n
for
u,
%le see that
R~x
holds ;
N o w take
any
z
~6 B r.
if
0
otherwise.
u<x~z,
Let
y<x~u~
definition
of
Case (b).
9u=0
x
in
(I),
where
Wsg(a~)(n,y).
(I).
then
R~x.
such that
~l(U,y) ,
Then there should be
u, y
such that
~ u(U,y) . This leads to a contradiction:
m~yW0(u,y) ,
so
conflicting
with the
9; ;
This establishes 2.6.10.
, assume
z=u
9u= I ;
for
u< z & Zy< z ~ u W0(u,y )
x~z
Case (a).
I
and define
1
~u={ Then
n+y+
thus we have e s t a b l i s h e d
hence
similarly.
(2).
ECF(~
Corollary.
is continuous~
and
ICF(@)
but not u n i f o r m l y
c o n t a i n a type 2
continuous
v~
x v~(~= ~x ~ ~ = ~ )
~z
V ~ 6 B r V 8 6 B r ( ~ Z = ~z ~ 9 ~ = ~ )
on
object
9
which
B r 9 i.e.
but
Proof°
We take for q~=n
~
ym= 0
is by
variables the type
of
& ~E n.
such that
ym=n+1
[t]TC~ , [t]ECF.
f o r terms
of type 0
Y
~ mRm)
if
Rn & V m ( m < n
~ mRm)
We d e f i n e pseudo-terms
[t]ICF
in all other eases.
2o6o11. D e f i n i t i o n
(~ [t]ECF) ,
9:
~ llu & V m ( m < n
The r e p r e s e n t a t i o n & n~m
•
0
t~ N-~*,
built
from constants of O
~-~
O
,
O
and variables of type I . Let re, Vl, v2, ... be I I I and let Vo, Vl, v2, .o. be the type I v a r i a b l e s
variables~
of our theory° For any constant defined in 2.6.1. F u r t h e r m o r e we put
c
of
N - H~A ~ , we put
[C]IcF ~ [c],
where
[c]
is as
145
O
O
[vi]iC F ~ V i , I
I
[vi]IC F ~ V i , and inductively [tt']iCF m [t]iCFI [t']ICF,
if
t, tt'
[ t t ' ] I C F m [t]ICF ([t']ICF) ,
if
t'
[tt']ICF m [t]ICF[t']ICF,
if
t e I,
2.6.12. Theorem. and type
Let
-~ Proof.
t
I variables
are both types
is of type
Xl,...,Xn,
/ 0 , tt' e 0 ;
t' e 0 .
be a term of type
(say
/0;
0 ,
constructed from type
~1,...,~m)
and constants.
0
Then
+ ACooF t ~y ~-~ [t]icF ~ [t]~cF ~y.
The proof is very similar to the proof of theorem 2.4.14.
We construct a code number ~1,...,~m
rt~
for pseudo-terms
t , containing at most
free, as follows
~(~,)~ ~
= j(2, j ( ~ , ~ , ~ ) ) = j(3, j(r~,~t,~)) where
r[C]ic; : j(O, C)
c
is some code number for
the constant r~ i
= j(4+i, 0).
As a result, r [t]ICF
can be computed effectively.
We now construct a predicate (la)
N-H~+ACoo
and for
A(x, ~, al,...,~m)
~ A ( r : 1, ~, a l , . . . . am) ~
such that, for
tE0 ,
t ( x 1 . . . . ,Xn,~ 1 . . . . ,am) " ~ 0
t/o
(Ib)
~_~+ACoob
Here
abbreviates
t(~ 1.... ,~n,~ .... ,%),
A(r{l
~, ~1 ..... ~m ) ~ t ( x l
t(~1,...,Xn,~l,...,~m).
A(~, ~, a I . . . . . %)
by
.... 'Xn'~1 ..... ~m ) ~
For convenience we abbreviate
t(~l,...,~n),
A(~,~)
respectively. A(x,~)
is defined such that
A ( ~ , ~ ) *-->[~ = j ( o , ~ )
& ~o = o] v ~ ( ~ , ~ )
v
V [JlX= 1 & ~YIY2[A(jlJ2X,Y1) &A(j2J2x,~2 ) & ~ y 1 1 Y 2 ] ]
V
V [JlX= 2&ZY1Y2[A(jtJ2x,Y1) &A(j2J2x, Y2) & ~ 0 " ? l ( Y 2 ) ] ] V [JlX= 3 & ~ 1 ~ 2 [ A ( j l J 2 X , ~ 1 ) V ~ [ x = j(4+I,0) & ~ = ~i] , where
B(x,~)
& A ( ]lJ2X'~2 ) & ~ 0 ~ Y I ( ~ 2 0 ) ] ]
is a disjunction of all clauses of the form
= j(O,~) & ~= [o] for
o
a constant of
N - H A~ m
~
e~0
o
"
V V
146
(I) is now proved by induction on the complexity The remainder of the proof is entirely parallel The addition of
ACoo to
~-H~ w
1969).
In fact,
QF - A C o o 0~(H~A)
2.6.13o The equivalence between For Kreisel's
ECF K) , relativized
that
to be available,
HEO .
continuous
and
functional
functions
HE0
and improvements,
1959, Kreisel - Lacombe - Shoenfield are in the privately
circulated
1963 ) and the unpublished
ECFK(~ ) )
the same class
literature ; publish-
1959 ; more details,
Stanford report
course notes Kreisel
Via the equivalence between Kreisel's notion of hereditarily functional,
(let us say
(i.e.
represent
(Full details are not in the published
ed is only Kreisel
so
of recursive functionals
and
ECFK(~ )
E~L
would have been enough.
to recursive neighbourhood
One can show classically of functionals.
1963, Harrison
ECF(~)
concept of hereditarily
t.
to the proof of 2.4.14.
was to insure
that we could rely on Kleene's formalization (Kleene
of
and Kleene's notion of countable functional
(Tait
1958 B.) continuous
(Kleene
1959~see
Hinata and Tugu~ 1969) the result then also holds for Kleene's notion. fact, it is technically Kleene's countable recursive
associates),
In subsections
even simpler te formalize
functionals
(hereditarily
or for our model
In
the result directly for
restricted
to functionals with
ECF(~) .
14- 19, 21 below the materials needed for the equivalence
proof are given ; the proof can actually be carried out in have refrained from paraphrasing theorem for the countable
~+MpR
Kleene's proof of the recursive
functionals,
for our model
. We density
ECF(~) ; the proof,
as it stands, is unperspicuous ; it is to be hoped that adapting the more informative
a~d perspicuous
arguments
ECF(~) , and extending the discussion more satisfactory
of Tait 1963 , Kreisel
1958 B to
to impure types also, will yield a
exposition in the future.
Because of the coding,
for extensional
functionals,
of objects of arbi-
trary types of our type structure by objects of pure type (1.8.5- I°8.8) we may restrict our proof of equivalence
to the pure types.
Kleene's treatment of the countable functionals has, when compared with our introduction
of
ECF(~) ,
a different
has really a classical hierarchy ("neighbourhood
functions"
of functionals
in Kreisel's
by talking exclusively about associates the functionals our notion,
themselves.
hoods of higher type, as a consequence functions
background:
in mind,
or neighbourhood
Kleene
coded by "associates"
terminology) ; ECF(~)
A somewhat artificial
as compared to Kreisel's,
of the neighbourhood
conceptual
is introduced
functions,
not abo~t
aspect of Kleene's and
is the fixed ordering of neighbourof the fact that only initial
(which correspond
segments
to Y~eisel's neighbourhoods)
147
play a rSleo However, ECF
From a topological
point of view, this is indeed arbitrary.
there are definite technical
instead of Kreisel's hereditarily
similarities
and differences
(and the topological relevant for available
analogy between
continuous ECF
and
advantages
functionals ;
ICF
2 ° ) formalizing of Kleene
HR0, HE0
2.6.14. Definition°
Let
V
10 ) the
is simpler for
ECF,
ECF,
is much less
we can use the
1969 ; 3 ° ) there is a heuristically
on the one hand, and
in using
are readily described
point of view, which is natural for
ICF) ;
apparatus
between
and heuristic
ICF, ECF
useful
on the other hand.
be a set of total recursive functions.
We put
def E(V)
is to be the set of (g~delnumbers
of) effective
operations
defined on
V , i.e.
V
is said to have a recursively
recursive and
Vn(On E V*)
v~ v~ 2°6°15.
Theorem
has a reoursively
dense basis,
~
(Kreisel - Lacombe - Shosnfield dense basis enumerated
m VzE E(V)
recursive function of
{z}(u)
k, y, z y}(x)
®
is
t
®,
In
~ . + %1{ ' if
where
~o(k, y, z)
V
then there is a partial
= {z}(y))
=
o
function
{~(k,y,z) },
D
a primitive
as follows :
if
(Vn<x)-~(Tkkn&Un=O)
;
o t h e r w i s e , where
{@m }(x)
We abbreviate
if
E(V) , i.e.
We define a partial reeursive
I
by
1959).
VyE V*(.T{m}(z,y) & VuE V*(~ul({m}(z,y))
: {-YT({~}(~,y))-
Proof.
®,
~({e~l(~) = ~x) .
recursive modulus of continuity for
KLS
enumerated by
and
u =" minv[Tkkv & Uv = O] . as
Pk'
and define
$(y,z) ,
$
primitive
recursive
by
{,(y,z)}(k) ~ We abbreviate
ro if '{~}(Pk) ~ {z}(Pk) {z}(Y) Jl if :{~}(pk~ I~}(pk) i{z}(y) Lundefi~edif ,{z}(p k) . $(y,z)
as
Assume y~V ~, ~ ( V ) . Suppose
-7{ql(q)=O,
q.
We first establis1~, using %~, then
Vn-7(Tqqn&Un=O)
, hence
that {q}(q)=O.
{pk}= {y};
but then
148
hence Therefore
~m
{q}(q) = 0
the partial
therefore (r{~R) {q}(q) = 0 o
recursive
function
with index
m ~ A~F. mi~T(~(y,z),~(y,~),v) is defined
for all
NOW suppose We will
n
Z6 E(V),
y 6 V*.
y , u E V*,
Mzy m {m}(z,y) ° Note
Put
that
{Ul(Mzy) = {y}(~zy) .
show
(I)
{z}(u)
Since
z 6 E(V),
®
= {z}(y) .
enumerates
a recursively
dense
basis for
V 9 we can find a least
such that
First we show that
{z}(~n)
(2)
Suppose
~ (2)°
{~1(~).
=
Then since
n : minm({em}
6 [y}(Zzy)
we have by the definition {z}(pq) ~{z}(pq)
&
{z}(pq)
(2) holds.
(3)
{q}(q) = 0 ,
so by the ~efinition of
This contradicts
{z}(y) .
:
{z}(~n)% {z}(y)) ,
{pq}= {@n} , and hence
of
/ {z}(Y) . B~t
o {z}(~)
&
m(2) ,
so
mm(2)
~,
, hence
Similarly
{z}(~) = {~}(u).
Thus we obtain (I) from (2), (3). 2.6.16.
Remark.
original Beeson
The proof as presented
paper and in Rogers
1972.
Another
1967
hers is close to the proof in the
(pp. 962 - 364) ; the modification
proof is in Gandy
is from
1962, which we find less intuitive
however. 2.6.17 . Theorem
(Refinement
Let us say that
vy~ v* ~ ( f ( b theorem
f
is a normal
in
associate
•
as follows : for each
(provable
KLS ; Kreisel, ,V.m[f.~O
l(n)~ = {~ I(Y) + I)) ~ T h e n
with a recursive!y
Proof.
of
dense base,
m+I%R
We define
for -~
Laeombe, z6 E(V)
~[n~mY=
Shoenfield if
f
z 6 E(V) , V
a set of total recursive
there exists a recursive
h
is total,
F.)
we may strengthen the precedin~ associate
)o
a recursive
1959)o
such that
(m
as in 2.6.15) :
for
functions z
149
I+ {zl(y)
if
~ u < n T~yu,
y
~u'
Vj -<- {m~(z,y)
~k
~Lu'
ht/O~
ht=h(t*n)
.
Then, for suitable
(I)
T~
(2)
T(Iml(z), y, u')
(3)
j < {~l(~,y) - Tyjkj
(4)
{~l({~l(~,y))
t ,
ltht > m a x ( u , u , , U u ' , k
Corollar 2.
for all
If
V
is a set of total recursive
(Existence
~L.)
For each
Consj(x,y)
Izl
{zl,z6 E(V) on
j~ I
dense basis for
function
(ii)
Consj(x,y) ~ (Xz.extj(z,x,y))Ith(x) If we put
Consj(x)
~def extj(y,x,x)
kx.extj(x,y,z)
Proof.
= x & (Xz.extj(z,y,x))lth(y)=y.
~def Consj(x,x) , and
Consj(x,y)
1959.
.
may be read as : x, y represent kz°extj(x,y)
and
neighbourhoods
~z.extj(y,x)
are elements
to this intersection.
The proof is given in detail in Kleene 1959~,pp.
Kreisel's
such that
then
with a non-empty intersection ; belonging
provably
E W!3 & Kz.extj(z,y,x) E ~!
Co~sj(x) ~ ~y.e~tj(y,~) ~ ~! ~ Xy.extj(y,x) ~ Intuitively,
ECF,
~.~)
Consj(x,y) ~ kz.extj(z,x,y)
extj(y,x)
functions with a
there is a partial recur-
there are a primitive recursive predicate
(i)
Corolla~.
m~j : (t)j. ht : I+ I~l(y).
V.
of a recursively
and a primitive recursive
(provably in
the~
j < Iml(~,y),n)~
coinciding with
2.6.19. Theorem
such that
from the definition we see
dense basis, then to any
sire functional
in
Iy I E t
t , h t % 0 ; moreover,
recursively
u,u',kj,n
: ~yl({~t(~,y))°
Now we can always find a
2.6.18o
= {YI(j))],
otherwise.
{YlE V*.
For this
,
& U k = (t)~] , j
related
(and in fact equivalent)
notion,
Since the proof is not very informative
86- 89 ; for
the proof is in Kreisel
and rather long, we shall
omit it and refer to Kleene 1959A. Kleenets
set of associates
of type
not coincide with it ; let us call it
j correspond to our W~J , but does W! for the time being. The principal D
~5o
m
~
distinction between
W: and J the additional condition
If we wish to use
W1 3
is that the elements of
W:
must satisfy
instead, we have to make the appropriate changes in
J
Kleene's proof. 2°6.20. Theorem. for
ECF
Proof.
QF-AC
T
(provably in
Suppose
(11
Vx~ ZyT A(x,y), quantifier-free.
Let
F, F,, O
to pure types, described in We can find a functional ~T (2) ~A x ~ y =0 ~ ~A
E - H ~ w)
holds
We have to make use in an essential way of the recursive density
theorem.
A
(relative to the language of
EL).
a term of
be the mappings effecting the reduction
1.8.5 - ~
~A
such that
A(r,x ~ ,
r,ya~).
~_~w
By the recursive density theorem there exists an element ating a recursively dense base for the objects of type ~6 W~D¢)(QT) 0
represent
~ 6 W I(O)OT D~
in
enumer-
ECF.
Let
~A o
Let us define
Note that for (~I~)(6) = 0
~
W ~a I ' it follows that there exists a
(by (I), (211.
Hence there is a
u
Now this implies
such that
I 6E W OT
(~l~)~z = I
such that
for suitable
7Ilw.u 6 ~z ; and therefore
~(~)
z.
is
defined. If we put
~ A1~.(71
~z. ~(~)),
then readily
~,(~1~)(~(~)) and therefore also in
= o ECF
~(~)~ Vx~ A(x, z~) . 2.6.21. Theorem.
HEO
and
ECF(~)
are isomorphic w°r.t, extensional equal-
ity ; i.e. we can find, for the pure types, functions
gn' hn
such that (if
application according to
Xn'X~ ~ ~n'
ECF(~) , i.e.
Xn-~ ~ ~-~' Y~'Y~ ~ '
n ~ I , partial recursive
W*n~ =def- Ix I Ixl£W11m ' and
[x](y)
denotes
[x](y) ~def {x}({y}) 1, then for
Yn-~ ~ ~n - l :
151
gn(Xn) E W'n, hn(Yn) E IVn , and (i) n
[gn(Xn)](gn_1(Xn_1 )) =
(ii) n
lhn(Yn)}(hn_1(Yn_1))
lXn](Xn_I)
= [Yn](Yn_1)
(iii)~ In(hngn(~n), x n) (iv) n
I~({gnhnYn},
(v) n
In(Xn,X~)
(vi) n
I~({Yn},
{Yn })
-- I~({gnXn} , {gnX~}) {Y~}) -- In(hnYn,
Proof°
By induction on
Assume
gk' hk h
For
I < k
and (i)k- (vi)k to have
of a partial recursive function such that
{hI(x,y) ~ [x](gn(y)) . By the
that
immediate.
k
be the godelnumber
such that
n= I
to have been defined for
been proved for Let
n.
hnY~) .
s-m-n-
theorem there is a primitive rec.
{h}(x,y) ~ {~(h,x)}(y);
take
{hn+1(yn+1)}(xn) ~ [yn+1](gn(Xn))
'
hn+1(x) ~ ~(h,x) o It f o n o w s
hence for
yn+1~ W*n + 1
'
hn+1(Yn+1) E Wn+ I . Now let F be the operation on elements of
W 1 defined by n WI F(IYnl ) = {Xn+1](hn(Yn) ) . By (Vi)n , F is really an operation on n' not just on W*. As we have proved, W I possesses a recursively dense n n basis, hence by theorem 2.6.18 we can extend F to a partial recursive functional
F'
such that
F'(~) ~ UminyT(%,~) (K!eene 1969, "34.1 on page 69). q0n =
~hen
F'(~)
~
Now define
iUm+1 if T(zo,m ) & m ~ n 0 otherwise° ~-~
is uniform]~
for some
m
~y(~(~) : x+1) .
recursive in
Xn+ I , so
gn+1(Xn+1) ; this is the required
~
is a function with g~delnumber
gn+1 "
Then [gn+1(Xn+1)](yn) = F(IYnl ) = [Xn+1}(hn(Yn) ) . Now the verification of (i)n+1 - (Vi)n+1 is completely routine:
(v)n+ I, (vi)n+ I
follo~ from
(v) n, (vi) n ; (i)n+ I by (iii) n, (v) n
(ii)n+1 by (V)n , (~i)n+1 : {hn+Ign+1(Xn+1) }(Xn) : {hn+Ign+1(Xn+1)}(hngnXn) (by (iii)n) = [gn+1(Xn+1)](gn(Xn) ) (by (ii)n) = {Xn+1}(Xn) (by (i)n) ; SO
In(hn+Ign+1(Xn+1),
Remark.
Xn+1) o (iV)n+1 similarly.
For Kreisel's definition of
in Harrison
ECF(~)
the proof was given in detail
1963 ; here the situation is even simpler.
~52
2.6.22. The models Let
~L
ICFr(~) .
be a universe of functions which is a model for
hybrid between ECF(%L)
ECFr(~),
ICF(@)
and
ICF(U) ,
EL.
is obtained by defining the set of objects of type
sire elements of the set of objects of type tively.
A kind of
and similarly between ECF(~)
Let us call the resulting models
a
in
~
ICF(~),
ICFr(Z/),
ECF(~)
ECFr(IL)
and
as the recurrespec-
respectively.
More formally • (a). 11
V r~ ~ V~I 0 R, to
~
w~r = WI~ 0 ~,
Ir~ is the restriction of
XW r
(b). The objects of type c~E V r ,
for
~ in ICFr(i~), ECFr(~/) are now the pairs (~,a) , (~,c) , ~ E W~r respectively ( ~ to be replaced by a number
and
e = 0 ). It is easy to see that
E-H~A w
respectively.
ICFr(~),
ECFr(~/)
are again models of
Especially interesting is the case where
Li
N - H ~ ~, is a
universe satisfying bar induction ; hybrid models for such universes we shall often simply denote by The
ICF r, ECF r.
G - r e a l i z a b i l i t y of Neschovakis 1971 may be viewed as "abstract"
modified realizability relative to
ICF r
objects of finite type as elements of
(cf° 3 . 4 . 2 ) by interpreting the
ICF r
(cf. ~ 4 ~5)
Q
2.6.23. A variant of
ICF
and
ECF .
Sometimes it is a disadvantage that the definition of the application operation is not uniform in all types for
ICF, ECF.
be removed by considering the following variants I We redefine the species V~ by ~E
of
V 0I ~ (~: x~.~0)
~6V (~)0
I _= ~ ( ~ = = O,
~p) , where (~)(~)
~
is defined as follows:
= 0
1BY
if if
~0 Note that
This disadvantage can
ICF*, ECF*
Ith(n) > y lth(n) _< y °
( ~ ) I kz.x = ~z.px. for
Application is now always interpreted as
or
TWO
"I" °
The interpretation of the constants is then adapted as follows : (a)
[0] --- Xx.0
(b)
IS] -= ~S,
so that the numeral
(In general, a function
8
n
is represented by
is represented by
~8 .)
hx.~°
ICF, ECF.
~53
(c)
[ 1 ] , ~ ] = A'~A~I~.~
(e)
[%]
~s f o r t a e s u b e a s e
Equality between terms of type
(~,o) As a pleasant
2.6.24.
s
is interpreted
~)
as
(~,o) ~-~ v~(~ = ~ ) .
=
corollary we have :
(~,~)
have
i~ ~,6.~ ('out now f o r a l l
~/0
( ~ , ~ ) ~-,
=
Remark.
Vx(~
~x).
=
The latter pleasant property would not hold, I V I by
if we would
bluntly defined
~ v ~I
-def V~ 6 V 0~
~
Vo~(~l~)
since then each function may be represented by many different elements of I VI . The redefined model, ICF*, is not isomorphic to ICF, for types more complex than corresponds ~I~ =~ ~, ECF*
O, I ; i e
to a single element of V I in ECF there usually I an infinity of elements of V~ in ECF*, since the predicate °
for.gQven
•
~, ~
does not determine
is the analogous variant of
2.6.25.
Pairing operators in
ECF,
~
uniquely°
with the obvious definitions.
ICF, ECF, ICF*, ECF*o
In case we wish to extend our type structure with cartesian products obvious interpretation
[D0,0]
[%,1]
~ ~xy.j(x,y),
[D$,0]
~ Jl'
^°~°A1~(~'J(x,~z))
~ [~,o]
ICF*, EC~*
The systems
Similar to
all
ICF ,
HRO-, HR~0
for
[~,~],
we d e f i n e
G~T ~ 0 ~ b u t new f o r
2.6.26.
~ A o ~.j1~O,
~ #~`~'jt
~'
IDa, T] ~ A l ~ . ~ z . j l ~ ,
[~'~,~] ~ A I ~ . ~ . j 2 ~ , case
~ J2
A1~.J2~o.
IDa, T] ~ A I ~ A I ~ . ~ . j ( ~ z , ~ ) ,
For
[D~,0]
[95,1]
,
[Dl,o] ~ A I ~ A l x ( ~ ' J ( ~ ' ~ ) ) IDa, o] ~
~,~ % 0°
[~,~],
~ro"o,~]
as f o r
ICF.
we can describe two extensions
als, or precisely the intensional ¢* 6 (~)I,
~C~, EC~ i n the
G, m .
express that the objects of finite type are intensional
adding constants
the
of the pairing operators would be given by :
for
continuous functions
~ ~'
~:T
of
~-HAA w
continuous
which function-
respectively,
6 ((~)T)(a)1
for all
by
~,m6
154
and axioms
G*I
* x ° = ky.x °, ~ x I = ~x 1 ~0
G*2
~ . x ~ = ~ ~ y~ ~
~'3
~o) x (~)T
G*5
V~ ~ V I ~ ~ ( ~ ,*y ~ ~
where
VI
I ~*
as i n 2.6.23)
x ~ = ya
y°
=
4"~ x(~)~y~
~)
is defined in 2.6 23.
the d e f i n i t i o n b e c o m e s
(~
U s i n g the original
slightly more complicated,
definition
of
V1
because we have to distin-
guish more cases for the axioms.
N-K~(~+G*I-G*4
is
It is also possible practical
interest.
IC~F-, N - H A W + G * I - G * 5 to define
systems
is
ECF-, E C F ,
ICF. but they are without
155
§ 7.
Extensionality
2.7.1.
In this section we bring
extensionality
rules and axioms
on computability 2.7.2.
and continuity
and the models
Extensionality
Extensional
in
together
=e
some results
on continuity
in
N - HA ~ , as applications
of
N -HA~.
and hereditary
equality
~_~w.
and
of our results
extensionality.
for type
s ~ (~I)...
(an)O
is simply defined
by o o _ o o xa=eya =def x ~ly an x =e y ~def VZl "°" Zn (xz1°''Zn Hereditary
extensional x o~yo
equality
is defined
°
over the type structure
by
~def x° = Y °
~(~)~ ~ y ( a ) ~ Note
~
= YZ1'''Zn)
_
a a
=def VXlYl
(xl~Yl
~ xxI~YYl
) °
that x o ~ y O @__> x o
yO e-~ x °
o
=e
= y
and x t ~yl ~
vzOuO (z o = u o ~ x 1z o = y luO ) e-~VzO(xlzO=ylz° ) e--ex a
(and similarly
x
1 a
~y
1
=ey
a
¢->x
The axiom of extensionality EXT,, T
x
a
=e y
a, T
Assume
EXT
structure Let
x
=e y
for all
art
a
x a
(o)o).
~(O)~yO. extensionality
z(~)Tx a ~ z(O)~ya
is equivalent
that a
e
axiom of hereditary
xa ~ y a ~
for all
~ (o)(o).o.
if
states
a ~ ~(~)~x a =
The corresponding (I)
a
=e y
to
a
a
=e y
e-~ x a
EXTa, ~
a, T ,
*--ex
~ ya
x
for all
a, • .
then we prove by induction
a
~y T
=e y
T
e-e x
T
~y
T
Then
( ° ) " :e u (°)'" ~ Also,
xa ~ya e_~x ~
hence and since
(a)~z
=
X=eX,
=
e y
u(o),,e so
a
v°x (~x :e ~ ) --~
~
x~x
u(a)Txa
a a
=
e
U
Vx y ( x ~ y ~
, z~u
~
Vx°(z:': ~ u x )
(a)T y a
~
,
tlx~uy
zx~uy) e-ez~u
implies
Vx(zX=eUX)
.
over the type
~56
Therefore
(I) is implied by
Conversely,
assume
for it holds
EXTa, T
(I) for all
for
a, T .
~ = 0 , and if
We note that
~ = (T) p,
xc ~x ~
for all
~ ;
then and the right hand side
of this equivalence Now assume
holds because
x s ~ y q d . _ ~ x G =e ya ,
of (1).
x v ~yT ~
xV =e yT •
Then x(~)~ =e y(~)~ Also,
if
z~u,
then
which is equivalent Conversely,
"
Vz~(xz =e y~)
yz ~ y u
to
"
(since
VJ(xz
y~y)
;
~yz) .
w~ua(~ ~u- xz~zu),
therefore
x ~y.
~~(a)T 4--) VZlZ2(Z I ~ z 2
x (a)~
xzl ~yz2 ) v~1(x~1 ~YZl) " vz1(x~1 =eY~1)
"
~o ~(~)~: e y(~)~ Therefore 2.7.3.
also
(I) implies
Theorem.
and type Proof.
0
In
variables,
instead
~ t~t
built from constants,
type
I
.
of
t
t., then t~t' l verifies, with the help
One readily x 1 ~ x I, S ~ S , :
~-~m
~t~ (I < i < n ) , and 1 l -- -tl,...,t n , and t' is constructed
from
z
t
N - H A~ w , for any term
(WoA. Howard)
We note that if
0
EXTc,T .
H~H,
Z ~Z . R~R
0
~xxlYYlZI(X~Xl
2.7.4.
(i)
& z
t , but everywhere
as
of this remark, is established
0
O
= z I ~ Rxyz
by application
0
that
0 ~0,
proving 0
~Rxlylzl)
with
t! 1
x ° ~ x °,
by induction
on
.
CorollarFo
~-~p where
x~ ~ y ~ t[z ~]
variables, (ii)
& Y~Yl
is constructed
t
~-H~W~
~ t (a )~ Ix a] ~ t
is any term constructed z
in particular,
(i) is i m m e d i a t e ;
2.7,5.
Theorem.
t, s, F[t], and
variables,
type I
, and constants.
Proof.
types 0
from type 0
V x ( ~ x = ~x) ~ t[~] ~ t [ ~ ]
and hence
where
(° )~[yS]
I,
if
t
is of type
(ii) follows
We have the following
F[s]
0
by our remark derived
or
that
F[t],
F[s]
~=e
8 ~
~ 8
rules of extensionality
are terms built from constants
and where
I :
and variables
are of type O, I or
2.
of
•
157
First proof.
Note that the case where
other cases;
for if
to
F[t]~ = F[s]~,
in
FIt], F[S].
F[t] where
~
Similarly if
be conservative
is of type 0
then
is a new type I
attention to the case where and a conservative
F[t]
is of type 2,
FIt]
So we may restrict our
Now we use theorem 2.6.12 on
extension result to be proved in
over
N-H~A w
is equivalent
variable not occurring free
is of type I.
F[t] 6 0 .
includes the
F[t] =e F[s]
for universally
5 66(iD,
quantified
ECF ,
implying
AC ~o
equations between
terms of type 0 ° If
~t
=e s , then
t, s are represented
functionals ; and since
ECF
in
ECF
is extensional,
by extensionally
equal
it follows that
[ - H ~ w b [F[t]]EC F " [F[S]]ECF" Combining this with (2.6o12) [-F~. w ~ [F[t]]EC F " F[t] _~wb
[F[S]]EC F " F[s]
-~b
Fit] = F[s].
we obtain
Remark.
For the case where
not contain variables HE0
instead of
F[t]
is of type 0
of type I
~
or
I,
and
t, s , F[%]
do
we may also use theorem 2.4.14, using
ECF ; we do not need the conservative
extension result of
366([i) in this case. Kreisel's notes appealing fulness of instead.
HE0
F
wor. t. numerals
contain a sketch for another proof, not (2.4.14 or 2.6.12) but only to faith-
and using partial reflection principles
F[t], F[s]
closed,
F
of this proof.
is of type 2
to the ease
This argument may be combined with one of the
for the case where
Let us, for simplieity~
functionals
reconstruction
(For reducing the ease where
is of type I °)
other arguments
t, S ,
1971A)
We failed to find a satisfactory
Second proof. where
(Kreisei
to the uniform faithfulness
F
is of type I °
once again restrict our attention to the case F[t]
of type 2 o
F[t], F[s]
represent
of type 2 , which have a provable modulus of continuity
b~ 2.Z 8 . Hence there are terms (I)
{"
Now let
t2
tl, s I
of type 2
such that
~ ~(t1~) = ~(t1~ ) ~ F [ t ] ~ = F [ t ] ~
be a term of type
{ t2nx = (n)x t2nx = 0 Then obviously,
for
x < lth(n)
elsewhere. by (I)
(0)I , such that
t2n
is defined by
constant
~5s
-~®b
-~
henoe
b
V~(F[t]~ .... F[s]~) ~
=
F[t]~
-~®b
Vx(F[t](tZx) = F [ s ] ( t 2 x ) ) ,
follows from
F[s]~
F,[t]x : F,[s]~
where 2.7.6.
Counterexample
tion that
(H.P. Barendregt).
Fit], F[s]
a counterexample
F[t] E O,
Take e.g.
defined Then
as
t, S
~-m
Fs
(where
2
, then
CTNF'
only, we can give
x--'x may be supposed to be
= Rx(Xuv.(R0(Xu'V'oV')u))x),
and l e t
F ~ ~yl x 2 y l
x2
obviously holds ; but
N-~W
would have to hold in all versions of x
I
~bF[t]:eF[s]
s =- kx.x'--x
Rx(kuv.prd(u))x
Ft =x2t,
or
closed.
t =- bx.0,
N-H~A * ~ t = s
of type 0
to
N-~bt=eS with
If in 2.7.5 we remove the restric-
contain variables
[t]IIRO. =" [S]HRO.
can be embedded
Fs,
for then
HR0 , i.e. if we take
H~O~
in the
(2°5.5).
~Ft:
HRO-versio~nto
Since
2
x2t=x
(Ax.x,2)
s
for
which the term model
t, s have different normal
forms,
this is obviously false. 2.7.7.
Counterexample
(R. Statman).
We can find a closed
F
of type 3 ,
such that
y2 ~ F x 2 = F y 2
N-~-~x2= ~
Take 2
F
~
t o be
x 2 = (m, 2)
e
kx2.x2[kz°.(x~XwOoz°))]
of the version of
HRO
,
and choose two e l e m e n t s
described in 2o5.5~
distinct indices of the same I - I total, recursive sented by
(P, 3)
for suitable
~.
Since
Fx 2
such that
function.
F
2
-
x l = ( n , 2) , n, m
are
is repre-
is in full
x2[z(nx2)(~(nn)(nz°))] it is obvious that since the version of equality axioms Z(~x~)(Z(HH)(Hz°))
IEI,
that
HE0
considered
Z(~x~)(~(g~)(Hz°))
in the model are different,
satisfies the
and hence also
Fx~ % Fx~
in
the model. A similar counterexample 2.7.8.
Theorem.
has been given by H.P. Barendregt.
Every closed term
modulus of continuity in
t6 2
of
~-HA ~
N - H~A w , i.e. a closed term
( ~, ~ variables of type I ). Proof. From 2.3.13 we know that
possesses a provable t'E 2
such that
~59
(I)
N-HAW~
where
x
process
I
t~:y
*--) SRED(~,rtx1~,ry ~)
is the type I
variable
to which
~
is assigned in the reduction
(cf. 2.3.8).
Also, because of the derivability
of computability
for terms of bounded type
level (Cfo 2o3.11)
(2)
~- m ~ ~ v ~ : y SR~D(~, ~t~1',~ ~) . S~.~(~, x, y)
Now
expresses: of
x
z to
Note that
where
is the (number of a) standard reduction
S~(~, x, y, ~)
sequence relative to
y.
f , defined by fn = maxIni I 1 < i < k l
where
~ Sa(~, ~, y, z),
may be written as
nl,°.°,n k
+ I
is a list of all numbers for which
been used in the reduction recursive function of
sequence
n,
a~ i contr ~m i
has
may be taken to be a primitive
no
Also (3)
N-H~w~
~(fn)=~(fn)
-~ (SR(~,rt~,%~,n)
~-~SR(~,rt~,rs~,n))
o
Combining (I), (2), (3)-
Now, using a result from the next chapter (closure under a rule of choice, see
5 yd(ii)) we find that there must be a
N-~,
t' 6 z,
t'
a closed term of
such that
~ - m ~ ~- v~(~(t,~) : ~(t,~) - t~= t~). 2°7. 9 .
Product topology.
Already at type 3, N-HA ~
the functionals
are not necessarily
represented
continuous w.r.to
by closed terms of
the product topology.
Take for example F - XZ2oZ2[Xx°.z2(~o,oX°)] F
is discontinuous
~1'''''~k
t~
the product
we can find a constant t2~i = 0
while
w.r.t,
F(k~.0)
{= o = m+1
t2
topology at
k~.O,
since given
such that
(I_< i_< k) , F% ~ / 0 ,
= 0 .
if
o
t2
is defined as follows:
~i ( ~ i i _ < k a otherwise,
where
~i ~ ° ~k) m = max{~i(y) 1 1 < i < k ,
0
.
Now
F t 2 / O ; for
them, say
(n o nO)k .... , (H e nk)k
are all distinct, hence one of
( ~ ~k )k (0 < k
therefore
~I k, .°., ~k k,
and thus
Ft 2
takes the value
2.7.10. "Floating product topology".
m+1.
In Kreisel 1971A, Kreisel noted the
following continuity property for functionals of
N-~,
definable in
F 6 ((~)0)0,
N-
HA w , for types
closed terms of
t1[~
N-~w
((q)O)O.
Let
at arguments t6 (~)0
be
then we can find a finite number of terms
(~)o], .... tn[~ (=)°]
of type
o
s~oh t~at
Vz(a)°( l
.Ft
Fz)
The proof is an elaboration of Kreisel's sketch. Proof.
Let us call a term not containing
t6 (a)O
term of type 0,
and let
F[t]
as a subterm
by a standard reduction sequence Fm,... ,F °
are
F[z (a)°]
F[t]
Let
t-free
reduces to a numeral, say
[z/t]F m .... , [z/t]F ° = ~,
x,
where
t - free.
t~,..
t' be the set of all terms of type "' n duction sequence, and let t1[z ] ..... tn[S ] be ti[t ] = t:
t - free. be a
be closed.
By the results on computability,
Let
t
be a closed term in normal form,and let
I < i
Given
F[z]
and
a
occurring in this re-
t-free
t , tl,...,t n
terms such that are uniquely de-
termined. Now we shall prove, for fixed reduction sequence of
Fit],
t , by induction on the length of the standard for all
t - free
F[z]
such that
Fit]
is
closed, that (I)
Vz (~)° (
A (tti[t ] = zti[z]) ~ F [ t ] = F [ z ] ) . 12i<_n If the reduction sequence for F[t] has length I,
B~.
F[t]
is a
numeral and we are done. ~duc~ion_~po with
F[t]
Now assume
Assume (I) to have been proved for all
closed, and standard reduction sequence of F[z]
to be a
t - free term of type 0
with
t - free Fit] F[t]
F[z]~ 0
of length 2 k . closed, with
a reduction sequence
(2)
[~/t]F k, [~/t]~k_~ ..... [~/tFc =
and let
(3)
A
lii<_n
fore'terms
t.
(tti[t] : zti[~])
obtained from this reduction sequence in the manner described
i
above.
By the induction hypothesis ~ [z/t]Fk_ I m Fk_ I
((5) implies the
hypothesis with respect to [z/t]Fk_ I
Fk_ I ).
is obtained from
[z/t]F k
by application of a contraction to a
subterm of one of the forms
ZSlS2S3,
RSlS2S 3
or
Eels 2.
We have to dis-
tinguish two cases. (a) The leftmost occurrence of
~, H, R
part of a subterm of the form For example, let si ~ [z/t]s~,
ZSlS2S 3
s~1
be contracted into
t - free.
Then
a corresponding occurrence of F k = Fk_1,
it f o n o ~ s
t,
so
[z/t]Fk_ 1
subterm of the form and
ti[z ]
is
ts,
for
s1[t ] = s1[z ] .
s1[~] Now
= slit]
Fk_ I = [z/t]Fk_1;
s1[z ]
in
Fk_ I = Fk;
Fk_ I
by
of type
el[t],
in the subterm contracted does not
~ ; so
[z/t]F k s
by contraction of a
must be of the form
tilt ] ,
= tti[t] since
implicitly contains a reduction se-
hence the hypothesis
(3) for
Also
zti[z ]
since also
by contracting
Now obviously
Let the result of the contraction be of the form
contained in the hypothesis thesis,
Fk
' ' ' . s~s3(s2s3)
F k = [~/t]F k-
Z, 9, R
st[t], s1[z ] being t - free. The reduction sequence for [z/t]Fk_ i quence of length ~ k
is obtained from into
is obtained from
s
t - free.
SlS3(S2S3) , and let
and since by induction hypothesis
that
(b) The principal occurrence of belong to
Fk_ I
Zs'IS2S ' 3'
[z/t]F k = [z/t]Fk_1,
[ ~ / t ] F k _ I = Fk_ I ,
in the subterm contracted is not
t.
(3) for
s1[t ]
is
F[t] ; therefore, by our induction hypo-
tti[t ] = zti[z] , hence under our assumptions
: ~ti[~]s1[z ] 6 0 , replacing the relevant occurrence yielding
Fk,
does not change the value, so
[z/t]F k = [z/t]Fk_1 , we have
[z/t]F k = F k.
162
§
8.
Other models of
2.8.1.
Contents°
models of
w.
N-HA
In this section we rather briefly comment on some other
N-HA ~
occurring in the literature.
given by a concept of importance sire functional Kleene
1959,
of higher type~ as described by his schemata
1963A)o
investigated
The first of these models is
in its own right : Kleene's notion of recur-
Unfortunately,
$I - 9
from a classical point of view,
e.g. Kleene assumes his function-
als of type
(a)O
("classical"
in the sense of "existing in say the intended model of
theory").
to be defined on all classical
It would be interesting
S I - 9.
functionals
of type
meaningful
is
model for the theory of
Because of this lacuna in the literature,
be rather brief and restrict ourselves
a Z F - set
to know which class of functionals
singled out from a given constructively finite types by
(see
this concept has up till now only been
we shall
to some remarks on the recursive
functions of finite type. Two other models for the theory of finite types were introduced Scarpellini
1971A.
in
They were introduced because they provided models for the
theory of bar-recursive
functionals
(cf. 1.9.16).
See below,
in 2.8.5 and
2.9.9-2.9.12. Two models introduced by Howard are briefly discussed in 2.8.6. 2.8.2.
The schemata
$ I - 9.
Kleene uses in his description let
~' 2'' "'°
$I - S 8
(Kleene
1959) only variables
be used for sequences of variables
of pure types ;
of pure types.
The first
schemata are as fellows.
s~)
~(x°,~)
s2) s3)
~(~)
=
sx °
=
o
=
~(o,~)
s4)
~(~) = ~(x(~),~) ~(o,~) = ~(~) ss) I~(sx;,~) : o o = _ : x(x ,~(~ ,~),~)
S6)
~(~) : *(~I)
( ~
not empty,
consisting of
21 obtained by shifting the
k+1
variables
k+1 st variable
and in
to the front) S7)
~(yl,x°,~)
$8)
~(yk+2,
Permuting
: yx
~) = yk+2(xuk
x(yk+2,uk,~ )
the order of variables
of different
provided only the order of variables $I - $8
characterize
type is regarded as immaterial,
of the same type is retained.
Kleene's primitive recursive
functionals
of finite type.
163
In the schemata
~
is always the functional to be defined,
~, X
are
supposed to have been defined before ; all functionals considered are assumed to be numerical valued.
(The functionals are considered throughout from the
extensional point of view.) To each functional we can assign an index ; the index for uniquely determines the intended functional, introduced by
$4 - $6, 88
$I, $2, $3, S7
the index for a functional
can be computed primitive recursively from indices
X, ~. The primitive recursive functionals are generalized to partial recursive functionals, by reading
~
for
=
in
91 - 8
(where
~
intuitively means
the same as in elementary recursion theory) and adding a schema which permits "self-reference" by introducing the index as an argument:
~(o,~,~,)
sg)
Of course,
~
~
in
{xO}(~) . S9
itself also obtains an index, primitive recursive in
the numbers of arguments of each type in
z
and
=
$I - S9
z' =
may be viewed as constituting a generalized inductive definition of
the relation
{xOl(~) = y O .
Kleene shows that the class of functionals determined by under the minimum operator,
SI - 9
is closed
and definition by cases (Kleene 1959, XVI, XVIII);
but closure under the minimum operator cannot replace
$9
(in contrast to
the theory of recursive functions). 2.8.3.
Recursive functionals as a model for
From inspection of $I - $9
E - HA w .
it will be clear that the only fact which needs
to be verified and which is not immediate from the definitions, under definition by recursion as given for numerical
is closure
types 1.8.9 (iv).
Via
the methods of coding finite sequences of pure types into pure types, as in 1.8.7, this is equivalent to showing closure under the schema
{~(o, yJ~ ~) = ,(yJ, z) ~(Sx°,yJ,~) = ×(x°,y~,XyJ.~(x°,yJ,z), ( ~
the functional to be defined,
X, ¢
~).
given functions.)
But this is precisely Kleene's XXIV (section 4.5 in Kleene 1959)o 2.8.4.
Remark.
That the functionals generated by
81 - 89
do not contain a
function representing a modulus of uniform continuity has been proved by R.O. gandy (unpublished). 2.8.5.
Scarpellini's models.
(8carpellini
1971R,1972R)
The starting point for the definition of these models is based on the following axiomatic characterization of convergence of sequences:
164
Let
X
(0)X
be a set on which a relation and
X
is given (we write
~
of convergence between elements of
n ~ p
or for short
the class of infinite sequences of elements of I)
If
n ~ p , and
2)
If
Pn = p
3)
If not
k I< k 2< ...
for almost all
n,
then then
i
is
X ) such that
i ~ p , n ~ p ,
n ~ p , then there is a sequence
such that no sub-sequence of
Pn ~ p; (0)x
k 1 < k 2 < k 3...
converges to
p ,
4) If n ~ p, n ~ q, then p = q. Then (X, 4) is called an L - space° Let (Xi, ~i) ~ i = I, .... s from
(X I ×... ×Xs)
to
Y
and
(Y, ~)
be
L - spaces.
f
is said to be continuous, if
f(Xl,n, o.., Xs,n) ~ f(xl,...,Xs)
whenever
<Xl,n>n ~i xi ' for
Let us denote the species of continuous mappings from C(XI...,Xs,Y ) .
A mapping
C(XI,...~s,Y)
is made into an
I_< i_< s .
X I X... × X s
to
Y
by
L - space again by defining
n ~ f ~def V<Xl,n> n... V<Xs,n>n Vx1"''Xs((<Xl,n>n ~ xl & ' ° -
...&<Xs,n> n Now Scarpellini~s first model
X s ~ (n ~ f ( x l . . . . ,X s) ) ) . MI
(called
scribed as follows, for the type structure SO = N
~
in Scarpellini 1971) is de-
~0
(cf. I°8.9).
are the objects of type 0 , with the following notion of convergence : x ~ def ~q
<Xn>n
S¢IX...)~ p
=
S¢ I
X..o
MS
~p
,
S
(¢I X...>~p)T -=def C(S¢ I'
..
the notion of convergence obtained from the notions in
., Scp, S )
with
S¢I , ..., Scp, S T
as
described above. MI
can be shown to be a model for the bar-recursive fttuctionals. The description of the second model (called
K
in Scarpellini 1971~)is
too long to be reproduced here, but may be viewed as a refinement of the first model. 2.8.6.
Compact and hereditarily ma~orizable fu~qtionals.
W.A. Howard has described two other models for
~_~w.
that of the compact functionals over the type structure
The first concept, T , is defined as
follows: (i)
A species of natural numbers is compact iff it is finite.
(ii)
A species
X
of functionals of type
compact species
Y
Ix(¢)Ty~ I x~ X ~ y ~ Y } (iii) A single functional
(~)T
of functionals of type
is compact
iff, for each
¢ ,
is compact. t~
is compact iff
It can then be shown that the functionals of
It~] ~-i~ w
is compact. are compact (in fact,
even the functionals of the Sheory obtained by adding bar recursion of type 0 (BRo)
are compact).
165
The second concept,
that of hgreditaril~
majorizable
functionals
is intro-
duced as follows. We introduce a concept
~j(xl,x2)
over the type structure
(i)
}~o(Xl'X2 )
~def
T
( xI
"majorizes"
x 2 ) by definition
as follows :
XlZX2
(ii) ~j(=)~(xl,x2) ~ Vyly~(~4(yl,y2)- ~4(xly1,~2y2)). We now define : a class
X
class
of
Y
of functionals
of functionals
is hereditaril~ ~
~x6 X ~[v6 Y ~aj(y,x) .
that the class of fttuctionals of
~_~w
is hereditarily
Kcward B makes the following application translation
by a
It is not hard to show majorized by itself.
of this concept : the Dialectica
(§ 3.5) of the simplest non-trivial
case of the extensionality
axiom Vy 2 V~[Vx(c~x= 6x) ~ y 2 ~=y28] is of the form
(1)
~X Vy2 V~[~(Xy2~) = ~(Xy2¢~) ~ y 2 ~=y2~] ;
and it can be shown that by a functional
from
of the extensionality from
N - H A w.
~-H~
X
satisfying w
(I) cannot be hereditarily
, and therefore
axiom has no Dialectica
For more information,
majorizable
the simplest non-trivial interpretation
instance
by a functional
see Howard B (Appendix of this volume).
166
§ 9.
Computability and models for extensions of
2.9.1.
Contents of the section.
N -HA m .
In 2.9.2- 2.9.4, 2.9.6 we describe exten-
sions of computability arguments ; in 2.9.5 , 2.9.7- 2.9.11 various extensions of our models for
~_~w
are described ; see especially 2.9.9, for a simple
model for bar recursion of higher type. 2.9.2.
Extension of c omputabilit 2 to (the functionals of) ~ - ID~Bw
and
related theories. For the theories of first-and second-order trees, computability is discussed in chapter VI. In Howard 1972 , a first-order theory tending
HA
U
is considered, obtained by ex-
with a species of "abstract constructive ordinals", introduced
by a g.i.d..
~U is embedded in a theory
~V analogous to
objects of finite type over natural numbers and ordinals. computability of terms in
~
~+I(I)
°
, for
by means of an ordinal assignment, thereby
determining the "proof-theoretic ordinal" of 1950)
q f - W E ~ - HmA~
Howard shows
U
as Bachmann's (Bachmann
U~ is proof-theoretically equivalent to
IDB.
In Troelstra 1971A, there is a proof of computability for the closed terms of
~ - ID~Bw
which differs from the method used in chapter VY .
Although the method in chapter VI
is more elegant, we thought it not without
interest to demonstrate the other method also, for the case of (cf.
~ - ID~Bw
2.9.6).
The contraction rules are as for
N -HA w
with product types and pairing,
and in addition I(@It)s
centr St
I(%2ts)t ° contr It(<s>* to) l(@3t)O
( %,
c2
contr O,
are constants of
l(@3t)(St' ) centr l(t(C1t'))(C2t')) ;
~-~
sueh that
<%>.
(C2t) = St
for all
t E 0 ), and furthermore
where and
>' X*v
2.9.3.
If
ItO ~' Ss,
If
ItO ~' 0
then
, then
~tt't"
contr tts
T tt't" contr t " ( X * v . ~ ( ~ 2 t v ) t ' t " ) t
means standard reduction, defined in terms of "contr" as in 2.2.2 indicates the combinatorially defined
k- operator.
Co mputabilit 2 for bar-recursive functionals
( ~ - I ~ w + BR) o
Proof of computability for bar-recursive functionals are to be found in Tait 1971, Luckhardt 1970 , 1973 and Scarpellini 1971R; another exposition is in Girard 1972 (following Tait's proof).
167
The methods of Tait 1971 and Luckhardt 1973 are essentially the same. A straightforward extension of the computability proof as given for - ~
does not work, because the induction hypothesis is too weak.
Specifically, suppose we wish to show to show that for computable
B~
t, t', t"
to be computable ; then we have B~tt,t"
is computable.
would have defined the computability predicate w.r.t, the assumption that
t, t', t"
But if we
closed terms only,
are computable is weaker (at least prima
facie) than the assumption that they are computable w.r.t, larger classes of terms ; e.g. a functional of type 2
which is computable for reeursive
arguments need not be computable for arbitrary arguments, but it is functionals of the latter kind we are intuitively thinking of. Taitls solution is to throw in additional terms according to the clause: If
~
is a function from natural numbers to natural numbers,
then the pair
(~,~)
is a term of type
(0)T .
Luckhardt's formulation (Luckhardt 1973 ) is similar. The intended interpretation of these additional
terms is as follows:
We assume all terms to be coded by numerical functions such that it is decidable whether a function codes a term of a given type or not. if
~
is a numerical function, and
~
preted as follows:
if
--
T) then
~
~m = kn°~j(m,n) ,
codes a term of type if
am
(~,~)~ contr ~ )
Scarpellini's method is slightly different.
~,
.
Whereas the original com-
at least for closed terms, only referred,
to the model of the closed terms themselves, ~-~+BR
is inter(m,~)(m) =
does not code a term of type is a fixed constant of type T , say 0 T "
(T,~)(m) = ~!~)
~m;
For the new terms~ we add contraction rules
for
(T,~) then
m
is the term coded by
putability argument,
T,
Then,
so to speak,
Tait's and Luckhardt's method
uses an embedding of the closed terms in a "generalized
term model".
A difference is, that Tait uses (informally)
DC~ and classic-
al logic ; Luckhardt formalizes his treatment in a system with intuitionistic logic + EBI D .
Scarpellini on the other hand uses instead of this
generalized term model his model
MI
(cf. 2.8°5) ; at a certain stage in
the proof, the computability of the bar-recursive constants is reduced to establish,
by means of bar induction,
model
(Scar~ellini
NI
197]A~page
spirit to the idea used in Troelstra
2.9.4.
truth of certain assertions in the
135) ; the method is thereby closer in 1971A, § 4 for the computability of
Computability for Girard's s~stem of functionals.
A first proof is in Girard 1971 ; a more detailed exposition is to be found in Girard 1972.
The contractions are as suggested by the equations
lg8
in 1 . 9 . 2 7 :
0(a)Tt contr 0 T,
D'(0 ~xT)
contr
Iv~[~], ~oVa~[~]tr co~tr 0°[~], Rtt'O
contr t,
D'(Dtlt2)
Rtt' n+1
contr tl,
oantr t'(Rtt'~)~,
Extensions
of
I - IDB w,
0T,
(Xx.t)t'
contr [x/t']t,
contr t2,
t (~[T])
oontr
HRO, HEO
Let us first consider for
contr
contr t ~ [ T ]
STe¢(~[~]) P(Iaa~[~], t l [T]) 2.9.5.
D"(0 ~xT)
St~t(0 a~[~]) contr t(0°[~]),
D"(Dtlt2)
ZV~[~],T D~t ~L~J)
0",
Pt I
•
to models for other systems.
HRO.
HRO
is easily extended to a model
~-HRO
by addition of
tx I t x } ~ t
vK
and representing
I
by
(~.~, (~)(o)o),
~1' ¢2 by ( t ~ y . Sx, ( 0 ) ~ ) ,
(~,.x(y,n),
(~)(0)~)
@3 by
and
(~ar.(l±y){Ixi(y)oi(tl(y)), The representation Let
Vl
of
Y
((o)~)~) .
is constructed
be the g~delnumber
by means of the recursion
cf a partial recursive
function
{wl(0) / 0 ~ t~ I }(Vo,W,x,y ) ~ txt(Iw}(o) ~ I ) t~}(o) = o ~ {~1 }(~o '~'X,y) ~ {t~}(~(%,~,~,y))
theorem.
such that
}(~)
where
~(v,~,~,y) ~ A~. If Ivl(I[%] I(~)I(~)l(x)l(y) , and
Au is to be chosen so as to correspond
-operator a
v
in the axioms for
~
to the syntactically
o Now by the recursion
defined
theorem,
there is
such that
t w t ( O ) / O ~ I I ~ t ( w ) t ( x ) t ( y ) ~ tx}({wt(O) "-1) , {~I(o) = o ~ {{ f ~ ( ~ ) }(x) J(y) = J y t ( Y ( ~ , ~ , x , y ) ~(~). By induction over
xEV(o)a ,
K
Wor. to
it can be shown that for all
e ~ w6 VK,
yE V((o)a)(~)~ , we have e = I~t ~ : f t f ~ t ( w )
For i f
e,
fwlel4,
[w~(O)/O,
t(x)l(y).
then o b v i o u s l y
' t l { V } ( w ) } ( x ) J ( y ) . Assume
169
{w}(O) = O, that so
and suppose
{{{~}(z)}(x) }(y)
{z~: k n . { w ~ ( x . n )
K-HEO
x.
for some
J{{{~}(w)}(x)}(y) .
Then
Hence we may take
z
such
Au.~(Vl,W,x,y) 6V(o)~ , and [Y ] = ~.
is defined similarly as a model for
the constants by the same numerals as in for
to be defined for all
~ - ID~Bw.
~-HRO
We may represent
; we define
Wa, I~
as
HEO , extending the definitions by W
~def V ~ ,
IK(x'Y) mdef l(o)o(X'Y) & x E V K & y E VKo The other developments in § 2.4, § 2.5 also carry over without essential change. 2.9.6.
A zplication of
~-HR0
: Computability q..f., the closed terms of ~ - IDB w.
The contraction rules have already been listed in 2.9.2. We define a closed term to be in normal form if no contractions are possible
and mt~no% of the form
%tt't",
Itt'
or
( t , %', t"
in normal form).
We then define "Comp" exactly as in 2.2.5, 2.3.7 (standard computability) with a clause added :
(iv)
Oomph(t)her Vt'(Oomp:(t') -- OompZ(Itt')).
Now, from the fact that
~-HR0
each closed term
a number
t6 ~
Itm £ ' For in If
m' ~
is a model n E V~
for
~ - ID~Bm,
such that if
we can find, for
CompS(t) , then
{nl(m) = m' o
K - HRO , natural numbers are interpreted by themselves.
[t]=n,
Itm ~' m'
i.e.
(n, K)
implies
represents
t
in
~-HR0,
I t m = m ' , hence in the model,
then obviously {nl(m) = m' , and similarly
in the ether direction. Now, if we wish to show that every closed term is standard computable, it is sufficient, as in 2.2.6, to verify that all constants are computable. ~e first note that a computable closed term of type O (i)
The constants of
N - }~
, D, D', D"
reduces to a numeral.
have been treated before.
I is trivially computable. (it) s _>' 0
~3 or
is computable ; for assume s _>' S~,
In the first case,
I(~st)s
and
% _>' t I
ComPio)~(t), CompS(s) o Then
for some normal
tI .
l(~3t)sr _~' I(~3tl)O contr. 0 ; in the second case
h' z(~stl)(s~) contr.I(t1(0j))(c2~)°
0 I, 02 are closed terme of
N _ H A.~-w , hence have already been shown to be computable, i.e. C2~ h' n2 ' so tions. reader.
C1n ~' nl'
I(~3tl)(S~ ) ~ I(tI~i)~ 2 , which is computable by our assump-
The verification of the computability of
~I' ~2
is left to the
170
(iii)
L
is computable.
Assume
We now define an operator numbers
n,
CompS(t),
ComPlo)~(t' ), Compi(o)q)(K)~(t")o
for each finite sequence of natural
@~n]~ (K)Z
as follows :
~°]t
-=def t
~[n*<x>]t2 ~def ~2 ( ~ n ] t ) ~
t h' t I,
Now let that
~tt't"
t' ~' t I
t' ~' t~,
is computable.
duction over the unsecured sequences of In I {[t]l(n):Ol
). (~.9.~S,
(tl, t~, t~
normal)
This is proved, for fixed {[t]}
We prove
t v, t" , by in-
(i.e. by induction over
(~)~, i.e. we show that
(I)
{[t]}(m) ~ 0
~ Comp"(~(9~m]tt't"))
(2)
Vx Comp"(~ ( ~ m * < x > ] t t ' t " ) )
.
= Comp"(L(~m]tt't"))
•
We note that
{[~m]t]}(=) : {[~]}(~.<~>)°
(3) Further
we
(4)
note that by
a
~m(Comp~(~m]t))
straightforward
induction on
lib(m)
•
Assume now { [ t ] } ( ~ ) = S [ , then by (3) {[9~m]t]}(0) = S[, and therefore by 2.9.2 ~ ( 9 ~ m ] t ) t ' t " >' t ~ [ , which is computable by hypothesis. Assume { [ t ] } ( ~ ) =0 , i . e . { [ ~ m ] t ] } ( O ) = 0 , and suppose Vx Comp(~ ( ~ m * < x > ] t ) ) t ' t " ; since I ( ~ m ] t ) 0 ~' 0 by our assumption, ~('~ m]t)t't''
--m' t~(X*x'~a(%~('~m]t#X)tl ~'''t'°l/1 '
which is computable by our
assumptions. With an application of 1.9o18,
(I), we conclude that
~ tt't"
~
2.9.7.
is computable, Extension of
hence
HRO, HEo
This extension of
HRO
to Girard's system of functionals.
appears first in Troelstra A, and is extended in
Girard 1972 to the intuitionistic ing extension of
HE0
(let us denote this analogue by objects of variable type. 0
(= 0a)
to be always
HRO HR02),
for Girard's theory of functionals the problem is how to interpret the
Noting that each type
a
is supposed to contain
it is reasonable to interpret each variable type as
a species of g~delnumbers least one element°
theory of types ; there also the correspond-
is described°
In describing the analogue of
a constant
~(%~°]t)t't" , i.e
is computable.
of partial recursive functions,
containing at
It is quite convenient if we could achieve this element
0 .
In order to do this, we note that for our standard pairing
j(O,0) = 0 ,
and we select a special godelnumbering for the partial recursive functions
~7~
such that (I)
{O}(x) ~ 0
for all
Such a g ~ d e l n u m b e r i n g g~delnumbering, be the
let
x°
may be c o n s t r u c t e d x
o corresponding
T - predicate
as follows.
be a g ~ d e l n u m b e r
In a given standard
of the f u n c t i o n
kx.O ,
to this g~delnumbering,
and let
and let ~
T
be
defined by x
~(x)
=
if
x = 0
0
if
x=x
x
otherwise.
I
o
°
W e obtain a n e w g o d e l n u m b e r i n g predicate
T'
T'(x,y,z)
%ef
and the r e c u r s i c n
and
we easily obtain,
as before,
special variables
T -
the
s-m-n- theorem
for species of one argument,
0 ; we suppose to each type v a r i a b l e V
x ° , and its
theorem.
Let us now i n t r o d u c e containing
0
T(~x,y,z) o
For the n e w g 8 d e l n u m b e r i n g
variable
by i n t e r c h a n g i n g
is d e f i n e d by
~
such a n e w species
to be assigned.
W e put f u r t h e r
Since
X 6 V(a)T
~def Vy6 V
x 6 V >~
~def J l X E V ~
0 6 V
by definition,
Zz6 V T ( l x } ( y ) ' z ) & J2xEV~
we r e a d i l y prove that D'
0
is i n t e r p r e t e d
Iva~[ff],r as
If
t[x I ,..o] 6 ~[~] ~
t'[xl,... ]
(Ax.x,
and
E
0 6 VT
for all
are i n t e r p r e t e d
T E mT o as before,
~ o (V~.~[ff])(~[T])) ,
I~[ff], T
is a term, not c o n t a i n i n g free v a r i a b l e s w h i c h then
t
is r e p r e s e n t e d
by a
such that
x16 V t~
for all
as
free in their type,
Then also since
So
D"
(~[~])(~.~[~])).
~.~,
contain
a s ( O ,o)
is i n t e r p r e t e d
,
V I , V 2 ~ o..
do not d e p e n d on
V
~
a ..o -- ~t'[xl .... ] a t'[x1,...] 6 V V ~ [ ~ ] .
is seen to represent
DT~%.
p - term
~72
If
%[xl I . . . . ] 6 (m[~])T ,
not occurring free in
in a type of a v~ri~ble free in
t,
then there is a
T,
and not occurring
p-term
t'[xl,... ]
such that
x1~V =
a...-~t,[~
which represents It follows
.... ]at,[~
t[x11,...]
in
.... ]~v(=[=])T,
HR02 .
that
y6V [~]&Xl~ V Then also,
since
~
& ....
~{t'[Xl .... ]}(y)&{t'[x1,...]l(y)6V~
V I , V 2 , ..., V
(y6 V ~ [ 4 ) ~ V ~
do not depend on
.... ~ t t ' [ x l , . . . ]
V
.
,
t(y) & { t ' [ x l , . . . ]
t(y) ~ V
So x1~ v~ I~°,. ~ :t,[x I .... ] ~ t ' [ x l , . . . ] ~ v ( ~ [ 4 ) whence
t'[xl,.o. ]
is seen to represent
The corresponding equivalence
model
relation on
Objects of type
~,
W
R~O 2
representing
I~O 2 ,
I(a)T , I X T
HRO =
2°9.8.
to
ECF(ZI) ° VI
HAS
(of.
For the sake of "homogeneity"
that
V 0I
0
(x) mdef I (x,x) . HEOo
(I [~](x,y)) .
of the models
of
More precisely,
V 0I
as well as
and VI
ICF2(ZL),
VI
II
it is of sequences,
for V I Hence 2 ECF (ZL) the second
The definition is in fact completely routine,
as an operation
ICF2(~),
ICF(LI),
the second definition in 2.6.23 of
we select for our definition of the analogues
I
W
to construct models
consists of natural numbers,
if we wish to permit substitution
definition.
relations with
as before in the ease of
which may be conceived as extensions
inconvenient
(~)
. Therefore
5.4.27)
is to be preferred for our analo~j.
defined
. Note that
W
can be useful in connection with an extension of modified
In a very similar way it is possible
ECF2(~) ,
the
W
defined as
Ia, I T
,
is exactly
for equivalence
is then automatically
I
equality between
relation on
I
mdef VI (I [~](x,y)) , Iz~.~[4(~,~}~cf/l
especially
realizability
the
W~
!
are defined in terms of
IV~.~[~](x'Y)
extensional
the field of
we only need to consider variables in their field;
ST~t.
is constructed by taking for
an arbitrary equivalence
in the definition of
T,
once we have re-
such that
~ . o II ~ ~ ~x.o
( s i m i l a r to the replacement of {x}(y) by { x l ' ( y ) ~def {~x}(y) in 2.9.7). An operation II satisfying (1) can be defined 5N ~11 p " ~ ~def F ~ I ~ ' ~ ' where
F
is given by
173
[ 2.9.9.
0
if
o/x
otherwise°
Models for
~x= I
N-HA w+BR.
The simplest model is presumably functions satisfying
EBI D
ICF(~) ,
(Cfo Io9o2~).
the classical universe of functions. E L+EBI D,
if
~
is any class of
For example, we may take ~
to be
The proof is given in the system
in 2o9o~O.
Other models are the term models of Luckhardt 1970, 1973 and Tait 1971, and the models of Scarpellini 1971~,197~A (efo 2.9.3). 3imilarly,
ECF(~)
is an extensional model for bar recursion.
for this is also given in
E L+EBI D
(2.9o10).
The proof
The corresponding result for
the term model of Luckhardt 1970 is proved in Luckhardt 1972 (1975). Scarpellini established the corresponding result for his second model (cf. Scarpellini 1971 @ i n 2.9. 10. Theorem. be a model for
~roof.
Set
Scarpellini 1972A.
If ~
satisfies
N-HAW+BE,
~
E~L+EBI D,
ECF(U)
VI( (~o ) o ) o,
~
for
( V;) ~
finite sequences of elements of type We define
7
ICF(~)
~-H~W+BR,
~ V ~~v ~ ),
then ( ~
in
can be shown to E L + E B T D.
denoting the type of
~ 6 VI~)~,
6 6 VI(~)7)(~)
.
by
{((~)x : (Y)sx for x < l t h ( ~ ) 7)Sx+u = ®9 Then we can find an
~(7)
< lth7
~(7) _> l t h 7
e
if
®
represents
0a
in
VI ,
such that
"* e I ( ~ ' el, fl, 6, %') =" fl [ -* e I ( ~ , o~, ~, 6, 7) ~ 6 I ( A ' ~ . ~ I (~, B, 6, 7 x ~h), %,) . e°
By the recursion theorem analogue (1.9o~6) we can find
such that
~(~) < l t h 7 ~ % 1 ( ~ , fl, 6, ,~) ~" ~ t '~ ~(~) >_ i t ~
~
% 1 ( ~ , ~, 6, ~) ~
61 (A1~o%i(~, ~, 6, y ~ ) ,
One then proves, by an application of "T(eo I (~, ~, 6, 7)), always defined if For the case of
and for
PT:
EBI h ~ taking for
~(~) < l t h T ,
that
~).
R : V I , for
~o I (~' 8' 6, 7)
Q7 : is
~, ~, 6, 7 satisfy the conditions listed in the beginning. ECF(~) ,
we must also show extensionality conditions
to be satisfied, but this can be proved in the same manner by an application of
EBI D .
2.9.11. for
Corollary
to the
~N-HAW+~ BR ° , resp.
proof°
ICF(~),
E~-HAW+~ BRo
ECF(LL) if
~[
c a n be shown t o be m o d e l s
satisfies
EL~ + B I D .
174
2.9.12.
Remark°
$I -S9,
yield a model of
by a recursion
Kleene's recursive ECF(~)
theorem analogue
partial recursive functional instead of before.
= ), and using
functionals, if
~
is supposed to satisfy
E L + BID;
(cf. 2.9.16 ) we can show the existence
satisfying the equations for BI D we can prove the functional
The form of the recursion theorem
found in Kleene
defined by the schemata
1959, XIV in subsection
to
3.12.
BR °
of a
(with
to be total, as
be applied in this case is
Chapter III REALIZABILITY AND FUNCTIONAL INTERPRETATIONS
§ I.
A theme with variations:
F IC °
Kleene's
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
5.1.1.
Introductory remarks.
the notion of
In Kleene 1952 (§ 82), S.C. Kleene introduced
F ~ _ realizability to obtain certain proof-theoretic results
for intuitionistic arithmetic, such as the well-known d~sjunction property ~AVB
= ~A
or
~B
closed).
(AVB
F~-realizability
was based on the
idea of combining certain deducibility properties with realizability (realizability is discussed in extenso in the next section). In ICleene 1962 and 1963, Kleene simplified his proof of the disjunction and existential definability property by introducing the ed by "omitting the realizability from wise,
FIC- relation, obtain-
F~-realizability".
Expressed other-
F ~ _ realizability may be viewed as the hybrid between realizability
proper (in the sense of Kleene 1952 , § 82) and the
FIC-relation.
As an introduction to the various variants of realizability, we shall in this section study the
FIC-relation,
its variants and generalizations.
For its model-theoretic equivalent, see chapter V. Contents of the section°
In subsections I - 10 the notion
the soundness theorem proved, and some properties of the soundness theorem are given. tion that
HA ~ I P ~ .
FIC
FIC
is defined,
and corollaries of
In subsection 11 it is shown as an applica-
Subsection 12 discusses
~IC
for
~+MpR
(~
empty
set). In subsections 15- 15 a variant of simple proof of the rule
IPR
FIC
is discussed which yields a very
(with parameters).
In subsections 16 - 18 Ereisel's method for dealing with derived rules with parameters, using partial reflection principles, is described and applied to obtain closure under Church's rule. Subsections 19- 24 are devoted to the use of (variants of)
FIC
for ex-
tensions of arithmetic. 5.1o2o
Definition°
We define
FIC,
for
F
a set of closed formulas,
a closed formula, by induction on the logical complexity of ("F I ~ A "
abbreviates
"FIA
and
(i)
F I P -= F ~ p
(ii)
r t A ~ B -~ r l ~
for prime
and
rtB
(iii)
F[ A VB ~ FI~A
or
Ft~B
F ~A" , P
"F ~ A "
C,
abbreviates
C
as follows° "HA+ F ~ A " ).
176
(iv)
r lA-s
(v)
F I VxAx ~ F 1 A :
~ riga = r I B for all numerals
:
(~i)
r l:Ax
fo~ some numeral
:.
~ rlbA:
A(Xl, .. o,Xn)
If
write
F I A(xl,...,Xn)
3.1.3.
Lemma.
tl,...,t n
Let
iff
free, then we
F I Vxl...XnA(X I .... ,Xn).
A(Xl,...,Xn)
contain at most
be a set of closed terms,
under the standard interpretation.
~i
Xl,...,x n
free ; let
the numeral corresponding to
ti
Then
(i)
F ~ A(tl,...,tn)
iff
(ii)
rl A(tl,'",tn)
iff rl A(:I .... 'in)
Proof.
x I ,... ,xn
is a formula containing at most
F ~ A(~ I ..... ~n)
(i) Straightforward,
by induction on the logical complexity of
A.
For the basis we use the fact that all true closed prime formulae (hence in particular ti= ~i' ~i= ti ) are derivable in ~ ° (ii) is proved similarly. As an example of the induction step, let
A m B-- C ,
and assume rIB(t I .... ,tn)
iff
rls(tl,...,tn)
Let
duction h y p o t h e s i s
the induction
rlB(~1,...,~n) , FIc(t I ..... tn)
rlc(~1,...,~n) .
" C(t 1 . . . . . t n ) , and let F ] ~ B ( : I , and ( i ) , rib B ( t I . . . . , t n ) , h e n c e
rle(:l,...,:n).
hypothesis,
-- C({ 1 . . . . ,~n) !
iff
riB({1,...
Thus
. . . . :n) O Then by inr I c ( t t . . . . . t n ) ; by rlS(: 1 . . . . , : n ) -~
) -- C ( { 1 , . . . ) = F [ B ( t 1 . . . . ) -- C ( t l , . . .
)
i s shown
similarly. 3.1.4.
Theorem
(Soundness theorem).
(all elements of
r~A Proof.
F
We show that, if
r Ic
for each
= FIA.
By induction on the length of a deduction of
We select again Godel's formalization
A
from
F
in
PL2).
F ~A, Let
tIenee, f o r all
A E r,
then
r ~Ax,
by the hypothesis of the theorem.
F ~Ax-~Bx,
riyal
:,
F IC
,
FlAx ,
FIAx -~ Bx°
vn(r I B:)
therefore
; therefore
For simplicity, we omit parameters in most other cases.
PL3). Assume PL7).
Let FI~A. Let
:t follows that
F~B--
F~A--B,
Then
rl~B
FI~A&B'~C rlFA&S,
, ,
C, hence
rlA
-- B ,
FIC ;
and assume
hence
~.
(1.1.4) as the basis for our verifica-
tion. Let
CE F
closed)
rle;
rl B-- C.
so
FI~A--C °
r I~A ,
rl~B°
so
rlA~(B~c)
I'lBxo
177
PL8).
Similarly, in the other direction.
PLg)o
Assume
F I~I=O ; then
closed prime formulae
P.
al complexity of
A,
PLIO), 11), 12).
Immediate.
sL13). Then
or
r I CVB
so
that
FIA
Assume r l t A - S , F It C
QI)o
F
is inconsistent, hence
for all
A,
hence
r I
I=O-~A
for all
under t h e a s s u m p t i o n
rltc--Ax,
°
FItCVAo
and let
F I~A ; hence by our first assumption,
Assume
FIP
Then one readily proves by induction on the logic-
FItCVA
scfora~l
,
i.eo
[,
r It C
F I CvA-~
rlA{,
i.e.
or
F It B.
CvBo
r l v xAx.
So
F[ C-~VxAx. Q2).
Let
hence if
if
~(m)
F ItVxA(x,Y) , then
t(y)
is a term containing only
,
and w i t h lemma 3 . 1 . 3 ,
r tt v~(x,y)--A(t(y), Q3). (t
F ItA(t(y),
Let
F I ~o~A(x,n) .
Q4).
y) ,
y
free).
F It~xAx ;
Let now
F ItA(t(~),
t(~)
is closed, and
) .
Thus
then
[)
for all
numerals
Then also by lemma 3.1o3,
This holds for all
rl~Ax~C,
Assume
FIA(t([),~
then
£,
so
rltAg--C
i.e.
being the only variable free in
fie.
free,
n, m ;
y)
containing at most
(x
y
for all numerals
is the corresponding numeral under the standard interpretation,
rlA(~([),E)
hence
F ItA(~,~)
r IWA~
F ItA({(~), ~) ,
r I A ( t ( y ) , y ) - ~xk(x,y) o
for all
Ax -~ C )o
for some
~,
hence by our assumption
r I XxAx~C.
so
The verification of the non-logical axioms is mostly trivial ; consider e.g.
Sx/O
also
F t I= O
The
, i.e.
S x = O -~ I = 0 .
which implies
FI~S~=O
F II=O , so
, F
is inconsistent, hence
F I Sx= 0 -~ I = O o
only non-trivial case which remains is the induction axiom°
FItAO&Vy(Ay-~A(Sy)) . duction one proves 3.1.5.
If
Then
FIbA0 ,
r t t VxAx ~ so
Corollari~So Assume
FItA£~A(S~ )
for all
E;
Assume by in-
r t AO & Vy(Ay--A(Sy)) -- V~Ax.
B, C, D, ~xAx
to be closed.
If
C I C , then in
HA: (i)
~C-~
(ii)
tc-BvD
Snm~x iff
iff =
(iii)
tC-~
Proof.
(i)
c lbA~
for some numeral
(ii)
ZxAx
tC-~
tc-s
Assume
A~
for some numeral
or
tC--D
tZx(C-~lx) o C I C,
tiC--91Xo
[~
hence
Then
C ff 9 i x ,
so
can be p r o v e d i n t h e same way, b u t can a l s o be o b t a i n e d
consequence of (i) and
C t XxAx,
i.eo
bC ~ A [ .
tSVD~--~ ~[(x=O--S)~(x/O--D)].
as an i m m e d i a t e
178
(iii) is an i~nnediate consequence 3.1.6.
Lemma.
mC ImC
Proof°
Assume
mCI~C.
C I I=0 . 3.1. 7.
Therefore
Corollary.
IPRc
for closed Since
C. implies
mC ~C
m C k I=0 , we have
mC I mC . In
3.1°5 (i), 3.I.6°
3.1.8.
Theorem.
Let
C
pc-~xAx
be closed.
If for all closed
D
such that
(i)
For prime formulae
(ii)
If
D
~o~Ax
for some ~,
~bc-A~
C ~D ,
By induction on the logical
D m DI & D 2 ,
~xA
~ ff ~ ( ~ C - - A ) .
Proof.
then for all closed
C,
H~A, for closed
ff ( ~ c ~ ~ )
Proof.
of (i).
also
complexity
C ID o of
D °
the assertion is obvious.
then
O kD
implies
C PD1,
C PD2 ,
cl~1&ciD 2, so c i r . (iii) if~D I v D 2, the~
0pD
implies by hypothesis
hence
cbD1
or
Opt2,
hence C IPD1 o r C I P D 2 , so C I D 1 VD 2 . (~e use h e r e the f a c t t h & t the a s s u m p t i o n o f the theorem a l s o i m p l i e s
bc-c lvC 2 ~pc~c
I
or
~bC~C
C1VC 2 ~--~ ~ x [ ( x = 0 " C 1 ) & ( x % 0 " 0 2 ) ] °) Remark° This theorem shows t h a t a I C dition
for 3.1.5
3.1. 9 . If
C IC
(ii)
If
C
Proof.
a~ai~by con-
(i).
(i)
and
C~
C' , then
C' I C, o
is a Harrop formula,
then
By 3.1.5 and 3.1.8,
taking
Harrop formulas
Remark.
satisfy
m~Ce--~C
C IC o C
itself for (IoI0o~)
D.
; then use (i) and 3.1.6.
We do not know of a simpler and more straightforward
invariance
of
C IC
3.1.10o Example. equivalent C
Cl v c 2 ,
Corollaries.
(i)
(ii)
2 forciosed
etc. etc. i s a n e c e s s a r y and s u f f i c i e n t
way to obtain
under equivalence.
We wish to show by an example that the class of fermulae
to a Harrop formula is properly included in the class of formulae
such that
C I C.
The example is taken from T°T. Robinson
By the ~osser version of G~del's first incompleteness
1965.
theorem, we c~n
construct for any system H containing a sufficient amount of arithmetic, o El- sentence G such that ~ G , ~ G on assumption of the con-
a
sistency of
~.
179
Let
GI
be a tosser sentence of
let
G2
Let
A ~ 7G I ~ G 2 V m G2 .
Then
~
mG11mG
~A, I
~
~
mA.
~+
For assume
it would follow that
H~+mG I pG 2 Assume
HA c , then
be the tosser sentence of
~
~.~
H~ ~ m ( ~ G I " G 2 V mG2) ,
mG I
is consistent ;
~ mG I-G 2 V mG 2 ,
m G I ~ G 2 or
l~+ mG I p mG2 ,
or
~+
mG I o
then by
H~ ~ m G 1 ~
~G2 ,
i.e.
contrary to our assumptions.
then
H~Ac ~ % , ,
i.e°
HA c ~ ~ G 1 ,
contrary
to our assumptions. Now assume
~
A<--eB,
B
a Harrop formula ; then
( m G I ~ G 2 V m G 2 ) <--~ ( m O 1 ~ m ~ ( G 2 V m G 2 ) ) ° But then contrary to what we just proved. Finally, we wish to show
b mG 1 '
H~A~c + A
plication
i.e.
A ling I ~nd
3.1.11. Theorem. Proof.
O1
Let
A IA o
I~ c h mG1 ' ~+A
Assume
~+
m m A < - - ~ A , hence
H~ b m G I ~ G 2 V m G 2 ,
A ~ mG I ,
~ :G I =A
( A p p l i c a t i o n of
~G 2 V mG 2
F I C )o
~ ~IP:
be a rosser sentence for
then also
Therefore the im-
which is impossible.
is vacuously true°
.
HA c , G 2
( IP: : closed
IP O o)
a tosser sentence for
H~ + G 1 o Then
(I)
~e p %
~ MG I ~ G2 v ~G 2
For assume
HA bG1 -* G2 v mG2 .
~ G2 v ~ % o
Then, since
G1 I G1
(G I
~'~ P GI "~ G2
or ~ p OJ-* m a R ' i . e . t ~ + G 1 pG 2 or which has been excluded by our assumptions. Let
(2) For
BtBo B IGI
and
~+B
b G1 = (B I G2
and
(sl~ %
and
~+B
b G2)
or
~+~b~2) assume }j~+G 1 -
(G 2 v~G 2) fig 1
H~Ac p G I which contradicts our assumptions.
Now consider the following consequence of
IP~R (= IPpR
closed formulae) :
(3)
~ + G 1 p mG2 ,
B m G I ~ G 2 V m G 2 . Also
i s true because the premiss i s vacuous : then
being negative)
B ~ [ ( C l o G 2 ) V ( G I ~ raG2) ] .
Assume . ~ b (3) .
Then, since
B IB
t ~ + B p GI~G 2
or
~+B
b G I ~ mG2
n °P%-%
or
~cbG1
ioeo
.~%
which implies
~e+GI pG2
or ~c+GI ~_~G2
restricted
to
180
which is false by assumption. 3.1.12. Addition of
.
~PR"
We can extend the soundness assuming
~7~(3)
Therefore
~ - consistency
of
theorem for
~-~+~PR'
]A
from
and using
~
NpR
to
on the meta-level.
For this extension we have to show that for any instance I
holds.
Let
F
F
of
}~R '
be -- ~ A ( x , y )
~V~A(x,y) IF
HA+%R,
is equivalent
.
to : for each numeral
n ,
which in turn is
I b ~Vx~A(x,~) = ~m(IbA(<~)). NOW assume
the
~ ~VxmA(x,~) ,
®-consistency of
~A(m,n)
t h e n n o t f o r all
~+l,~)
for some n~meral
~,
hence, using
~,
hence also
~ mA(m,~)
~,~
I~ A(m,~)
( b e c a u s e of
metamathematicany, (assuming
A
to be
prime). Thus we may establish 3.I.13.
Definition
DP, ED
for
(of a variant of
This variant was introduced
~+
F I C )o
ity of
closed.
, and enables us to obtain
in de Jonah
some results also for open formulas. necessarily
MpR
The definition
Below, of
E IA
E
is by induction on the complex-
A.
(i)
E~P
(iii)
E IAVB
: [EiA
(iv)
E!A--B
~ (EIA)&(E--A)
(vi)
~ i~
~ ~((~!Ax)~(~'Ax)).
Note that
~ E~D
E [A
for
P
prime
& (E~A)]
is represented
V [(E!B)&(E'B)
by a formula of
property
3. t . 1 4 . Theorem.
=~ffsis-~iA.
~bE'A
]
- E!B
which is a metamathematical
Proof.
is a single formula, not
of
.~,
in contrast
By induction on the length of derivations,
E Ik ,
one shows
The details are very similar to those in 3oI.4, and are therefore or the reader may consult de Jongh
to
E,A .
omitted ;
181
3.1.15. Corollaries. (i) (ii)
In
HA:
h A ~ ZxBx
= A~A
h Zx(A~Bx)
hA ~ BVC
~ A~A
h (A~B)
h ~A~xBx
= h~x(~A--Bx)
h ~A'BvC
~ h(mA'B)
Proof.
(i)
hA!A
Assume
(IPR)
V(~A--0) .
hA--~xBx,
" Zx((AiBx) & (A--Bx))
V (A~C)
then
b A
so
A!A
,
!A
- A
i~Bx,
hence
~ Zx(A--Bx) ; the second assertion
is a direst consequence of the first one. (ii).
Note that
h -~A~-TA ,
since
~A~A
& ~A-~A
implies
~ A -~ I =0;
apply ( i ) . 5.1.16- 3.1_.18o A method of dealing with variables using partial reflection principles. 3.1.16. We shall describe a simple instance of a method, first used by Kreisel in Kreisel
1959R,
to obtain proof-theoretic closure conditions for
instances where parameters are present, by means of partial reflection principles° Let If
Prn(Y) mdef ~xPr°°fn(X'Y) ' A(x I
function of
,Xn)
is a formula,
Xl,..°,x n
where
Proofn
~A(~I,..,
is defined as in 1.5.1.
n)
(a suggestive notation is
may be conceived as a rA~(xl,...,Xn) ; but then
it should be tacitly understood that only closed formulae have a godelnumber, formulae with free variables have a function assigned to them). in
~
the formalized
plexity ~ n
(notation
variables free in
(i)
In ) as follows
F,
( Xl,...,x n
We define
restricted to comcontaining all the
P, A, B) :
n P(Xl'''''Xn) ~ Prn(rP(~l . . . . . ~n )~) n (Aa~) ~ (In A ) a ( l n B ) (AvB) (In A ~ P r n ( ~ A ( ~ , . . . , x-) ~ ) ) V ( I n B& Prn(~B(~ I ' " " ,~)~)) n (A--B) ~ (ln A) a P r n C A ( ~ , . . . , ~ n )~) - InB n Vmx ~ VX(lnAX) n ZxAx ~ ~ ( l n A X & Prn(rA(~,~ 1 , . . . , x n ) ) .
(ii) (iii) (iv) (v) (vi) Then
F I C- relation, for empty
we prove
5.1.17 . Theorem.
If
~n
denotes provability in
~,
restricted to formulae
of logical complexity ~ n , then ~-n A = ~ Proof.
h (In A) •
Completely parallel to the proof of 3.1.4 ; we have to use repeated-
ly the fact that whenever More details in de Jonah ~
~n A(Xl .... ) , we can also show
hPrn(%(~1,...)])0
182
3.1.18.
Closure under Church's rule
CR.
Now we are able to make applications. I~ ~ H ~ , y )
. Then also
~(InZYA(x,y)), Vx~y~z
i.e.
~n~YA(x,y)
assume n ; hence
b~y(InA(x,y)&Prn(%(~,~)1)).
Proofn(z,rA(~,~)l)
predicate,
For example, for suitable
, which implies,
since
IIence we find Proof n
is a recursive
that for some numeral
By the partial reflection principle,
Thus we have shown closure under Church's rule For further applications kind,
(CR) o
of partial reflection principles
of the same
see chapter IV, § 4.
3.1.19. Extension
and generalization
In Moschovakis tuitionistic
1967 ,
analysis
soundness theorem, uncountably
F IC
of
F IC
to higher-order
s~stemso
has been extended to certain systems of in-
containing
function variables.
In order to prove a
one usually has to extend the systems considered with
many additional
function
constructed from a denumerable
symbols
(since in the usual systems
set of symbols, not every function has a name
in the system)° Friedman A considers a generalization higher-order
of
F IC
systems with species variables,
types with impredicative certain definitional
comprehension.
extensions
applicable
to various
such as the theory of finite
Here also one has to consider
of the systems one is interested
in, to
provide enough "names" of objects for establishing a soundness theorem. In 3.1.21- 3.1.23 we have described Friedman's
method for the simplest in-
teresting case : HAS o F IC
in its original
form, without complicated
extends to
HAS
+PCA o
formula of
HA,
with axioms
tricks,
quite easily
To see this, we add species constants
CA ,
A
a
CA(xl .... ,Xn)(Xl .... ,Xn)+-~A(x I ..... x n) to
HA~o+PCA
of
HAS
; the resulting
+ PCA .
system
~
is obviously a definitional
We then add to the clauses for
F IC
in 5oI.2
( ~
extension referr-
~ O
ing now to
~) :
(vii)
r l VXA(X ) ~ F1 A(CB)
(viii)
F I ~XA(X) ~
Clause
F I A(OB)
(i) is extended by
for all
and
CB
(with the right number of arguments)
~ b A(CB)
for
some
C3 o
F I CB(x I .... ,Xn) ~ F I B(xl,..O,Xn) o
~83
If
A(Xl,...Xn,
x1,...,Xn,
X1,...,Xm)
X1,...,Xnl
is a formula of
free,
then
F I A(x1,°'',x n, CBI,''',C B ) CBI,..o,CB~.
H
containing
F I A(Xl,.oo,Xn,
for all numerals
at most
XI,...,Xm)
is defined as
Xl,...,~n,
and all constants
with the appropriate number of arguments.
The idea of this extension is simply this : in order to prove a soundness theorem, we need at least a name for each definable family of all arithmetical
species is a model for
species°
Since the
bIAS + PCA,
it is obvious
~ O
that the arith~etical
species are exactly the definable
constants for each arithmetical
ones ; hence we add
For an application
species.
of this exten-
sion, see 3.1.20. Extremely
similar is an extension
to
~,
-
~- ~&~, K- a . 9 ,
where we
put
r lvx~,Z
iff
r I At=
for all closed
ta
for some closed This
yields
DP, ED t for
N-ZA ~
The methods of ~oschovakis IDB
routine,
see our remarks in 3.1.24.
Theorem.
IDB 1
(i).
etCo
1967 can also be readily extended to the
systems
3.1.20.
and
(1.9°18).
The proof is for the greater part
The soundness theorem f o r
d e f i n e d i n 3. I . 19 (under the assumption t h a t
(ii).
~or
Proof.
-
~ ff(.*-c)
zxs(x)
(i)
extends to
for
H
as
E e F )o
or
ff(-A--D) --
xx~...x)
~s(x)),
We have to verify the quantifier rules and axioms and the schema.
QI).
F~C-*
Assume
CB ,
so
A(X) ,
Pl C-~ A(X) °
Let
FI~C,
then
El A(CB)
r I VXA(X) o
F t C-'~ VXA(X) .
Hence Q2).
FI E
~ t- - ~ A - ~ x ( v x 1 . . . ~ n ( F ( x ~ . . . % ~ )
comprehension
for all
F] C
~AS +PCA ( ~ A , ~Bx, CvD, ~XE(X) closed):
W'*~cvD
~- ~A
ta o
F I VXA(X) -~ A(T) , where
verified
as f o l l o w s :
let
T
is a species variable or constant,
FlffVXA(X) ,
then
F I A(Y)
and
is
F I A(CB)
by
definition. Q3)-
F I A(T) "~ ~XA(X) , T
Q4).
Let
suitable P A).
F I~A(X ) -* C,
CB,
constant
a speCies variable or constant, assume
We restrict ourselves
species variable
X
F I~XA(X)
and s i n c e a l s o
o
Then
F I k A ( C B ) -~ C ,
to the case where
A
free, and a single numerical
is also immediate~
F I~A(CB)
for a
F I c.
contains a single variable
x.
(unary)
Consider the
184
instance
~x :: Vx[A(X,x) ~-~ Yx] We have to show
vc B :%, ~x(F I~A(%,~) This is simple : replace in result is a formula Now let
A(X,x)
every occurrence of
Xt
by
at . The
B'x.
F I ~ A ( C B , ~ ) ; since
rI~,~, hence r l ~ % ~ C o n v e r s e l y , if F I b C B , ~ , (ii)
= r L~ %,~)
A(CB,~ ) ~ B ' ~
then
is provable, we obtain
FI~B'~
etc.
The first two assertions are proved in the same manner as before.
The third assertion is established as follows : Let
~ ~A-
(using that
ZXE(X) , then also 7A IRA)
;
3.1.21- 3.1.23. Treatment of 3.1.21. Definition. adding constants
H~A
therefore
We define a conservative extension
CB, V
n
~+--A
~ A I~E(CB) , hence
I~XE(X)
~ ~ ~A~
E(CB) .
of
by
HA~S o
for every formula
variables free, and for all sets contains
~ X X E ( X ) , hence
H+
V
of
B
of
~
H
HAS
not containing species
n - tuples of closed terms, if
B
numerical variables free, and adding axioms
CB,v(xl,...,x
n) ~ B ( x
1 . . . . ,Xn) •
is obviously a definitional extension. on the complexity of
A,
as follows ( ~
Now we define
R(A)
by induction
referring to deducibility in
(ii)
R(CB,v(tl,..,tn) R(A&B) ~ R(A)
(iii) (iv)
:(A v~) ~ (R(A) and hA) or (R(~) and bB) ~(~-B) ~ R(A) and ~A = ~(B)
(v)
R(VxAx) ~ For all
~) :
) m (t 1 . . . . , t n ) 6 V and R(B)
~, R(A~)
(vi)
: ( ~ x ) ~ For some ~, :(A:) an~ bA(~) (vii) R(VXA(X) ~ For all CB,V,R(A(CB,v) ) (viii) R(~XA(X) ~ For some CB,V,R(A(CB,v) ) and
bA(CB,v) .
We shall p u t
,xn, X1, . . . . X
containing
CBI,VI,... )
for all numer-
R(A(Xl,...,Xn,
all the variables free in als
Xl,...,x n ~ all constants
3.1.22. Theorem Let
X I . . . . . Xm) ) A)
HAS ~ A ,
if
(X I . . . .
R(A(:1,...,~n,
CBI,VI,...,CBm,V m
(Soundness theorem ; adapted from Friedman A). then
R(A) .
185
Proof.
The verification for the axioms and rules of arithmetic and predicate
logic is completely similar to the proof of 3.1.4, with
F
empty.
It
remains to verify the comprehension schema. For simplicity,
consider the instance
W ~X Vx[A(Y,x) ~ A
Xx] ,
not containing species variables free besides
We have to show for all
Y .
CB, V
R(~X W [ A ( C B , v , X ) ÷ - ~ X x ] ) . Now t a k e any
CB, V
and l e t
B ' x ~ A(CB,v,X) ; we wish to show
R(V~[A(CB,v,X ) 4-->CB,,~ x]) & ~ W [ A ( C B , v , X )
~
CB,,W x]
where = {t I R ( A ( C B , v , t ) ,
Hence, s i n c e
t closed}.
~Vx[A(CB,V,X) <--~
Cs,,~
x]
i s o b v i o u s , i t r e m a i n s to be s h o ~
that R(A(CB,v':)) & b A ( % , v
First,
assume
~A(~B,V,~),
bC~,,W x~
Then a l s o
~(cB,,~ ~)
'~) = R ( % ' , W
~) ~ b % , , w
~"
R(A(CB,v,~)) •
and s i n c e
R(Cs,,W ~)
~ ~ W ~R(A(Cs,v,~)),
too.
Conversely,
assume
R(A(CB,v,~)) ~
R(CB, W ~) & ~ C B , W x ; then obviously
also
R
may be motivated as follows.
If we attempt to extend the original definition of case, we encounter the following problem. for
X,
would be: I A(B) .
I VXA(X)
IB
I
The natural formulation for
is not well founded, i.e.
for formulae
B
as having logical complexity R(CB,v(t I .... ,in) )
R
B
IA
with complexity less than 0. iff
CB, V
B
can be arbitrarily is not defined in A o
are introduced ; they are treated
Instead of asking R(B(t I ..... in)) ,
which would be expected in view of the axioms for make the definition of
to the second-order
iff for each substitution of a predicate
For this reason, the constants
to determine
IA
But since the logical complexity of
large, the definition terms of
hence
~A(Cs,v,~) .
Remark on the,,,p,r,0ofo The method of defining
1 VXA(X)
~EW,
CB, V , but which would
not well founded, we use the (arbitrary)
set
V
R(CB,v(tI,ooo,tn) ) .
Using the freedom made possible by the set erence to the notion
R
V,
we defined a
V
by ref-
itself (an example of an impredicative procedure),
186
in order to establish
R(F)
for instances
We cannot establish the soundness
of the comprehension
theorem in the form : ~ ~ A
since then we would be required to show for arbitrary numerals
F
schem~.
= R(A) ,
R(CB,v(xl .... ,Xn) e-~ B(Xl,...,Xn))
Xl,..,~n , leading back to the problems we so care-
fully tried to avoid. 3.1.23.
Corollary.
(i)
p~x
(ii) (iii)
ff~vc~ffB
HAS
In
~ pA~
(~¢kx, B VC, ~XD(X) c l o s e d )
for a numeral
ffc
or
p ZXD(X) = p ~ X [ V X l o . . X n [ X X l ° . . X n = A ( x I . . . . ,Xn) l a D ( X ) ] ,~ors~,~ ~-
Proof.
Completely
similar to the first-order
In (iii), we first obtain
case for (i), (ii).
for a closed species term
~ ~ D(T)
T ; then the
assertion of (iii) readily follows. 3.1.24. Extension
of Moschovakis
The idea of Moschovakis
1967 to
iDB , ID~BI o
1967 is to construct
suitably conservative
ex-
tensions of the theories considered by addition of new (possibly uncountably many) function symbols so as to have "names" for all functions which can be shown to exist in the system. if
F I A~
for each closed functor
if
FI~A~
for some closed functor
IDB , ID~BI
= IA
for of
of
IC
for instances KI, K2
C
of
(induction over unsecured
let
do not cause us trouble.
The
(for a proof
1970 , 3.2.1):
sequences).
let us assume
~01,~2,°°.
(2)
KI, K2, K3 , to obtain
may be verified with the help of the 1emma
Ka a V n ( a n / O - P n ) & V n ( V x ( P ( n . £ ) ~ P n )
For simplicity,
~°
we have to add to the work done in Moschovakis
~ ~ IDB , ID~B1 .
K3
see Kreisel and Troelstra
~;
are extended with :
F I V~
1967 a verification pA
FIA
F I ~ei~ In the case of
instances
The clauses for
Q
-- PO
We use this lemma on the meta-level.
to be a formula with a single free variable
range over closed functorSo
A s sume
t}- K~I o
So
(3) Let us put (a)
If
<01~n =
~1~ / O,
Xm.~l(:. m). say
~1~ = t p ,
then
ff~l,~(x)
= Sp,
so
V~(]~1,~x=$p)
,
~87
ice°
Ik V X < ~ I , ~ X = S p ) °
I~ AK(Q'~I,~), hence (b)
If
Assume
by
V~Y I Q(~I,~.
Hence also
) °
~ 1 , 5 0 / 0 , %hen ]k Q(~1,5)
~ V y K ~ I , ~ . ,
I~ SyVx(~I,~x=Sy) ' ice.
(5) IQ(~I,~). by (a).
Le% ~1,~0=0 o
By (I) and (2)
~ V y Q ( ~ n , ~ . ) , and since also by hypothesis
lb
V~y I Q ( ~ I , ~ . <~>) , it follows that
I? ~I,£ 0 : 0 & VyQ(~I, £. ) .
VYQ(~I,~e) ' hence also
~ow apply (3) and we find
l q(%,g).
Therefore
Apply now our lemma on the mete level, and we find
I Q(~I) . Therefore
Q. eodo Remark.
TI(<)
can be dealt with in a very similar way.
3.1.25. ,q,o,,ncludin~remarks~ F IC
and its variants demonstrate various devices and phenomena in a
simple context which one meets also in realizability notions and normalization theorems for natural deduction systems° For example, the original non-formalized version of Kleene's ability was similar to
F I C , and could not deal directly with derived or
admissible rules involving parameters. E ! C
F~realiz-
However 9 while the formalized variant
yields certain results with parameters,
the existential instantiation
rule does not generalize thus ; but in the case of its analogue
q - realiz-
ability, discussed in the next section, we obtain such a generalization : Church's rule.
In this section, we used a formalization plus partial re-
flection principles to obtain this° The introduction of constants
CA,V,
means of a concept generalized from
in Friedman's treatment of
HAS
by
F I C , is very similar to the use of
Girard's "candidate de r4ductibilit@" in Girard 1971 , Prawitz's assignments in
Appendix B of Prawitz 1971 , and the use of computability predicates
in Martin-L~f 1971~.
188
§ 2.
Realizability
3.2~I°
notions based on partial
Introduction.
S.C. Kleene in Kleene of intuitionistic constructive
Realizability
recursive function appl~cationo
(by numbers)
was first introduced by
1945, and was intended as a kind of reinterpretation
arithmetic,
interpretation
so as to bring out more explicitly
of the logical
cperatOrSo
viewed as a variant of the abstract interpretation by Heyting
(see Heytin~ 1 9 ~ , ~ g 5 6
Kreisel 196~, Troelstra
Kreisel
1969,
~ Z
1965, ~ 5 ).
~),elaborated
the intended
As such~ it may be
schema first introduced
and made mere precise ix
( for an informal
description,
see e°g.
As we shall see from the definition and results in
the sequel, Kleene's notion is not just a variant of, but essentially from the interpretation
intended by Heyting.
Hence,
make the intended meaning of the logical operators more precise. "philosophical
reduction"
is also only moderately interpreted
of the interpretation
As a
of the logical operators it
successful ; e.g. negative formulae are essentially
by themselves.
On the other hand, realizability
possesses
which provide it with some matematical portant,
differs
it cannot be said to
realizability
very convenient
interest of its own;
but more im-
and the many variants deriving from it turn out to be
tools in the development
In this section,
some nice formal properties,
we restrict
of intuitionistic
attention to realizability
proof theory.
and variants based
on partial recursive function application ; in the next section we turn to realizability
notions based on continuous
might have been developed
simultaneously
function application.
tion, but, as long as an elegant axiomatic application
is lacking,
~ny
details
for those two concepts of applicatheory for partially defined
only at the cost of considerable
notational
complex-
ity, thereby obscuring the simplicity of the underlying ideas. Contents of the section° realizability,
Subsections
the soundness
In subsections
9 - 19
theorem,
2-8
are devoted to the definition
and some direct consequences
~r- realizability
is characterized,
and
of
of these.
ECT O
intro-
ducedo Subsections HA+TI(<)
2 1 - 22 extend this to
, and
provable in
H~+~+
~+N,
TI(<) . Subsections
HA c , with an application.
discussed in 2 7 - 28.
Subsections
Subsection H~ c
The non-realizability
IDB, HAS.
32 describes possible generalizations,
is not finitely axiomatizable
Subsection
33 compares
~ -
and
2 3 - 24 to
of
2 9 - 31 are devoted to extensions
results to some other systems such as
that
subsections
2 5 - 26 describe realizability
over
~.
~ - realizability.
IP
is
of the l~jk~
with as an application
~s9 3.2.2. most
Definition. X1,o.o,X n
For each formula
A(xl,..°,Xn)
of
H~
containing
free, we shall construct another formula of
~,
at
denoted
by X~pA(xl,...,Xn) containing (at most) X,Xl,...,x n free, x ~ {x I .... ,Xn} ; x~(xl,...,Xn) is to be called the (P-)realizability predicate of A. Here HA,
P(A)
is assumed to be a property of
containing at most
the logical complexity ~p(i)
X~pQ ~def Q
x1,...,x n of
free.
A
expressible
The definition
by a formula of is by induction on
A.
for
Q
prime
~p(ii)
X~p(A&B)
~p(iii)
X~p(A VB) ~def[(JlX=0~(J2X~pA)&P(A)) &(JlX/O~(J2X~p ~&P(B))]
~p(V) ~p(Vi)
X~p(~yBy) X~p(VyBy)
3.2.3. A).
~def (Jl x ~ P A )
& (J2X~p B)
~def(J2X~pB(JlX))&P(~(JlX)) ~def Vy ~z(T(x,y,z) &Uz~pBy) o
Examples.
P(A)
is universal,
e.g.
P(A) ~ (0:0) .
omitted (modulo logical equivalence)
In this case,
in the definition
of
is a formalized version of Kleene's original realizability version was first developed for
x rpA,
in Nelson 1947).
P
{p.
may be The result
(a formalized
In this case we write
and we speak about " r - realizability"
instead of
x~A
P - realizabil-
ity°
B). x~A C). D).
P(A) ~ A. for
We call the resulting notion
~-realizability,
and we write
X{pA.
P(A(xl,...,Xn) ) ~ Prov(rA(~1,...,~n)~) o Let us call the realizability notion to be introduced in this example
- realizability,
and let us write
x~A
for
X~pA . Then we define
P(A(x I .... ,Xn)) ~ Prov(r~(x~A(~1 ..... ~n))~) • In other words,
P(A)
and
x~A
have to be introduced
simultaneously.
notion was introduced in Beeson 1972 ; for an application
This
of a slightly
modified notion see § 9 in this chapter. E)°
Generalization
3.2.4° (i).
Theorem Let
P
of (B) : P(A) ~ C ~ A , C
fixed,
etc. etc.
(Soundness).
be any property defined relative to every formula
and expressed by a formula
P(A)
of
~,
(~) r ~ p ( A ) , rbp(A-B) ~r~P(B), or eq.ivalently b P(A)~P(A--B) - P(S). Then~ for closed A : (~)
closed,
~ ~ A = ~(~
~ ~[J) •
such that
A
of
16~ ,
190
In fact, the numeral
(ii).
Let %0
H ~ H~+ r HA,
HA.
~
in (I) does not depend on
P.
be obtained by adding a set of closed axioms
and suppose (A), (B) to be fulfilled w°r.%.
~
F
instead of
Assume moreover
(c) A ~ r ~ ( ~ ) . Then, for closed
(ii:).
(2)
~ ~ A = ~ ~ ~x(xr~oA ) .
Let
~
be as in (ii), and assume instead of (C) :
(D)
~
is conservative over
Then, for closed
(E Proof.
A
HA
Z oI - formulae.
A
in fact not depending on
(i).
w.r°t, closed
P)o
By induction on the length of deductions.
described in 1.1.4 for our verifications.
We select the system
The induction step (automatically
also including the basis step in this case) splits into a number of cases corresponding to the axiom schema or rule applied to o~tain the end formula of the deductions. ~(H~ ~ ~pF*)
For each instance
(where
F*
F
of an axiom schema, we establish
denotes the universal closure of
any application of a rule : FI,...,F n = F "'" ,
n' nI:pF~' ''°'
(for
all to hold, we can establish in PL2).
Assume
VxAx,
Vx(Ax~ Bx),
- ), '''''nk for some
~pF*
~pVXAx,
Thus
and
P(<),.
""
n.
m rpVx(Ax~ Bx),
V~(P(A~--Bx)). Then {:}(~)~pA~, {:}(X)~p(AX~Sx), {:}(X):pAX & P(Ax) , therefore
and for
we show that assuming
some
H~
F ),
VxP(Ax),
so
{l:}(x) }({:}(x))~pBX.
Ax.{{:}(x) }({:}(X):p VxBXo
VxP(Bx).
Also
Below we shall usually consider the cases without additional free parameters, for simplicity° PL3).
Assume
if we assume
A~B,
B--C,
~pA~B,
~pB--C,
[X~pA)&P(A) , it follows that
P(A~B),
p(B~C) .
P(B) , (and hence
Then,
P(C) ) ;
also
{:}(~):pB, {~}({:}(~))~pC, so t~.{:}({:}(~));?--C. PLy).
Assume
Assume
xr~,=
It follows that p(A~(B~A&B)),
A&B
-- C,
P(A),
:~pA&B
y~pB,
-- C,
P(A&B
-- C) o
P(B) .
P(A~(B~C))
, hence
P(C) ° Also
P(A), P(B)
we have
P(A&B)
j(x,y)~&B;
, hence
Thus Axm.{:}(j(~,y))~pA~(S--0) o PL8).
Assume
A ~ (B~C),
::pA ~ (B~C),
P(A&B~C)
o
also by
{~l(j(x,y))~pC.
x~pA&B,
Let
P(A&B) .
Since
P(A&B~A),
P ( A & B ~ B ) , it follows that
P(A), P(B) ; thus {{~}(jlxl(j2X)~pC. Therefore and e.g. A x . 0 r p V x ~ Ax) P~9). 0rp~k~A ~o).
Ax. J 2 X ~ p A V A ~ A
PL11).
Ax. j(O,x) ~p A ~ A VB,
PL12).
Ax.j(1~Jlx,J2x ) ~pA VB--B vA,
PL13).
Assume
; Ax.j(x,X)~pA~A&A
.
AX.JlX~pA&B~A
•
ax.j(j~x,jlx) ~p A & B ~ B & A . Let
X~pCVA.
If
jlx= O,
hence
A ~B, Then
P(A~B),
~pA~B.
Jlx= O&(J2X~pC)&P(C ) , or
X~p C VB ; if
jlx/O&~2X~pA)&
JlX/O , I~l(J2X)~pB
and
P(A) .
P(B) ,
j(jlx, I~l(j2x))~pCVB.
Thus
~.[(l~jlx)x Q1). Let Then
+ sgjlx, j ( j l x ,
{~}(J2x))]~ P C VB.
Assume VyVx(C~--A(~,y)) z~pCy,
Vy~P(Cy--A(x,y))
~rp Vy~(Cy~A(x,y)).
P(Cy) .
{~}(y,x,z)~pA(x,y) , so
Ax. {~}(y,x,z) ~p VxA(x,y) , hence
~ A ~ . {[ }(y,x,z)ip vy(cy- VxA(x,~)) ~). ~.{~}(t)~ ~xAx-At. ~).
~.;(t,~)~pAt--~x.
~4). Assume Then, if
V~(Ax~C) , ~[~V~(Ax--C),
U~p~lx,
W(~(A~--C)).
P(~xAx) , it follows that
J2U~pA(jlu) ,
P(A(Jlu)) ;
hence {~}(JlU,J2U)~pC, so Au.I[}(JlU,J2U)~p(~UxAx~C ) . The equality axioms are realized as follows : AXl...AXnAU.O~p Vxl...Xn(Xi-Xi
~Au.OI~ vx(s~/o), ~ m ~ . O ~ p
.,xi,. ........ vx~(s~:sy-x=~) o
i
.... ));
All defining axioms for primitive recursive functions are realized by hence their universal closure by Assume u ~ p A O ~ V x ( A x ~ A ( S x ) ) , partial reoursive function
~
AxIAx 2 .... 0 . P(AO &Vx(Ax~A(Sx))) °
0
We can find a
such that (recursion theorem)
~(u,o) ~ j ~ , ~(u,Sx) ~ { ~ l ( ~ , ~ ( u , ~ ) ) . By induction we show
~(u,x)
to be defined for all
x,
hence
AuAxo ~(u,x) ~p [AO & Vx(Ax~A(Sx)) ~ VxAx] ° Assume
~+
F ~A,
formulae from
(ii).
~°
Hence by
from
A
closed;
then
H~ ~ F ~ A ,
F
a conjunction of P(F)
(C) : ~ ~ ~x(x~pA) , using
(obtained
(A) and (B)).
( i i i ) . Similarly, i f
~÷FbA,
we find ~ b F - - A ,
F
aconjunotionof
192
formulae from NoW l {~}(m) HA
F, in
hence H,
~ ~ ! {~l(m) & {~l(m)~pA.
hence by (D)
all provable closed
HA ~ ! {n}(m) , and therefore, since in
E I - formulae are true,
~
~ m o ~{~I(m) ,
so
P ~o ~P A Remarks.
(A). Under (ii), (iii) it is sufficien$ to assume (A), (B) for
and to add, besides
(C),
rasp.
(C'), (D) ,
A c r = ~p(i). For assume
~ + ~I ~P(A),
HA+~ Fo + FI ~P(A) , Then
H ~A ;
F I .... ,F 6 F.
then
( Fo_CF,
This establishes
(B)
HA b (FI~ (F2~...
Hence with
(A)
n
(B)
for
'
[ ~P(A).
hence
( r I finite) ; then
HA+__ re + F I ~ P(A--B)
H~A+ F° + F 1 ~P(B) .
Now let
H + FI ~ P(A--B)
ro
for
(Fn~A)
HA
finite).
~. ...) , where
H A + P ( F I ) + ... +P(Fn) ~P(A) ,
~
This establishes
(A)
for
~.
(B). The following variant of (iii) in the theorem also holds, as will be clear from the proof: (iii)'
Let
(C")o
H
be as in (it), and assume instead of (C)
A 6 F = ~t(~ ~ t~pA)
where
t
is supposed to range over ~A
3.2.6.
Remark.
~I,~2
gives a quite elementary
(primitive recursive)
for each deduction of a closed formula
such that
~HA ~ {trpA)&= ~t .
A
in
HA,
method a
HA ~ P r o ° f ~ A ( X , % D
-
Proof~A(~Ix, Ct
r ~ A ~) ~2 x ~
A , where
t 2x
denotes the
p - term with number
On the other hand, we do not have provably recursive functions
~2 x. ~I,$2
such that
(I)
~ p Proof~(x,rA ~) -- Proof~(~Ix," ~2x~pA ~) .
This is seen as follows : take for We now construct
~
A : VxZyT(~,x,y) .
as follows :
~z rroof~A(u,~Vx~yT~xy~ ) ~ ~z
~(u,x) ~jII~2ul(x)
ProofHA(U,rVx~yT~xyl ) -- ~(u,x)~
Under the assumption that (I) holds, ~'x =def- U@(x,x) + I
~
0.
is provably total ; hence also
is provably total, which yields a contradiction by a
simple diagonal argument.
p-
Formalizing we find primitive recursive
such that
for closed
A :
The proof of the soundness theorem by induction on the
for constructing, t
Then for closed formulae
= ~t(~ ~ t~pA) o
length of deductions, term
p - terms.
~93
3.2.7°
Verification
of the conditions
of the soundness
theorem for our
examples. For examples For example predicates
(A), (B),
(C) we use the well-known for
be verified
3 • 2.8.
of the conditions
properties
is immediate.
of the canonical
proof
H~.
In the case of example
itself.
(E) the verification
(D), the conditions
simultaneously
for the soundness theorem have to
with the inductive proof of the soundness theorem
See § 3.9. Lemma o
(i)
V u ( u ~ A ) * ~ Vv(~v~A) ~
so
ur~Ae-~Vv(~v~A) .
~u(u~A) ,
(i:) u ~ A * - ~ w ( w r A ) . Proof.
Straightforward
by application
3.2.9- 3.2.19. Analysis 3.2°9.
Definition.
contain for
v , and
of
of the definition of
r - realizability.
r - realizabilit~.
A formula is said to be almost negative if it does not
~
only in front of an equation between terms
(i.e.
~x(t=s)
HA).
Note that, modulo logical equivalence, are the formulae constructed 3.2.10.
Lemma.
For all formulae
ly equivalent Proof.
xr(A ~ B) ~
HA
the almost negative
A
in the language of
~,
formulae
V, &, x=rA
is logical-
to an almost negative formula.
By induction on the complexity
For example,
for
Z oI - formulae by means of
from
of
A.
assume the lemma to be proved for
Vu(u~A ~ ~v Txuv& Vw(Txuw~UwrB)
A, B ; then using
we can rewrite
xrA~B
as an
almost negative formula. 5.2.11.
Lemma.
let
be a string of number variables,
a
Let
A(~)
be an almost negative formula of arithmetic,
A.
Then there is a partial recursive
of
HA )
containing all the variables
function
~A
(expressed as a
and
free in p - term
such that
(iii) ~-u:A~u~A. Note that (i) and (di) together imply
~ b ~u(urA) ~--~A for almost negative
A.
Proof.
(i),
al complexity of
A,
(ii), of
(iii) are proved simultaneously A ;
as follows :
~A
by induction on the logic-
is defined by induction on the logical
complexity
194
(a) (b)
*t=s(ff) " o ° If A ~ ~ y ( t = s )
(c)
Zf
A ~&C,
take
*A(ff) ~ J(*S(ff)'*C(ff ))°
(d)
If
A
~ Vx~x,
take
*A(~) ~ ~'~S(~)(ff 'x)"
(e)
If
A
~
B--C,
take
,
(A).
Let
of
(i),
A ~ ~x¢(x)=s(x)) o
~u(J2u ] ( t ( J l U ) = S(JlU)) ) This e s t a b l i s h e s ( i ) . Now assume S u ( t ( u ) : s ( u ) ) . t(u')
then proves
: s(u')
,
0) .
CA(~ ) ~ Au.¢c(ff) . (i), (ii), (iii). ( i i ) , ( i i i ) are o b v i o u s .
Now we turn to the p r o o f For prime formulae,
$A(~ ) e j(miny[t=s],
take
~u(u~x(t(x)=s(x)))
which i s e q u i v a l e n t Then
and j ( u ' , O )
implies
to
~v(t(v)=s(v)) .
minz[~(z)=s(z) ] m @~u(t=s)(ff)
is defined,
realizes
call
A(ff) .
u';
it
This
(ii).
(iii) is obvious.
(B).
If
Let
A m B-*C, and S ,
u~,
~U }(¢B =(a)) r C. this establishes Conversely,
and t h u s Finally,
and assume ( i ) , ( i i ) , ( i i i ) for B, C . t h e n ,'~S(ff), ' , B ( f f ) f f S , so 'iu}(%(ff)),
Therefore
C
holds,
and thus
~u(ur(B-~C))
-~ ( S - - C ) ;
(i).
assume
:*c(a),
B-* C , and let
,C(ff ) r C o
u r B.
Then
Therefore
B
holds, hence
,~B~c(a)= =rB-~C .
C
holds,
This p r o v e s ( i i ) .
u~(S~ C) ~-~ W(x~S~ .'tu t(x) & lu t(~) E S) *-~
(Induction hypothesis
is used thrice in the second equivalence : xrB ¢-e xqB,
{u}(~)~ce-~{u}(x)qC, and ~B~(~rs)~s.) (C)°
The other cases:
3.2.12.
Lemma.
equivalent
(in
recursive
$
If
A
HA)
A ~ B&C,
is an arithmetical
formula,
for which
then
If
z
rive recursive function Now by (i),
is provably
(i) and (ii) of the previous lemma are provable, (ii)
(i), (ii) to hold for
p - term.
A
iff we can find a partial
Aa ~
~a
A.
~
does not occur in
(3.2.11).
may be supposed to be represented ,
Az.$~
can be given as a primi-
q~_o
(ii)
Aa~-~
~a
& ~arAa
A~ ~
[~uT(~, o, u) & Vv(T(~, o, v) - - ~ A ~ ) ]
hence
the right hand side of this expression
i.e.
& Sara
The "only if" part has already been established
Now assume by a
are left to the reader.
to an almost negative formula,
rA) ~ A, Proof.
A ~ VxB
is obviously almost negative.
195
3.2.13.
Remark.
provably fied
almost
3.2.14.
that
Note
equivalent
to an
negative
H~ ~ A e - ~ x ( x { A ) almost
negative
for
formula
a formula
or an
A
iff
existenti&lly
A
is
quanti-
formula.
Definition.
Let
ECT o
denote
the following
schema,
for
A
of
ECT
almost
negative : ECT ° (y
VxEA~yBy] not occurring
Note that for
" ZuVx[A~[v(Tuxv
free in
& B(Uv))]
A )o
A ~ 0 = 0 , we obtain
CT
(Church's
thesis)
O
CTo ECT
vx~By stands
~uVx(~v Tu~v &
~
for "extended
B(Uv)).
Church's
thesis".
O
3.2o15o
Lemma.
is a numeral
For any universal ~
~ A p ~ r ~A ,
Proof.
Consider
For simplicity, B
besides
closure
A
of an instance
there O
such that
~~ + E C T
an instance
we assume
x, y .
of
W~qA O
ECT ° ;
that there are no additional
free variables
in
A,
Assume
u~ V~[A~yBy] and abbreviate
vx[A
t ~ {ul(x,¢A) ;
- :t ~ t ~
then
~yBy]
or equivalently
AX'Jlt'
Put ~I Then
~2 = minuT(~l'X'U)'
~(u) = j(~1,AxAw. J($2,J(O,J2t)))
°
~(u)~r ZzVx[A ~ ~v(T(z,x,v) & B ( U v ) ) ] ° Hence
Au.~(u) r (I).
Similarly
in the presence
of additional
3.2.16.
Theorem
(Idempotency
Proof.
By lemma
3.2.10
variables,
of realizability
and remark
3o2°13.
or for
; Nelson
q - realizabilityo
1947).
196
3.2oi 7 . Notation.
Let us abbreviate "r- realizability which is provable in
the formal system
~"
pression
" ~ - ~ - realizability".
"H-~-realizable"o
3.2.18. Theorem ~+ECT
(ii)
HA+ECTo ~ A
O ~ A ~
A : B--C;
r °
~ ~x(x~A) .
(B--C) ~
r e q u i r e d an a p p e a l t o
(~x(x:B)
ECT
O
A.
Consider e.g.
-- ~ y ( y : C)) ~ - ~ V x ( x r B - - ~ v ( y : C ) ) ~-e
~ :~(z.r B~ c ) . and lemma 3 . 2 . 1 0 .
The t h i r d
equivalence
The implication from right to left follows from (i) ; the implication
from left to right is verified thus:
let
I ~ + E C T ° ~ A,
a conjunction of universal closures of instances of
H~ ~ ~ X ( x ~ F ) ;
also
b ~Y(Y:~'~)
~
realizability for certain extensions H m ~+
(I)
F,
F
then
HA ~ F ~ A ,
ECT ° , hence
(by t h e soundness t h e o r e m ) ,
3.2.19 . Corollary to the proof of 3.2.18. Let
instead of
~lx(x: A)
= ~
~-~ : z V x ( x : B ~ : v ( : z x v & u v : c ) )
F
q
H A - ~ - r e a l i z a b i ! i t y ).
(i) is shown by induction on the complexity of
the case
(ii).
Similarly, we use the ex-
Similar definitions with
(Characterization of
(i)
Proof.
as
H
(Characterization of of
so
H-~-
H~A).
a set of (closed) additional axioms, such that
A G r = g~
~(x:A).
Then
(:)
:+ ~c~ ° ~ A +~ : ( x : A )
(ii)
H+ECT
~ A
= H ~ ~{x(x:A) o
0
Proof.
~
(i) follows immediately from 3.2.18 (i).
One direction of (ii) follows from (i)o For the implication from left to right, note that by assumption (I), and the soundness theorem for duction theorem for
~: HA)o
H ~ B = ~
~(x~B)
(using once again the de-
Then argue as for 3.2.18 (ii)°
3.2.20. Theorem. (i)
HA+ECT °
(ii) H~A+ECT ° Proof. (ii).
is consistent relative to is
w- consistent on assumption of the truth of
(i) is an immediate corollary of 3.2.18 (ii) or 3.2.15. Assume
H A + E C T ° ~ A~
H ~ + E C T ° ~ ~VxAx ° n
HA.
depending on
n;
by cur assumptions.
Then
for each numeral
IL~ ~ m n ~ A ~ '
also Now let
2,
and also
for each numeral
In ~ V u ~ ( u r~ VxAx) . Hence ~
~
and for suitable
Vu(~urVx~)
is true,
197
~(y) ~ Jlminz[ProofHA(J2z, r j l z ~ A ( Y ) ~ ) ] Then
Ay.~(y)~rVyAy
(since the truth of
reflection principle) HA+ECT
~
contradicting
~HA
•
implies the truth of a uniform
Vu(~ur VxAx) ; therefore
~VxAx
O
Remark.
ECT
cannot be generalized to arbitrary formulas o trated by the following counterexample. Obviously,
(I)
Iv~[ (~Tx~y v ~ ~y T x ~ ) -- ~z((z > 0 & T(x,x,z~1))
ECT
generalized
to arbitrary
A
A , as is illus-
V (~=0 & ~ y T x ~ y ) ) ] o
would yield, when applied to (I), the
O
existence (2)
of a partial recursive iVx[(~yTxxy
function with godelnumber
V mZy T x x y ) ~ w ( T u
u°
such that
xw &
& {(uw> o - T(x,x,Uw~ 1)) &°(~w=O~ ~ ~yTxxy)t}]. Vx m m (~yTxxy V ~ ~y Txxy) , therefore with (2)
On the other hand,
vx ~ w ( T ~
x~ & I(Uw > o & T(x,x,~w~1))
V (~w=O & ~ y T ~ y ) I )
O
NOW l e t
vO
be such t h a t
~WTVoXW ~-> {u O } ( x )
~ 0 ;
then
~WTVoVoW ~--~ {Uo}(Vo) ~ 0 ~--~ m ~ y TVoVoY , which is contradictory;
hence
(2)
is false. In fact,
this counterexample
even refutes a schema
W [ A ~ ~yBy~ ~ ~uVx[A ~ ~{ul(~) & B ( l u t ( x ) ) ~ . Later, we shall prove that 3.2.21.
Lemma.
Let
F
~+
CT ° ~ E C T °
be the universal
(3.~.I*).
closure of an instance
of Markov's
schema
Vx(A V ~ A )
& ~
~xA ~ 2xA .
Then there exists a numeral HA*M~ Proof°
~rF,
L e t an i n s t a n c e Vx(Ax V ~ A x )
be given, besides
~
such that
HA.M ~ ~q F. F & ~
of
M
~xAx ~ ~xAx
and assume for simplicity that x.
Assume
u~ Vx(Ax v ~Ax) & ~ x . Then
A
does not contain variables free
19s
Vx(~{jlu}(x) & ([j1({jlul(~)): o & j2({j~ul(x~A~] v v [j~(ljlu}(~)) ~ o & j2({jlu}(~))~ ~Ax])) ~ j2u~ ~ Let
~(u) " m i n x [ J 1 ( { J l u l ( x ) ) ' 0 ] .
V x : w ( w ~ : A x ) , equivalent to
(ii));
,
i.e.
j2u:~:xAx~-~:::w(w:A:)
o: the other hand,
(3.2.
Vx(J1{Jlu}(x ) / 0 )
VxVu~(u~Ax)
~x. would imply
Vx~Zu(u~Ax)
~:Vx
.
:::(w:~x)
hence contradiction.
Thus Vx@1{Jlul(x) l O ) , i.e.
~x~v(T(jlu,x,v) hence with Thus
M,
& jIHv:0)
Zx[J1{Jlul(x)=0 ] .
lq0(u) , and
J (~@), J2( {JIu l(~(u))))~ ~Ax, and so m.j(~(u),j2({jlul(~(u))))~F. Similarly for Remarks.
q-realizability.
(i). As we shall see later, not all instances of
M
are
KA-r-
realizable. (ii). In the presence of ~R for
~ A
ZxAx ~
CT ° , M
ZxAx
primitive recursive.
there is a
u
is equivalent to the weaker schema
such that
For let
Vx(Bx V-~Bx).
By Churchts thesis,
Vx~y[Tuxy& (Uy=0-~Bx) & (Uy~0 -~ ~ B x ) ] .
Hence
-7-~~xBx -~ XxBx is equivalent to -~-~ ~x~y[ Tuxy & Uy=0 ] -~ ~x~y[ Tuxy & Uy=0 ] which can be obtained as an instance of Note that in the presence of
NpR,
valent to a negative formula, by
MpR.
every almost negative formula is equiSxA ~-~-7 Vx-~A .
3.2.22. Corollaries.
(i)
~A A +_E C T_o + M b
(ii)
~A+ECTo+M
Proof. (ii).
to
is consistent relative to
,~.
(i) is immediate from 3.2.18 (ii) and 3.2.21. (i) implies that
proof of and
~ ~+~b__ ~(xrA)._
I =0
HA + N
in
~+ECTo+},{
~+ECT
o+M
gives rise to a proof of
is consistent relative to
(§ I. 1o).
is consistent relative to ~
since
HA c
I= 0
~+N in
(a ~+M;
is consistent relative
199
3.2.23. Lemma.
For each closure
~n(HA+ TI(<) b ~ r F Proof°
of an instance of
& ~qF)
TI(<) , we have
.
For simplicity we restrict attention to a closed instance V~((V~
of
F
~ ~u
TI(<) , and prove the lemma for
r - realizability.
Assume ~ Au) ,
wrVu((Vv
Vu({w }(~) r ((Vv
Vuw,(w, ~ (Vv
implies with
V v < u ( { w ' }(v,O) ray) , since
v
~c(X,U,V) ~ {0 x
quantifier-free.
if
u>v_
if
u
we can easily find a partial recursive
such that
~(z,w,u) ~ i w ~ ( u , A ~ . % ( { z l ( w , v ) , v , u ) ) ~
abbreviates
Now if we put
~
By the recursion theorem, there is an
Vv
o
such that
{:}(w,u) ~ I w i ( u , A v ~ . ~ e ( I : l ( w , v ) , v , ~ ) ) . We now easily prove u.
Vu(!{~}(w,u)) , and
{~}(w,u) r A u ,
by TI(<)
w.r.t.
E.g. for the latter assertion it suffices to show
vu(Vv
~ {~}(w,~)~Au)
which is completely straightforward ; etc. etc.
3.2.24. Theorem° Let
H be
(i)
H+ECT b A ~ n ( H b ~ A
Proof. (ii).
For A
(i).
close~,
O
~
or ~
H~A+M+ TI(<) o
and ~+ECT o b ~ q A )
Immediate from 3.2.15, 3.2.21, 3,2.23.
Immediate from 3.2.19 (ii).
3.2.25. Theorem ~c Proof.
HA+ TI(<)
Assume
(Characterization of
~ ~(x~A)
= ~+~+ECT
HACb~x(xKi)
negative formula obtained from
o Also x rA
~c_~_
realizabilit2) "
° p m~A.
~+M~xrAe-->Bx
where
B
isthe
by replacing every subformula of the
200
form
Zy(t:s)
by
is conservative
Hence
~+M
we find
~Vy~(t:s)
over
~.~
~ ~.x(xrA)
H~+ECT O~M
Conversely, if
o
Then
HA c ~ ~ V x ~ B x
~ ~mA
+~b
~A
then
O
3.2.26.
Theorem.
IP ° is
'
(3.2.8 ( i i ) ) ,
As an application
HA c
is
hence
Every universal
mc b m(~A)
, so
~
HAc b ~x(x~A) o
of this characterization
Vx[A V ~ A ]
and since
.
~m+SCT
HAc b mmZx(x[A)
,
w . r . t , negative formulae (§ 1.10), ~ ~ ~ V x ~ B x . . Since HA+ECT ~ A ~ - e Z X ( E ~ A ) (3.2.18 (i)), ~ O
we prove
closure of an instance of
& [VxA--~yB] ~ Z Y [ V x A ' B ]
HA c - ~ - realizable.
Proof.
Consider an instance of
IP
with one additional
parameter
z.
To
O
show that it is free in
~c_
realizable,
we have to show
in
HA+M+ECT
°
(y
not
A )
~Vz[Vx(A Let us abbreviate
(I)
~VzA
Because of (2)
v ~A) ^ ( W A - - ~ B )
this as
~
Assume
o
~Vz P ~-9~z~p ~
VzA .
-- ~(V~A--B)].
~z o A
, this is equivalent
to
.
Now assume
(3)
~ A= .
Because of
~ (P~Q) e--)m~p&
~Q,
~
(P&Q) ~-~ ~ P & ~ Q ,
valent to
- ~ vx(A v -A) ~ . - ( V ~ - - ~ y ~ ) ~ - ~y(v~-- ~).
(4) Assume
(5)
Vx(A v ~A),
(6)
V ~ A - ~y~ ,
(7)
~(V~A--B)
From
o
(7) Vy-, (V~--* B) ,
hence
SO
~VxA
& Vy-~B.
this is equi-
201
Because of (5), this yields VxA & V y ~ B which contradicts
(6), hence
[(5) ~ (6) ~ (7)]o Therefore
m (4) ,
This refutes
i.e.
m (3) ,
hence
m~z.A
,
and t h u s
m (2).
(I), and so the desired conclusion has been established
(in
HA
in fact X)° Remark.
It can be shown that
IP
is not in general O
Namely,
~
by a version of de Jongh's theorem
formulae
A, B, C
H A - r - realizable. O
(5.6.16), we can find
ZI -
such that
HA ~ ' ( m A ~ ( B
VC)) "
This formula is logically
((~A'B)
equivalent
V (mA--C))
.
to a formula of the class
F
(see O
3.6.3 ) for which
is conservative
H~A+ECT
over
HA,
hence the formula is
O
provable in 3.2.27.
~+ECT
°
(ioeo
H A - realizable)
iff it is provable in
HA°
Theorem.
(i)
HA+N~R+CT
(ii)
Not all (closures of ) instances of
+ IP
is inconsistent°
O
zP
(~A ~ ~yB) ~ ~y(~A ~ B)
are realizable
P~oof.
(i)
(1)
(2)
HA c - r - realizable).
By +%R ~yTx~y],
V~[~T~ry~
hence with
and by
(and certainly not
IP Vx~y[mm~vTxxy
~ Txxy] ,
CT O
This implies that (ii).
vx{ ( ~
(3) This is
SyTxxy
is recursive
in
x , which is contradictory.
Consider
~T~xy-- ~ T ~ y ) -- ~ ( ~
(I) -- (2).
Now we k~ow that
_~+~+ CTo+ ( I ) - (2) b ~ , hence also
~+M+ECT
o b ~ (3).
~T~xy-- T~x~) }.
202
~e~ce ~ c ~ m ( x g ~ ( 3 ) ) , 3.2.28. Remark.
by 3.2.s (i).
Kleene proves a slightly stronger theorem (Kleene 1965~) :
he shows that the universal (I)
~(x~(3)),
so ~ c b
(mA--BVC)
is for certain
closure of
-- (mA--S) V ( m S - - C )
A, B, C
is not realizable.
by slightly refining the argument As our instance of (I) we take for
~T~xy
We can also obtain this result
in 3.2.27, mA~(B
sub (ii).
vC)
- ~(x,x,O) v ( ~ > o)Tx~
which again follows from
~PR
Note that our disproof of realizability
is classical
but the disproof of provable realizability
(at least it uses
is intuitionistic
N );
(it uses : I~ c
is consistent). 3.2.29 . Extensions
to other systems°
An extension of realizability
to the language of
variables
is obtained by interpreting
recursive
functions, x r V~A~
~(viii)~
x~[~A~. ~ J2xrA(IJlx]);
A(Itl)
eliminating
& jlxsV I o
is shorthand for a formula
t'[~"]
of
~
in
A*(t) , obtained by systematically
A(~)
by application
~t"=,,
(t'
not c o n t a i n i n g A~
~
by
~)
as abbreviations
by
Itl,
p - terms, and the "prime formulae"
interpreted
of
= t'"~-~ ~u[~t"=u & t ' [ u ] = t " ' ]
In short, if we replace in contains
as ranging over
~VYSVl(~Ix}(y ) & {xt(y)~A({y}))
each occurrence
and r e p l a c i n g
with function
so we put :
~(vii) Here
E~L
the function variables
as in Kleene
If we wish to extend this further to
]v(T(t,t",v)&Uv=u)
by
~,
then
of the form
1969 (Cfo ID~B,
°
A(Itl)
t ~t'
must be
Io3o10)o
we must put (cf. Kreisel -
Tr,oelstra 1970, 3.7.1)
~(i), where
x~K~ ~ x ~ =
{x},
$ = {xJ mdef Vy([zTxyz & Vx(Txyu--Uu:q~)) o
For higher types we may extend realizability higher-order
quantifiers
as ranging over the
V~
similarly, of
HR0
interpreting or the
~
the of
HE0. Realizability
can be extended to the language of
associate with each variable write
X*
for
V~.+I 1
if
V~
X ~ V~ o 1
of
HAS
a variable
HAS as follows. ~e ~'~ I V~ + ; below we shall
203
~r X(%,...
r(i)"
, t n) -= x * ( x , % , . . .
, t n)
r(i~) ~ r ¢'.~(x) ~ x ~ * ( ~ ( x ) )
r(x)
For ~n application of this extension see 3.2.31. 3.2°30~ Realizability for treatment for
H~,
For any formula
IDB°
The treatment is very similar to the
hence we give a sketch (following Troelstra 1971A,§ 5)°
A
of
IDB,
let
Ar
denote the formula obtained by re-
lativizing function quantifiers to recursive functions, and let us define almost negative formulae as before (3.2.9). negative, then also
Ar
Note that if
A
is almost
is (equivalent to) an almost negative formula, since
where, as befere
: Ixl ~def V y ( ~ x y ~
~ vu(~-~u~)),
and
(~(t [~] = s [ ~ ] ) ) r ~--~ a~c v l ( t [ ~ ] where
f
t[~]
in
= kx.(n)x
(1.3. 9 C).
= s[~]) ~
~(t[f n] = S [ f n ] )
(As a lemma we have to show for any term
n
IDB
that it depends continuously on
One then proves that for almost negative numerical variables,
~ ~ %''°''~n =
existence of a partial recursive numerical variables,
~
A(~,~)
(provably in (x
IDB)o)
a sequence of
a sequence of function variables) the
~A(~,~)
( y ~ Yl .... 'Yn
a sequence of
~ 0 ~ = ~ ) such that
:u(urA) ~ A r Ar(~'~) & ~I = IYll &'.° & ~n = ~Yn 1 (A direct corollary, by the preceding remark, of the analogues~3.2.11o) Using this, we find also for each universal closure IDB ~ ~
F
of
ECT °
in
~(IDB) ,
F , in view of which we then obtain, after extending the soundness
theorem for
IDB
as in K r e i s e l - T r o e l s t r a
1970, 3.7.2
ID~ +~c~ ° ~ A ~-, ~ ( x ~ A )
~e might also have considered
q - realizability, and treated it similarly. = For an extremely detailed treatment, see Celluqi" 1971o 3,2.31. Theorem°
UP
uF
HAS+
CTQ
+UP
is consistent relative to
(the uniform principle) can be stated as
vx ~ A ( x , x )
-
~ vx A ( X , x ) .
HAS ,
where
204
Proofo to
Straightforward, by extending the soundness theorem for
HAS+
CT o
+ U P ; cf. e.g. Kreisel- Troelstra 1970, 3 ~
H A + CT
o
, and
Troelstra A, § 3 . 3.2.32. Some generalizations. A generalization inspired by Lauchli 1970 (there in the context of
a
notion closer to modified realizability) consists in the introduction of a family of realizabilities, be a c o l l e c t i o n ~
as follows.
Let
{Ui I i6 I},
( I an index set)
species of natural numbers such that
n {Ui I iE I} = ~. We now define ~(i)'"
r (i)
realizability by the clauses
xr(i) A m ~A-- xE U.
[(ii)-{(vi) ({(i) e~erywhere
and
ability with respect to the family
replacing 5), and then define realty{Ui I i E I}
by
x rI A ~ ViE I (x r (i) A). = def = Technically, this concept is not so easy to characterize as ordinary realizability.
r-
=
We may expect, in view of the result of Lauchli 1970, some-
thing like (an approximation to) completeness with respect to the schemata of intuitionistic predicate logic (if the concept is treated classically). But
CT anyway remains valid. o result is of doubtful interest.
Yet it seems to me such a completeness
Another, rather obvious, possibility of generalization is of greater practical interest : the use of replacing
{t}
unary predicate HA+Vx(AxV
A - recursive functions
{t}A
everywhere
in the definition of Ax.
The r61e of
x =r B , for an arbitrary arithmetical HA is then taken over by
~ A x ) ; the soundness theorem becomes provable for this system.
This idea has been used in Smorynski ly unbounded over
~
to show that
I~ c
is essential-
HA.
Sketch of the proof : Define functions for
~,
n,
Z° - realizability as realizability by n+1 H ° - set. Then establish a soundness theorem n and show that for any A with alternating-quantifier complexity {xlA , A acomplete
there is a primitive recursive
~A
such that
~c
~ A(Xl,...,Xn)<___>
e-~A(Xl,...,Xn)_~A(x1,o..,Xn) . Finally, note that for a predicate Ax 6 Z~+I -Ao n+l ' ~ .c.. ~- Z y [ y ~ m Vx(Ax V ~Ax)]. Then it readily follows that for each set of axioms ~ c + A / H ~ + A,
so
A
H~ c
For similar uses of
of bounded alternating-quantifier complexity, is an essentially unbounded extension of
HA.
A - recursive functions in the context of modified
realizability, see 3 ~ 3 ~ .
2O5
3.2.33° Comparison of
~-realizability
with
=r- rea lizability.
q- realizability does not admit a simple characterization in 3.2.18 for for the
~-realizability.
~- realizability of
H~ ~ Xx(x~A VA) ~ HA b A.
A VA
implies that
Hence, if
provable (e.g. a suitable instance of H~_ b ~x(x~A-- (A VA)) , but not
such as given
It is not even closed under deduction, A
is
is true, and formally
H~-~-realizable,
CT ° ), then
HA b X x ( x ~
A
but not
HA ~ ~x(x~A) ,
VA~.
- realizability may be viewed as a "hybrid" of the " F I C " relation in § 3.1 and realizability.
206
§ 3.
Realizability notions based on continuous function application°
3.3.1.
Introduction.
Realizability by functions (in the present sense) is
first introduced in Kleene & Vesley 1965 , not formalized. are discussed in k~eene 1965
Formalized versions
, Kleene 1968, Kleene 1969, Part II.
For many
details we shall rely on these publications, especially Kleene 1969. The general development is similar to that of the preceding section. 3.3.2.
Definition.
We define the
realizability) for formulae in the language of of
I (~p-
P - lrealizability predicates EL.
P(A)
is again a formula
EL,
with its set of free variables contained among the free variables of I The realizability predicate ~ ~p A contains, besides variables free in
A .
A , the new function variable
~.
As before, the definition is by induction
on the logical complexity of
A .
I . ~p(Z)
prime,
I ~pA
~
A
r1(ii)p
~p
r~(iii)
~r~ ( A V B )
for
A
~ "
3 1 ~ p A1
I (A~)
. I , & 32~rpB= •
I
~ (J1~O:O -- g 2 ~ r p A
& P(A))
& (J1~ojo - j ~ r I ~ & P(B))
2 ~P
IB),
1 ,.i2~pA(Jl,:,O)
& P(A(.il~O)),
IA(~))
(vii) ~(viii)
~ p I ~8A8
~ j.2 ~ p AI( J 1 ¢ )
& P(A(JI~)) .
Note that this is Kleene's notion of in
Kleene
1969,
Part
II,
if
we t a k e
~-realizability P(A) ~ 0=0,
and
and P(A)
q-realizability ~ A
respectively,
disregarding a difference in the choice of pairing functions and codings of sequences of natural numbers. 0=0 1
~p-
(so that we may omit
realizability for
notions 1
have
3.3-3.
Theorem
~q
P(A) ~ A
practical
interest
~-
altogether) as as
for
realizability for ~
1
1
q - realizability° = us.
We w r i t e
P(A)
- realizability, and 0nly these two
correspondingly
~
IA ,
A.
(i)
Let
that
EL ~ A ,
in
Let us indicate P(A)
A
EL
(Soundness)° be the system as described in 1.9.10. Then, for any there
such that
is
a
p - functor
~
containing
free
only
A
variables
such free
207
(ii)
Assume
AI,...,A s
Then there is a or variables
to be closed,
p - functor
~I,°.o,~ s
~1~ 1 AI'
~,
and suppose in
such that in
AI,...,A s ~ A . A
EL
~s:r ~As~ ~ a
"'''
EL
containing free only variables free in
(~ I~) (~a).
qIA I, ..., % q I A s ~ ~ & Corollaries° (iii) Let (closed)
H ~ E L + F,
F
p - functers
~
a set of closed additional
axioms.
If there are
such that
~r~b:~(~
1A)
then
p - functor ~ containing free only variables free in I I q instead of r .
for some
A , and
similarly with (iv)
Let
~'
, and
H ~ E L + F,
it follows that for
H' ~ E L + F, ,
1969, Theorem 50 A, B (page 80).
In comparing the formulations
assumptions constant
AI,Oo.,A s
Let
was interpreted
with the variables terminology),
not act as proper parameters
3°3.4o
it should be remembered
AI,°o.,A s ~ A
(in Kleene's
deduction
Then, if
~ ~ 1A) .
(i), (ii) : See Kleene
Kleene's usage,
sets of closed axicms°
~'
~' b A ~ ~ ( ~ b ~ Proof°
F, F,
as:
that in contrast A
occurring free in
AI,°°.,A s
i.e. the variables free in
of the rules
QI, Q4 lot
to
is deduced from
VI, ~I
held
AI,..°,A s
do
in natural
systems).
Theorem
(Special instances
E L + (~, ~, y)
= 0, I, 2, 3
be extensions
of
of soundness). EL
according
to the following
code:
corresponds
to the addition of nothing,
ACo0~ , ACo0 , AC01
corresponds
to the addition of nothing,
FAN~, BI!
respectively ; = O, I, 2 where
and
FAN2
BII
is obtained by replacing
= O, I, 2, 3 tively
respectively,
is
( C-N~
corresponds is as
C-N
V~x
in
BI D
by
to the addition of nothing, but with
V~x
replacing
V~:x ° C-N:, V~x
C-N, C-C ,
C-C
respeccan be
208
Then, i f
Proof.
H -= EI~, + ( 1 , ~, 0 ) ,
a detailed
H - EL + (I', p, 0) , where 3.3.5.
Definition.
language of '
I
it
follows
~ (i),
inspection
that
I'
o f t h e arg-ament shows t h a t
stands for
QF-ACo0
We extend the definition of
(1.9.10), would suffice.
~rIA
and
~qIA
to the
by insertion of clauses : = K~
I~=_~. ql
as b e f o r e
as
~r
1
A = A ,
~ql
(i)
may be s t a t e d ,
3.5.6.
Theorem
(Soundness for
Proof.
By 3.3.3 (iii) and 3.}.4 it is sufficient to establish the
A
r
IDB v.~ O
~rlKp
~I(i) , So
( ~ > O)
Kleene 1969, 5.10 (page 10~-104).
Remark. A c t u a l l y ~
rl(i)
H' -= EL + ( ~ , ~, 7)
A ~ A
for
prime.
realizabillty for
IDB ).
KI, K2 and each instance of K3 (cf. 1.9.1@). We g i v e I I r -realizability ; the argument for q - realizability
the verification for
is obtained by slight additions. (i)
(ii)
A~A~. ( ~ x . O ) r 1 V~[ Vx(am = Sy) -~ K¢] .
Assume ~ 1 [~=0 ~ W ~ ( ~ . . ~ G . n ) ) ] .
Then
~O=0 ~ Vx(J(j2~)l~y.x ~ (j2~)lky.x~ 1 K ( ~ . ~ ( ~ . n ) ) Hence
r I - a,a q'-
~0=0 & VxK(kn.~(x.n)
, hence
K~,
.
and therefore
(iii) Assume
(I)
~ I [ vyQ(~,x.sy) ~ v~(~o=o ~ v ~ ( ~ . ~ ( ~ ,
(2)
~/ 1 K~
(i.e.
K(~
n)) ~ ~ ) ]
holds).
Then
(~)
v~(:(~l~)t~z.~ ~ (~1~.)I ~ . ~
(4)
V~(:(j2~)i~ & j2~l~r I (eO=O & VxQ(Xn.~(x.n))
(4) is equivalent to
~(~.s~)) -~ Q~)) .
209
(5)
v~(z(j2~)I~ & [vY(J2YrI~ VxQ(~n.~(:.n)) & ~O:0) - ~ ( ( J ~ ) I ~ ) I Y ~ ( ( ~ ) I ~ ) I Y~Q~]) •
There
exists
a functor
~/0-where
fl(~,8)
=
such that
~l(~,~,y)
~
kno~(<60>*n)
Using the r e o u r s i o n
~o-
~
theorem
~l(~,~)
~1
kz'(~:l)
.
analogue
Io9.16 , we find a
@
such that
~ ~ I ~ - ( ~ )
~ = o - ~1(~,~) " ( J 2 ~ l ~ ) l J ( ~ . ~ , ^ < ¢ l ( f l ( ~ , ~ ) ) ) Then one proves by induction
_
over
K
w.r.t.
~:
-
K~ ~
I~I(~,~)
&
~Ko
91[~,~)~ 1''Q~ (of. Kreisel- Troe!stra¥}°7.2). 3 . 3 . 7 - 3.3.13o 3.3°7°
Almost
Characterization negative
of
formulae
r
1 -realizabilit~o
are d e f i n e d
as in 3.2. 9 .
Similar
to 3.2.10
we have Lemmao
F o r all f o r m u l a e
in the l a n g u a g e
A
equivalent to ~n almost n e g a t i v e 3.3.8.
Lemma.
Let
A(~)
of
(i)
A.
be an almost negative formula of
Then there is a
is l o g i c a l l y
p - functor
@A'
EL,
and let
containing all the variables such that
EL ~ :~(~:IA) ~ A
Proof. of
x rIA
formula.
be a string of number~ a ~ function variables, free in
EL,
Quite similar to the proof of 3.2.11.
We indicate the definitions
~A :
(a)
~t=s(~) -~ef ~:.o
(b)
9Zx[t(x)=s(x)](=a) ---def j(kz°miny[t(y) = s(y)],
(e)
~'~[t(o,)=s(~)](~) -def J(ft' X~.O), ~here f n = X~.(n)~ a~ ~here t : minn[t(fn)=S(fn)]
(e)
~'v~A~(~) = A~x" %~(~,x)
(f)
%~A~(~) =- AI~. ~A=(~,~)
(g)
~A-~(~) -= AI~" ~B(~)
(of, Kleene & Vesle~- 1965 , Kleene 1965).
kz.O)
210
3.3.9.
Definition.
negative,
~
GC
denote the following schema, with
A
almost
not occurring free in
3o3.10o Lemma.
For any universal closure
exists a closed
Proof.
Let
p - functor
~
A
of an instance of
CC
there
such that
The proof is very similar to the proof of 3.2.15.
Consider an instance of
GC
not containing parameters, and assume
r I v~[~ - ~ ] .
Then
~(
r ~
3y
°
3.3.1-8
Now put
~ (61~)1(~AI1~) , then
Vc~[A-~
:~p &
We must construct a
J2~rl B jlqo] . p-functor
x~ 1 ~v~[A-, Then
X
"4~ ~ w ( ~ l ~ ~ - B~)]o
C(j1[(j2Xl~)l~'],
01X,~ ) 61rl "Yt~'°
where
C(81'Y'~) -=def
where
D(e', ~, ~, ¢) =def e~=rI B o o
.'yl~ and
there is a function
yI~='e
are almost negative.
take
Hence (lemma 3.3.8)
~,yl ~
:~'1 ~ ~'> : ~': ~1 ~(~', ,~) & '~':yt,~(',', ~) ,iI~=" ~ +-~ ~,(~, ~ I (~i~ ~)) . Now
such that
must satisfy A~' --
Note that
X (~ # 6 ] )
~ ('.~1 ~')
X such that jl X=" A~. j lq)
j1[(J2×l~,)l~,, ] ~" ~,yl,:,(jlx,~, ) == '~,.yl~,(A~,.j~,~,) ((J2[(J2xlo,)l~"])
I
~)~' ~" .i2~-
211
Hence we must take for
X:
X ~ j[A~.JI~, A~A~'.j{ ~I ~(A~°jI~,~), ~71 Then
A6. X
Similarly
r - realizes our instance of I
for
q
3.3o11. Theorem
(Characterization of
E~L+GC ~ A ~-~ ~ ( = r I A) .
(ii)
Let
H
°
GC.
- realizability.
(i)
be any extension of
r 1 - realizability)
has
b~(~ Proof.
AeA¢'.j2~}]
1
been
EL
r I -realizabilitF).
for which the soundness theorem (for
established,
then
IA) ~ ~ + a C b A . A , completely
(i) is shown by induction on the complexity of
similar to the a r g u m e n t
in
3o2.18,
sub
(i)o
(ii)
The implication from left to right follows from (i).
H+GC
~ A o
Then
F
H ~ F--A,
a conjunction
Also therefore
g ~ ~(~
3.3.12. Corollar~o
of universally
H ~ ~(~r I F~A)
Now assume closed
instances
by the soundness theorem,
I A) o
H+GC
is consistent relative to
H
for
H ~ E~L,
EL + BI~ 3.3.13o Remarks° To s e e
CT , GC
this, we note that if
(I) where
(A). Under the assumption of
Vx[A~ - ~y~xy] A
is almost negative, then also
(2)
W[A(~O) -- ~B(~O, ~0)]
and with
GC
~v V~[A(~O) - 'v1~ ~ B(~o, (~1~)ol], therefore,
by
CT
there
is
a
z E VI
such
that
W [ A ( x ) -- '{z}I~yoX & B(x, ({zllkY,x)O)] . Now we can find a
u
such that
and h e n c e
~u Vx[Ax- {ul(x) ~ B(x, {~I(~))] I% is open whether
ECT
and O
CT
imply
GC.
implies
ECT O
212
be
(B). Just as for of the f o r m u l a
ECT
, GC cannot~ggeneraliz~ so as to omit the r e s t r i c t i o n o b e i n g almost negative.
A
A counterexample
may be c o n s t r u c t e d
Take for
~x(c~x%0) V m ~ x ( a l x / O )
A(~) :
B(~,y) ~ (y=o ~ m ~ ( ~ / o ) )
Since
~A~,
of the g e n e r a l i z e d
~he~
GC.
However, of a
the premiss
~
is valid,
such that
& B(~,~(~)] .
~lse (~sing
V~ ~m a~t(~(S~) t 0 )
(3.2.20).
V~[A~ ~ ~ ( ~ ) a B ( ~ , ~ ( ~ ) ) ]
the c o n c l u s i o n w o u l d imply the existence V~[Aa ~ ( ~ )
ECT °
, and for
v(y/O ~ ~ ( y = I) I o ) .
V~[A~ ~ ~yB(~,y)] ~ ~ is a c o n s e q u e n c e
in a similar way as for
~ V ~
a Vy< z ( ~ ( ~ )
V~A,
~(P~Q)~
= O) a ( ( ~ ( ~ ) =
(--~P--~Q))
~ ~ 0 a ~x(~%O))V
V (~(~z)~ljO & ~(~(~z)=2) / 0 ) ) }. NOW l e t
Such a
6
6
be
8n=O
It follows
such that
~ 8 ( e ) 4---> y ( e )
is easily d e f i n e d :
~ 0 ,
~8(e) ~
e-¢ 7 n = O
8n= 2 i f
~.mnn(ym= I & Vm' <m('Cm' = 0))
6n=O
,¢rm~n('~n> 1 & V m ' < m ( Y m ' = O ) )
if
.
that
~6(~) & 6(8) > 0 ~-~ ~(6) = 0 *-~ m ~ ( ~ x j O ) and since
and
we put
~6(6) ~
~-~ ~ 8 ( ~ )
6(6) > 0 , we have a contradiction.
6(e)
> 0 .
but
213
§ 4.
~odified realizability. = = = = = = = = = = = = = = = = = = = = = =
3o4oio
Introduction.
Nodified realizability was first introduced and used
in Kreisel 1959, 3.52, and later in Kreisel 1962 under the misleading name of generalized realizability (see section 10 of Kreisel 1962).
Nodified real-
izability in its abstract form provides interpretations of the various
~w_
versions into themselves ; the interpretation may be specialized (to an interpretation in (a subsystem of) a version of
HA w
or into another system) by
specifying a model for the objects of finite type ; thus Kleene~s "special realizability"
(Kleene & V esle~ 1965 , § 10) may be viewed as (a variant of)
a specialization of modified realizability to
ICF 9 the intensional con-
tinuous functions (cf. 2.6.2). One of its most distinctive properties is that
MpR
is not validated by
modified realizability ; this was already noted and used by Kreisel (Kreisel 1959, 3.52, Kreisel 1962, Thm 6) to show underivability of of intuitionistic analysis.
~R
in systems
Kleene used his "special realizability" to the
same purpose (Kleene & Vesley 1965 , § 10). See below in 3.4.9. On the other hand~ modified realizability validates
(y
not free in
A)
(This fact is connected with its invalidating
MpR,
cf. 3.4.12 (i), 3.2.27 (i)). This property was used (although not yet in full generality) in Kreisel 1959 B, D
for proof-theoretic applications ("derived rules").
Vesley 1970 also depends on a weakened version of this property. Below we shall describe modified realizability ( m r - realizability) and a variant ( ~ -
rea!izability) which bears the same relationship to
realizability as
q - r e a l i z a b i l i t y does to
Contents of the section. mr-~ and
~-
r-realizabilityo
Subsections 2 - 6 are devoted to the definition of
realizability and to the soundness theorem.
In subsections 7, 8
m r - realizability has been axiomatized.
9 deals with variant formulations ; subsection 11 compares and
mr -
Subsection
r - realizability
m r - r e a l i z a b i l i t y where the objects of finite types are interpreted by
HRO. Subsections 12- 25 discuss the realizability and non-realizability of various schemata such as proof that
~+
~R'
CTo ~ E C T ° .
realizability for
HAS
CT, CTo, FAN, BIo, TI(<) , and contain a Subsections 2 7 - 28 are devoted to modified
(relative to
HRO ).
Subsection 29 applies modified realizability to obtain a characterization of the provably recursive functions of functions represented by closed terms of
H~:
they are exactly the recursive
N - H A wo
214
3.4.2.
Definition.
Let
A
tending the language of
be a formula in a language
~-H~ w
objects of finite type°
Let
that the free variables
of
~
by some (possibly none)
P(A) P(A)
be a property,
obtained by exz
constants for
definable
in
~,
such
are contained among the free variables of
A . We define a predicate variables string
x
(the variables
the variables
in
x
construction of
A
( x
P-modified
of
x
A
x
its free
A ) ;
of the
the types of
only. ~m~rpA
In the other clauses, x, y
A),
are determined by the logical
is by induction on the logical complexity
xm~rp= A ~ A ; x= is the empty sequence,
types in
realizes
A , and variables
not occurring free in
and the length of
The definition of mrp(i)
x mr
contained among the variables free in
assume
x~pA,
~mrpB
if
A
of
A ,
is prime.
to be well-formed
(i.eo the
are correct)
m.~_.rp(ii) mrp(iii) ~--m~p(iv)
z,x,Zmrp ( A v B )
Z mrp (A-~B) -= V x ( ( x m ~ p A ) &P(A)-~y=xmrp B)
~S(v)
~)
~p(VyaAy
mm~rp(Vi) m 3.4.3.
-= [(z=O-* (x=mrpA) & P(A)) & (z/O-~(y__~pB)&P(B))]
~ vy~ (~y ~ ~ p A y ~ )
z ,x__mrp (~y°'Ay~') Examples.
= __XmrpAz 0' & p ( A z ~') .
Actually,
there are only two examples which are of prac-
tical interest to us. (A).
P(A)
= 0=0.
In this case we may omit
P(A)
equivalence) ; we call the resulting notion xmrA=~ (B).
for
xm~pA
P(A) -= A .
write
x m~A
Notational A°
AoX
(i).
for
A I, A1x
xm~pA
for
~/_(xmrA) , x m r A
of
and we
in this case.
to think of
A° A,
~x(xm~A)
to write x
A
as the "modified realizability and of
AI
respectively. interpretation"
as the "mq - translatlon" " of
A .
Remarks° Note that the modified-realizability
is identical with P(A) ~ 0 = 0
A
xmr= ~ A e - ~
predicate
(not only logically equivalent)
is omitted,
This remark corresponds (ii).
and write
m~-realizability,
convention : It is sometimes more convenient
It is helpful
(modulo logical
in this case. We call the resulting notion
or "mr - translation" 3.4.4.
altogether
m r - realizability,
i.e.
x
, and
for negative
on the convention
is the empty sequence;
to 3.2.3 for
~(~ymrrA)
xmrA
and
that
A @-~ x m q A .
r-realizability. x= is again the empty sequence~
A
215
(iii). For all
A,
xmrA
is a formula in the negative fragment.
seen by induction on the logical complexity of 3.4.5 • (i)
Theorem
Let
This is
A.
(Soundness theorem).
P(A)
be a property as intended in the definition of
m ~ p - real-
inability, satisfying
or equivalently b P(A) ~ P(A--~) -- P(B),
where ~ - ~ HA w , ~-~®,
E-~®,
H~O , ~ - ~ .
Then, for any closed
for a suitable sequence
t
of closed terms of
(ii) Let
H
one of the systems in (i),
~'
be
such that for
H + £,
H'
also holds for
H. F
a set of sentences,
(A), (B) and
for suitable sequences Corollary"
A
t
of closed terms of
H' ; then the assertion of (i)
H'
In particular,
(iii) The assertion of (i) also holds for respect to either
mr-realizability
or
H,
and for
H'
as in (ii), with
mqq-realizability, if we replace (C)
by
(C')
A~
Remark. for
F
Proof°
r =~- t~A
In (ii), for
(resp. ~b ~ A ) .
mr-,
~-realizability,
(C) is automatically satisfied
consisting of negative formulae (by 3°4.4 (i))° Note that the assertion of the soundness theorems for
equivalent to the following assertion: containing
x__ free, then
if
A(x)
N-I~A w ~ A(x) = ~ - ~
a sequence of constant terms of
N-HAA wo
N-HA w
is of a formula of TxmrpA(x)
is N - HA ~
where
T
is
Now it is sufficient to prove this
assertion by induction on the length of derivations, ioe~ we show that the assertion holds for axioms, and secondly, if of a rule, and the assertion holds for then the assertion holds for
F
FI,o°.,F n =
FI,o..,Fn,
and
F
is an instance
P(FI),.o.,P(Fn) ,
(Again, we use G~del's system (1.1.4) for
our verification). PL 2).
so
with
Assume
==Txmr~Ax~-r = , =T'xm~rp=(Ax-~Bx)= = , P(Ax)= ! then
=T'x(Tx) m r == p B x ;~=
=
~,,~_~p B~.
~"- ~.~,~(~),
For simplicity in notation, we omit parameters in further cases. PL 3). If
Assume
T~pA-~B,=
x m~r p A & P ( A ) = , then
=T'mrpB-~C'~
P(A-~B) , P(B-~C) °
Txmr~B==~ & P(B) , hence
=T'(Tx) mrpC=_
. Thus
if
2~g
T" ~ kx.T'(_Tx) , then PL 7).
Assume
ymr~B)~
T"= m~rp (A -~C) .
TmmrpA&B-~C
P(B) , then
P(A), P(B) ).
, P(A&B--C)
x,ymrA&B
Therefore
and
.
Suppose (_xmrpAl&P(A) ,
P(A&B)
T_x_y=m~p C , hence
(since
P(A-*(B-*(A&B))
T m~A-* (B-*C) .
PL 8).
Similarly.
PL 9). PLI0).
The empty sequence realizes /k-* A : <>mrp ~ A . If ((z°,_x,y_}mmm~rp(AvA~)& P(A VA) , then T must be defined by cases
•_z°~_ __ :[xy if z°:o =
~.: =x,~=~p(ALet
z° % 0
T = ~zx~.R__x(k_u_vO .~_)z ),
(one may take PL11).
if
then
T~pA
VA-~A.
A &A).
xm~rpA&P(A)
~.[o _ ,x_,O _ _]~pA-
, then (O,_x,0__)m~rpAVB,
Av B.
(Here
0
hence
is a sequence of
OV's
of the appropri-
ate types°] Let (x,~mrp ( A & B ) ) &
P(A) ; then
x_m~rpA
hence
~XyoXmr~ (A&B-*A)
PLI}). ~et ~p(A--B), P(A-~). Assume or
z°,x,ym~rpCVA,= =
O
z ~0, y mrpA
Tym~rpB.
Q ~). ~et Assume Then Q 2).
take
P(A) .
P(C)
~tY =~-~zmr~Ay ~ , Let
xmrp=~ Vy Ay, V
y ¢y Ay
_~A
Then either
z°=0,
xmrpC,=~ P(C) ;
In the second case, also
P(B) , and
T' ~ Xz O Xyo[Z°,X,(Ty__)] .
~tym~C-AJ,
zmr_C,
_ ~x.xt ~mrP
Q 5)o
and
Hence,
P(C VA) .
bP(C-Ay)
Then ky°ty ~=
~-P(Ay) ,
hence
=~zmrp y Ay
P(Vy~Ay) ; then
~-P(VyAy) o
, and
kzy.tyzmr~C-*Vy= : ~_
Ay
.
xt ~ _ m~rp At ~ , so
t~
kx.[t,x] mrp (At-- Zy~Ay)
4). Let ~ y ~ p (Ay° -C) , ~(Ay~--c) and assume Then x ~ A y &~(Ay), a~d also t y p i C . Hence
y,~
~yAy~
~(~yAT)
~yAy~--C.
~o_~:~o~j_c_~__~Ao_m_s. Equality axioms and defining axioms for the constants are trivial purely negative). It remains to verify the induction Let
xmr~AO, ~ p
(because
schema.
Vz°[Az--A(Sz)]
F(A0 & VzO[Az--A(Sz)]) ~ We put T ~ Lxyz.Rx( where
u, ~
o
are of the same type.
Then
Tx~O = ~ ,
T~(Sz) =~z(T~z)
.
2~7 By induction on TmrpAO
z , we see that
Tx~zm~rpAZ , hence
& Vz°[Az--A(Sz)] -- Vz°Az .
~he preceding verification is an almost immediate
cf (i) holds alike for all systems
corollary,
~.
(:i)
theorem ; and
in view of the deduction
(iii) is immediate. 3.4.6.
Remark.
If we are satisfied with the weaker statement in the sound-
ness theorem
~' b A = ~, ~ ~ ( ~ A ) , we may replace in (iii) of 3.4.5 (C') by
~(~)
A~r =~b
(c,,)
3.4.7- 3.4.8.
Axiomatization
3.4.7.
(i)
Lemma.
of
m r - realizabilityo
For instances
Fx
(containing at most
=
ipW
(~A-- ~ U B )
free) of
x =
-- :y~(:A--B)
not free in
(ya
A)
or
Vx :yT A(~,y) - : : ( ° ) i v 2 A(x,zx)
A%,~
we can find sequences of closed terms
T
such that
=
~-~F~, where
H
Proof.
TxmqF~,
HAm
Trivial and straightforward,
account.
(Use the notation
taking for
~=XAo(X )
IP w
remark 3.4.4 (ii) into
for the modified realizability
inter-
pretation here.) (Characterization theorem for m r - realizabilit~.). Theorem HA w -~, ! - t ~ ~, HRO- or ~~ - m ~ or an extension in the be ~ ,
3.4.8. Let
H
same language for which the soundness theorem can be established. (i)o
H~
(ii).
H + IP w + AC b A
Here
AC -=
IPm + AC ~ A~--~ ~ ( _ _ x ~ A )
U
AC
~,TET
PrOOf.
¢~ ~ ~ ~x=(__xm~A) o
~T
(i). By induction on the complexity of
by t h e induction h y p o t h e s i s , Vx(XmrB-~Zy(y~C)) xmrB<--~xmrB,
(ii).
, and since hence by
The implication ~
H ~ F-~A
for
F
(B--C) ~ xmrB
IP w,
A -= B-~C ;
C))
is in the negative fragment,
AC:V_x(xmrB-~y(ymr
follows by (i).
a conjunction
A ; consider e.g.
(:x(xm~rB)~___(ymr
Assume now
C)) 4--->
:÷IP~+AC
of closures of instances
of
~ A , then
IP ~, AC ;
so
218
H ~ =tmrF, 3.4°9°
~H ~ =t'mrF-~A'~
Inessential
hence
~H ~ =t'tmrA ° =
(but convenient)
variants
of
mr-realizabilityo
First we note that we have rather arbitrarily mr-realizability of types)
(3 4°2)
should modified-realize
Instead,
we might have
is a string of variables same sequence
Secondly,
with an arbitrarily
-A
in the proof of the soundness
for
fixed
A
prime,
sequence
where
of types
theorem
if we had based ourselves of types,
xmrA
m~A
~ A
if
x
A
m~AvB
is prime
such that
operators
as follows :
(v)
x
(vi)
x ( ~ ) ~ m_r Vy q Ay~ -- V y ~ ( w ~
Kleene's
~
a*A
"special
types
T~
(i)
I
is
(ii)
if
~
(iii)
if
~, T
y
O"
0
(~
~ Vy~(ym=~rA-*xym~B) = (D"x) ~ A ( D ' x )
realizability" generated
0
, °. , 0 m
as in (i))o
o
o
A y~ )
o
is based on yet another
schema.
Pie only
by :
type;
a
is
a type,
then
are types,
We then define,
(~)0
(written
then so is
for all types
s, T
of
as
~ XT T*~
~+t)
is
a type,
(written by Kleene o,r
(representing
as
~* I ~ (a×l)O
(b) a . ( ~ ) o -= (~x~)o (e) ~* (~1 x~2) ~ (~ ~ ~I) x ( ~ . ~2) °
(~~,T x ~ * ~ " )y ~ x~*l
~,F:(~)%
:
that we can construct E T,
~,T
ky~kz°.x
~(~)~ C
0".1,.~
O
~*T ,
0
[~y z )
Xy°x°. x(~)n((~,y)(~,,y)) =
V
~
oX(~y~)
functionals ~,T
and
~
,
a,T '
such that
~'~,~ are inverses,
(a,m))~
(~)T)
follows :
It is obvious
type).
&
is a sequence fixed,
arbitrary
°
D'(D"x) m r A )
t~0 TI ooo 0 Tm 6 0 ,
x ( a ) T m r (A-~B) m~r ~y
some fixed but otherwise
m r B ) , where
(iv)
O'XT
(~
-= ((D'x)O=O-~
&((D'x)0/0-~D"(D"x)
~,(T)o
about
I.
x a I ~ 2 mr A & B ~ Dt x ~ I xc2 tara & D"x ~ I X~2 m r B ~
~I
(but the
and other results
on a theory with pairing
we might have redefined
(ii)
(a)
x
this would not have made
x
needs
of
sequence
true prime formulae.
stipulated : x m r A
(i)
(iii)
in the definition
(with the "empty"
realizability°
and products Variant
sequence
of types for all prime formulae) ;
any difference modified
that the empty
fixed,
and
as
219
C,, (T) J: (0 x'O 0 ~a, ("r lX~2) x
y
. ( (O'y), (D"y))
,~*('~-lX~D ('rlx'2)
~'o', ("r 1X,r2) x
Xy% D(%, TI (D,~:)y~) (%,~2 (]),,x)ya)
=
O"
= D(~y%,(xy))(~,y
D Tv (=y))
°
N o w we may construct Variant
(i)
II.
x mrA ~ A
(it),
(iii)
(if
(with
A
(iv)
~
(vi)
x a.~ ~mr ..vya.Ay c, -=
mr(A~B)
Kleene's the
is prime);
~ = I),
~
(v) as in v a r i a n t
%,~ Vy~( (~a, x)y~Ay) ~
"special r e a l i z a b i l i t y "
ICF
I;
B) "
may n o w be seen as based on variant
as model for the objects of finite
type
(modulo inessential
II, with coding
differences) ° The
O-realizability
slight m o d i f i c a t i o n ICF r
of M o s c h o v a k i s
of variant
as the model for the objects
It is u s u a l l y
convenient
1971 may be viewed as based on a
I (with
~= I
of finite
in clauses
(i),
(iii)), w i t h
type.
to use v a r i a n t I, when we wish to interpret
objects of finite type in the study of modified realizabilityo wish to interpret (iii) of variant in
clauses
N-:qA w
by
HRO
or
~0,
I ; if we wish to use
(i),
ECF,
we take ICF
or
~ = 0
the
So, if we
in clauses
ICF r ,
we use
(i),
~ = I
(iii).
3.4.10. N o t a t i o n a l
convention.
shall say that
is
A
If
H-mr-
~x(xmrA)
has been proved in
r e a l i z a b l e ; and similarly for
H,
we
H - m ~ q - real-
izableo If model
~(~m~rA) M~
holds if we interpret
we can say that
be e s t a b l i s h e d
in a theory
izable.
Similarly with
Remark°
The set of
M-mr-
closed u n d e r d e d u c t i o n w h i c h are fulfilled 3.4.11.
Comparison
Let us variant realizability, p r e t a t i o n of
is
M - m~r- realizable ; and if this fact can
H' , we shall replacing
~'
of
HRO-mr-
I of
3.4.9
HR0 in
HA;
examples
realizability
(with
A
A
is
H',N-mr-real-
H' - m r - r e a l i z a b l e
satisfies
~ = 0
as a model of
in
formulae is
some obvious requirements,
of
H' )o
and
r - realizability.
clauses
N - H A w.
(i),
(iii))
as follows.
a domain of d e f i n i t i o n
for
mr-
Thus we obtain an inter-
let us denote this i n t e r p r e t a t i o n
This can also be defined directly, each f o r m u l a
say that
mr.
realizable,
(provided
in all relevant
and HA
mq
A
the objects of finite type by a
also by
xmrA.
W e first associate with
D A , a u n a r y predicate,
by induction
220
on the logical complexity of structure of
A.
DA
depends exclusivel~ on the logical
A.
(i)
DAX = [x= x]
(ii)
if
A
is prime (x
not free in
(iii)
DA&~(x) = DA(jI~) ~ DB(j2x) o DAv~(X) =- (jlx=0-.DA(J2x)) ~ (j~x/0-~D{j2x))
(iv)
DA_.B(x) ~- V Y ( D A ( y ) - ~ . ' I x l ( y ) & D B ( I x t ( y ) )
(v)
DZxAx(Y) m DAx(j2y ) o
(vi)
DVxAx(y)~ Vx(:{y}(x) & DAx({Y}(X)) ) .
A) .
o
o
(A literal use of variant I would have required (iii') D A ~ ( X ) =- DA(jlJ2x ) a DB(j2J2x) but our deviation is inessential,
and gives a slight technical simplifica-
tion. ) Now we define
xmrA
by
-= A
for
xmrA
mr(i)
A
prime (x
not free in
mj(ii)
xm~A&B
=- (Jlxmm~rA & J 2 x m r B ) .
m r(iii)
xmrAVB
= (Jlx=0-~J2xmzA)
~(iv) ~(v)
xmr~ ZyAy = j2 x " m~rA(JlX ) ~
mr(vi)
x m r vyAy -=
A)°
& (Jlx/0-~J2xmrB)
.
o
Vy(~ txl(y)& {~t(y) ~ A y ) .
~ with a similar change in m r(iii), Note : If (iii) had been replaced by (iii'),Ywe would have obtained exactly HR0-mr-realizability
(say
by a routine argument that the logical complexity of
xmr' A).
We leave it to the reader to verify
HA ~ ~x(x mr A) ~
~y(y mr~ A) , by induction on
A °
As induction hypothesis one should use a slightly stronger assertion, namely the existence of numerals
~A' ~A
such that
~xm~A ~,{~A}ix) ~ {:A}(~)m~' A ~xmr'A-~.'I:A}(x ) & {:A}(x) mr and
b
A
DA(X)~ ~{~A}(X) ,
b :i(x)- :InA }(~) (here
D~
is the predicate obtained by using clause (iii)' instead of (iii)).
It should also be noted that
x mrA
automatically implies
DA(X )
by
our definitions. It is now obvious that the essential difference between inability and
r-realizability
the case of implication:
HR0-~-real-
consists in the additional requirement in
DA_~B(X) , so that
recursive function defined for each
y
Ixl
such that
is not only a partial y mrA,
but defined for
221
all
y
such that
DA(Y ) . So the effect of using types in the definition of
modified realizability
is, when interpreted
domain of definition is prescribed only on the logical
that a minimum
for the realizing operations which depends
of the formula to be realized,
Realizability
3.4.12-3.4.25. 3.4.12.
structure
in this model,
and non-realizability
not on its truth.
of various
schemata.
Theorem°
(i)
MpR
is not
HRO
(ii)
c~
is
HRO~mr-
CT
(w.r.t.
o realizable. Proof.
mr - realizable,
-
HRO-)
is
(i) Either direct:
translation
F°
nor
realizable,
EL , ICF - mr - realizable ;
but not
HEO-mr-realizable.
t:fR0- - m r - r e a l i z a b l e ,
if
F ~ Vx[~m~yTxxy
is logically equivalent
to
hence
also
-- ~yTxxy]
HR0 - m r -
then the
~zIvx°[mVymTxxy
--T(x,x,zx)] .
HRO - m r - realizability would imply the existence of a recursive Vx°[mVymTxxy~T(x,x,zx) ~yTxxy . below),
, which would imply the recursive
Or indirect : IP
hence by 3.2.27
Similarly,
and
(i),
CT °
~R
are
we cannot prove in
E~L
that
z
such that
decidabilily
HRO-mr-~ realizable
cannot be
EL
(contradicting
(ii) HRO-
The
(3.4.7 and (ii)
~Vx[~Vy~Txxy
~ T(x,x,arx)],
Since
AC
HRO - ~ CT
is
function
e.g. 3.2.30).
H R ~ O ~ m r - realizability
(2.4.10).
of
HRO -mr-~ realizable.
since this would imply a proof of the existence of a non-recursive in
mr-
Then obviously
of
CT
CT
is immediate from the axioms for
is also
HRO--mr-realizable,
CT
~.., H R O - realizable.
is
HRO--mr-
realizable
and
HA,
realizable. is not
}~0 - m r - realizable,
since this would require
[c~] ° ~ ~z2u(1)(°)°Vx I vy°(~(zx,y,uxy) we would require an effective
~ xy= ~ ( ~ y ) )
to hold in
~0
~ ie.
operation which would assign a g~delnumber
to
each recursive function,
depending on the extension of the function only, •
which is obviously false
(e.g. by the well-known
mTnnx, make
f x = I n
~yTnny
Shoenfield 5.4.13.
if
recursive.
theorem
in
n
of
f
n
f x = 0 n = kx.O
if would
Another argument appeals to the Kreisel - Lacombe -
(2.6.15)).
Corollar X.
(i)
MPpR
(ii)
HA+IP+
(iii)
-H + I P m + A C + C T ,
is underivable CT °
relative to
in
~
;
is consistent for
relative to
~H E HA m,
HA;
-N - H A~ m , ~I- HA ~ w , HRO
is consistent
~.
Proof.
Immediate by 5o4o12 and 3.4.8 (ii).
3o4.12,
see § 3.6.
below,
example with
Tnnx ; recursive decidability
An application
settling the relationship
For more refined consequences
of (ii), due to Beeson,
between
ECT o
and
of
is given in 3°4.4
CT o : ECT o is not deriv-
222
able from
CT
in
HA.
e
3.4o14.
~+CT °
(Beeson 1972).
Theorem
~ECT ° ;
~...+ECT° +IP
in fact,
is inconsistent. Proof.
Put
~Vn~T(~,z,n) - ~T(~,z,n), B(~,~) ~ ~Vn ~m(z,~,n) - T ( z , ~ , u ) .
A~ ~
A
is almost negative.
the first HA + ECT
assertion
We shall derive
of the theorem
a contradiction
then follows
in
from 3.4.13
~+ECT
° +IP ;
(ii).
In
: O
(~)
~z[Az--~uB(~,y)] - ~ W [ A ~ - - ~ ( T v ~ B ( ~ , ~ ) ) ] .
~ith
Az ~--~ ~uB(z,u) , hence by (I) there is a
IP
Since
Vz ~ A z
(3)
Vz ~
~
v
such that
also
~w Tvzw
and also
(4)
v~V~(T~w -- ( ~ : T ~
(To see this~ note that Z u T z z u -- ~ V u ~ T z z n Hence
V~uTzzu)).
ZuTzzu
implies
, it follows
Az , hence
when combined
with
B(z,Uw) ; and since
B(~,~w)
that
~(~,~,~).
~uTzzu~-cT(z,z,Uw).
Now apply
ECT °
to a rewriting
V z i ~ w T v z w --
of (4) :
( ~ T z z u V~uTzzu)]
and we see that we may assume
for some
v o
(~)
vz[~wTvzw ~ ~Wo[TVoZ % ~ (~Wo=O -- ~ u T z z ~ )
We define
v I = Az.minw(TVozw&Uw=O) ~WTWlW
let
Uwo=O
UWo=O
~ ~ZUTVlVlU
Hence
~wTVVlW
WCF is
E~L~
; if
,
then
~uTvlvlu ,
UWo ~ O ~
then
; but this contradicts
Theorem. V~ ~
. Then
~ ZWo TVoVlWo ;
TVoVlWo ; if
3o4.15.
~ (u~ ° > o - ~u T ~ ) ] .
CT
is not
but also by (5)
~UTVlVlU~
hut by (5)
(3).
ICF r - m r - realizable,
but
WCT:
Zz Vy ~z[Txyz & ~ = U z ]
ICF r - m r -
realizable,
and
EL,
ICF r - m q - r e a l i z a b l e .
ZUTVlVlU.
223
Remark.
This result is very similar to, and was suggested by, ~1oschovakis
1971. Proof.
ICF r - m ~ - r e a l i z a b i l i t y
of
CT
all objects of type 2 are continuous
t y of
is refuted utilizing
in
ICF r.
CT would r e q u i r e the e x i s t e n c e of
x2 ,
For,
the fact that
ICF r-m~r- realizabili-
z(1)(o)o
such t h a t
V ~ V y [ T ( x 2 ~ , y , z ~ ") & ~ = T J ( z ~ ) ] , 2
which of course would imply
x ~
to be a continuous
in ~, which is obviously
false. On the other hand,
(1)
WCT °
is
Vxl~Vy°Vvl~Vz°[T(y,z,w)
We also have, in
& U(vz)=xz]
.
ICF r
v~ I ~°~vl VzO [T(y,~,vz) a U(v~) = ~ ] , which may be weakened
wl ~o
to
-~-7 z v l v O [ T ( y , z , v z )
& U(vz) = x z ] ,
i.e.
vx I ~y°-~Vvl ~vs°[ T ( y , ~ , ~ ) which in turn implies The
Theorem.
Proof.
FAN
FAN
is
follows by observing that in
(EL+FAN),
ECF-mr-realizable.
may be stated as follows :
v~ ~x A ( ~ , x )
-* ~z W ~y V ~ ( ( ~ ) z
We carry out a derivation in Assume
V~xA(#~,x)
,
(I).
ICFr-~-realizability
3.4.16.
~ ~(~) =~]
.
By
WCT I <--~ WCT ° .
(~relsel ~96Z)
(@ =- X~.kx.sg(~x)) = (~)z
-. A ( ~ , y ) ) .
E-H~+IPW+AC+~%TC AC ,
EL
~z 2 V ~ A ( ~ , z 2 ~ )
(2.6.4). o Hence by the axiom
~JC :
v~v~((7~) (%oZ ~) : (Y~)(~uc z~) ~ s~(~) ~ ~ ( ~ ) ) and therefore
~z V~ ~y V8 ((~--~)z = ( ~ ) z -~ A(¢~,y)) .
3.4.17. Theorem.
(i)
WC-N
is not
fying
EL o
(it)
WC-N
is
EL, I C F - m r - realizable.
(iii)
WC-N
is
I DB, ICF r - 2m -
Proof.
(i)
2.6.5.
If we define
ECF(U)
- m r - realizable,
~[
satis-
realizable.
This can be shown by paraphrasing
A(~,x)
for any universe
-dee Vz<j1(x--" I ) ( ~ = 0 )
Kreisel's
a ~(j1(x--" I ) + I )
eounterexample
> J2(x~--1)
in
224.
(~A(~,~)-y=0) I)] ~hen obviously
For
x=O,
B(O,~,y)
V~B(O,~,~o~) ;
intuitively represents the
for
x~O
B(x,~,y)
~o
of 2.6.7
represents the
~
m ° = J 1 ( x : 1 ) , so if
Application of
~C - N
VxV~ ~ ~y
If WC - N y(O)(1)o
Now
Y0
were
x = J(mo,m ) + I,
--
/~
B(x,~,~m,1~/
of 2.6.~, for all
v ~ ( & ~ = ~ ~ B(~,~,y)).
ECF - m r - realizable, we would have to find
z(O)(1)o
such that
~ = ~ ).
@o'
Y(J(mo'm) + I)
to
~m,1
(as is seen by
Then, by copying the remainder of the argument in 2.6.~, it
follows that we cannot find a solution for (ii)
then
would yield
must be equal to
taking
tx°1~ I J2~
'
with
i.e.
We can show
WC-N
to be
Z.
m r - realizable in
N-HAW+MC
(2.6.3),
since x mr V~Zx ° A(~,x) means x ~ m r ~ x °A(~,x) so
x~
can be written as ~mr
(z2~,y~) , and
A(~,z2~) °
On the other hand,
mr-realizability
of
is equivalent to
(I)
~xYU v~ v~(~(x~) = ~(x~) ~ u ~ mr A (~,Y~)) o
If we take for
X
~mc z2 , for
Y
z 2 , and for
U
~8,
then (I) is satis2 being uniform in z , ~ ,
fied.
The construction of the desired
X, Y, U
WC - N
is
can be shown to be a model for
-H~ w+MC (iii)
m ~ - realizable. in
EL,
Since
ICF
the assertion of the theorem follows.
Similarly, using
IDB
instead of
E~Lo
3.4.18. Theorem° (i)
E ~ L + W C - N + IP I + W C T
(ii)
~_~W+WC_N+IP
is consistent relative to
w+AC+WCT
ID~B;
is consistent relative to
225
(iii)
N-HA w ~ + W C - N + IP w + A C + W C T I~ I
Here ip I
(mA~B)
Proof.
~ Z~(mA~B)
(~
FAN
is not
ECF(~)-mr-
A ). (iii) by
realizable.
We proceed similarly to the proof of 3.4.17.
represent the
~
Obviously, if
~
ranges over
~,
& ~E n.
V~ ~!n A(~,n) ; application of
yields a statement which is false in
ECF(~
3.4.20. Lemma.
BI M
is provably
FAN
then
by 2.6.10, hence certainly
E C F ( ~ - m r - realizable (if we keep in mind that
N-HAW+BR
New we use 2.6.10 and
defined there by
A(~,n) m Rn & V m ( m < n ~ m R ~ )
not
not free in
3.4.15.
3.4.19 . Theorem. Proof.
°
(i), (ii) are immediate by 3.4.17 (iii), 3.4.15, 3.4.7;
and
(ii)
~CT
is
Ao(~,n) ~-~A(~,n)).
~mr-realizable in a theory
+ B I D + continuity axiom ; the continuity axiom is formulated as:
~
O
(I)
vz2 Vx I ~ o Vu I (~(y) =~(y) ~ zx= zu).
Proof. BI M
The modified realizability interpretation of the four premisses of
takes the form
(t)
~ZX W Po(~(X~) ,Z~)
(2)
~Z° Vn°Z(Po(n,z)-- %(n,Zon~) )
(3)
ZZ1 Vn°m°z (Po(n,z) & m~n "~ Po(m, Z1nmz)
(4)
~z 2 run( Vy %(n. ~,uy) -~ %(n,Z2~)
where
P°(n) ~ ~zPo(n,z) ,
Q°(n) = B u Q o ( n , u )
taking single variables
z, u
the third hypothesis in
BI M
Let
fn
denote the sequence
instead of to
(for simplicity in notation
z, __u), and where we have modified
Vnm(Pn & m > n - ~ P m ) .
Ix(n)x °
The modified realizability interpretation of given
X, Z, Zo, Zl, Z 2
X, Z, Zo, ZI, Z 2
BI M
now requires that
as in (I)- (4), we can construct a
U,
uniformly in
such that
Qo (n,Un) . Such a
U
is constructed by taking X(fn) < lth(n) -~ Un = Zon(Z1(~n(Xfn))n(Zfn)) , X(fn) ~ Ith(n) -~ Un = Z2(kyoU(n*y))n.
By
BRo
we can find a
U~
such that
U'XZZoZIZ 2 = U.
satisfies our requirements, we note that, if we take for
To see that
U
226
X(fn) < lth(n)
Pu ~
qn ~ qo(n,Un) then we find, implies
hence
since
X
is continuous :
X(fn) (Ith(n) , hence
Po(~n(Xfn) , Z 1 ( ~ n ( Z f n ) ) n ( Z f n ) )
Qo(n,Zon(Z1(~n(Xfn))n(Zfn))) Pn
-~
QB
,
V~Zx(X(f~)
(x)
~n(X(fn)) < n ; also
i.e.
,
o Also,
Pn
Po(~n(X(fn)),Zfn)
,
and t h e r e f o r e
Qo(n,Un) ,
so
o
Also Pn V ~ Pn and finally, that
if
VyQ(n.y)
V Y Q o ( n * y ^,
U(n*y))
,
then this implies,
, therefore by (4)
assuming
X(fn) h l t h n
Qo (n,Z2( Xy.U(n * #))n )
, i.e.
Qo(n,Un) ; hence also
v y Q ( n . ~ ) ~ Qn° Applying
BI D
Remark.
yields
VnQn , which we had to show.
For the Dialectica
to interpret
BI N
by
interpretation,
BR ° , without
Howard
(in Howard
the additional
help of
1968 ) manages BI N
itself.
The treatment does not carry over automatically
to modified realizability,
however ; presumably
and also extended to bar
the lemma can be improved,
induction of higher types (under suitable additional of interesting
applications,at
present,
assumptions)
for lack
we have refrained from carrying this
OUt. 3.4.21. BI N
Corollaries.
is
Proof.
In
m r - realizable
E L + BI D
Combine 3.4.20 with 2.9.12(which
3.4.22. Modified realizability Let
it can be shown that in
(cfo Kleene & Vesley
~
for
be a sequence of length
N-HA w
U < V
~
Let us consider
•< ( Xv
~u
CXUV
o a* =
X
R<
(where
n,
with types <
!CF r 1971 ) .
ICF r)o
~1,o°.,a n.
of the natural numbers, C
Tor
any
we can
such that
0 a* ~ 0 °I ..... 0 ~n)
TI(<) , where
T<
such that :
= ~u(~v.c(~<~v)~)
the defined
and
o
~-H~ w+T~+
a new sequence
also applies to
a sequence of constants
u£v~£~uv=
ICF
, Moschovakis
H ~ + TI(<)o
given primitive recursive well-ordering define in
1965, 411
~- operator) o
(u,v ~ Then we have
o)
denotes a defining axiom for
227
3.4°23.
Lemma.
Proof.
A s sume
is ~-~®+T<
Tl(<)
+ TI(<) - mr - realizable.
wmr v~((Vv
hence
Vuw,(w, mr (Vv< u)Av -- ~ w , ~ A ~ ) , w h i c h is the same as (I)
Vuw'((Vv<
u)(w'vmrAv)
-~ wuw' m r A u ) .
W e wish to show R ~ w mr VyAy . By
TI(<) , it is sufficient
If
(Vv
to show
then also
v < t l - ~ __C(R%v) = R<WVo= =
Vv
Therefore,
C(R<_wv)vumrAv,
since
by (I)
_wu( kv. C (R<wv)vu) mr A u , --
so
--
=
=
R<wu mr A u o
3.4°24°
Lemmao
In
H A + TI(<)
HRO, HE0
can be shown to be models for
~-Z~+ T<+ ~I(<) ° S i m i l a r l y for Proof.
ICF, E C F
in
EL+TI(<)
.
HRO.
HEO
W e give the p r o o f for
tensionality
v e r i f i c a t i o n ; for
Because of the presence the case where
x ~ x
only r e q u i r e s an additional
ICF, E C F
of pairing in
ex-
the proofs are very similar. HRO
(a single variable),
we r e s t r i c t R<
R <,
our a t t e n t i o n to
hence also
C ~ C.
=
Let
[C]
in
HR0 .
be the numeral
such that
([C],a)
for suitable
Then
~_~ - {[c] }(~,u,v) - [o ~*] We wish to construct
{
such that
{{}(=,~) - {=}(u, Av.{[c] }({{ }(~,v),v,u) and
{ C V(~)(O)T,
7{e easily find an
if m
{m}(n,~,u)
x~ V,
R < 6 (~)(0)~' .
such that " {x}(u,A~.l[c]}(ln}(x,v),v,u)
;
~
represents
C
228
the recursion theorem yields TI(<) 3.4.25. (~i)
Corollary,
NpR
~
as required°
~6 V ( T ) ( O ) T ,
one establishes (i)
Then by an application of
°
TI(<)
is
HA+ T I ( < ) ~ HRO - mr - r e a l i z a b l e .
is not derivable in
~+
TI(<)+ IP,
etc. etc. Proof.
Immediate, by 3°4.23, 3.4.24, 3.4.12 (i).
3.4.26. For modified realizability in a context of theories with generalized inductive definitions iterated once or twice, 3.4.27 . Modified realizabilit2 for
HA So
It is possible to extend the "abstract" modified realizability to
HAS
see § 6.7, § 6.8.
(i.e. not relativized to a model)
by using Girard's system of functionals
described in 1.9.27 ; cf. the analogous extension of the Dialectica interpretation in the next section (3.5o21), which is even more complicated. It is simpler, and yields a more direct application, m r - realizability for above°
We use
fying Then
HR02
{O}(x) ~ 0
HAS
as an extension of the definition in 3.4.11
for all
D Vin x ~ U~J(n,i) ' where
U~, U~, U~ . . . .
unary species containing
0
is a sequence of variables for
(the addition of such variables is obvious-
ly a conservative extension of
HA~S)o
We shall write
for 1
Wx(DA(x)(X))
D~(x)(X)
~
(viii)
D~XA(x)(X)
m ~Dx(DA(x)(X) )
and adding to the clauses x mrV.(t.,, ~ ni ~ °°'in)
We shall write
X*
mr(i) - (vi) : m ~i +I (x'tl ..... in) & DV~( x) . "
mr(vii)
V~ +I corresponding to J x m r VXA(X) m VX*(xmrA(X))
m~(viii)
xm~KA(X)
for the
m ~X*(xm~A(X))
Similarly we may define
(i)
D
v~
(vii)
3°4.28.
mr(i) -
DA
U](n,i).
HRO 2 - m r -
satis-
x.
HRO 2 - m~r- realizability is obtained by extending the clauses
mr(i)'
HRO 2-
as described in 2.9.7, with the godelnumbering
(vi) in 3.4.11 by adding to the clauses (i)- (vi) for (i)'
to describe
ICF 2 - m r -
V~. ~ X . J
. realizability
(cf. 2.9.?)o
realizability as defined here is introduced in T r o e l s t r a ~ an@ 1971&. Corollaries.
HAS+
IP
+CT °
is consistent relative to
HAS
HAS+
IP
+CT o
is conservative over
w°r.t, negative first-
HAS
229 order formulae.
(ii)
MpR i s not d e r i v a b l e i n
Proof.
(i)
for
+
HAS
The consistency
+ CT ° , and
IP
HA~+
+ IPo
CT o
follows by establishing
the soundness theorem
HRO 2 - mr - realizability ; the conservative
ex-
tension result follows by proving OmrA
e-~A
for all negative formulae of
~(H~A)
(by induction on the logical
complexity
A).
of (ii)
~PR
is refutable in
relative to
3.4.29 . Theorem
~s I
Proof.
s
1
H~S+IP+
CT °
is consistent
, § 4). of provably recursive functions).
function of
N - HA w , and conversely.
H~ ~ V x ~ y T ~ x y
where
~
(Characterization
Each provably recursive type I in
H A + CT ° + IP , and
HA~SS (cf. Troelstra
HA
is represented
by a closed term of
Ioe.
= ~ s l ( ~ - H A = ~ Vxy(Tnxy~slx:Uy))
and conversely
~(~_~w b Vx~y(~xy ~ slx = ~y)),
ranges over closed terms of type I of
In one direction,
~_~w
we use modified realizabildty : if
then by the soundness theorem for
m r - realizability,
.~ ~VxZyTnxy
for suitable
,
tI
-~= ~ ~(~,x,t~x) o So we may take Conversely,
s 1 ~ ~x.U( tlx ) .
we may for example appeal to 2.4.14 and find that
~-~=b where 3.4.30.
([si],I)
t [ s l ] t(x) = s l x ' is the standard representation
The theorem automatically
the provably recursive functions type I of
~_~W+Tl(<)
+T<,
of
[s I]
in
extends to other theories correspond
HRO. such as
H A + TI(<) ;
exactly to the closed terms of
in view of 3.4.23,
3.4.24.
Quite similar results may be extracted from the Dialectica
interpretation,
discussed in the next section. 3.4.31.
The concept of
also be generalized
to
HRO 9 but relative to the ideas of 3.2.32 on can show that
HA+M
HRO-mr-realizability, HHO A - mr - realizability. A - partial recursive A - realizability,
as described in 3.4.11, Here
functions.
HRO A
Combining this with
and the argument
is not finitely axiomatizable
over
can
is defined as
in 3.4.12 (i), we HA.
23o
§ 5.
The Dialectica
interpretation
and translation.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
3.5.1.
Introduction.
The Dialectica
interpretation
first introduced in G~del 1958, for intuitionistic was to provide a consistency for classical
arithmetic)
by an interpretation
arithmetic.
proof for intuitionistic
by elementary
of an arithmetical
"logic"
and translation were The purpose
arithmetic
(and hence
(i.e. quantifiers
especially)
statement by a quantifier-free
formula in a theory of objects of finite type, where the concept of a constructive
(computable,
in Godel's terminology:
finite type was to be regarded as primitive Hence logic was to be eliminated object of finite type.
and intuitively
From footnote
3 in Godel
it seems that a concept with decidable . primitive was intended In Kreisel
1959, and in Spector
object of
evident.
in favour of a suitable basic concept of
below,
intuitionistic
"berechenhare")
1958, and in view of 3.5.6
equality at all types as a
1962, which apply the interpretation
analNsis , only equality between objects of type
0
as a primitive ; equality between higher type objects is interpreted tensional
equality.
Spectorls work°
Howard
1968 contains
(For a discussion
simplifications
to
is taken as ex-
and refinements
of the r61e of extensionality,
of
see 3.5.12-
3.5.15 below.) A characterization
of Dialectica
first given explicitly in Yasugi (weakenings of)
AC, IPo, M
interpretation
(cf° Kreisel
(Kreisel
interpretable
implied the equivalence 1959, 2.11,
ty of
M
1959, footnote
Yasu~i
1963, see the review Troelstra
Contents of the section.
formulae of
~ - ~
was
1963 , after Kreisel already noted that of a formula with its
3.5oi) and showed the interpretabili-
I on page 113)o
For a correction
to
19 72.
3.5.2- 3.5.3 are devoted to the definition of and
the motivation behind the Dialectica interpretation° In 3.5.4 the soundness
theorem is proved,
to W.A. Howard showing the decidability
3.5.6 gives a counterexample
of prime formulae
due
to be essential
for the Dialectica interpretation. 3.5.7- 3.5.11 are devoted to the axiomatization interpretable
formulae,
interpretability
of the extensionality
3.5.16 lists some miscellaneous ..... * This, however, since
~-H~ w
functionals
of the class of Dialectica-
3o5.12- 3.5.15 to the interpretability
and n o n -
axiom, with an application°
properties
regarding the Dialectica inter-
conflicts with the suggestion in the last line of Godel
requires for its Dialectica
(e.g. to interpret
interpretation
E a ) yielding functionals
the binary tree, thus not satisfying the fan theorem
1958,
non-e~tensional discontinuous
(cf. 3.5.6 below).
on
23'~
pretability
of
CT, CTo, C - N ,
3.5.17 describes
FAN, IP
the D i l l e r - N a h m
which yields an interpretation fragment of
~-~
~ - H A ~.
interpretation,
in au "almost quantifier-free"
N - HA m , and which does not require in the proof of the sound-
ness theorem decidability 3.5.18 describes systems.
of
relative to various models of
variant of the Dialectica
of prime formulae.
the extension of the Dialectica interpretation
to stronger
Since these extensions have already been discussed in some detail
in published literature, of the principles
we have restricted
of the extensions,
ourselves to a brief indication
and a fairly detailed heuristic account
of the motivation behind Girard's extension to theories with species variables. ~e have not discussed Parsons KA c
and subsystems
3.5.2.
Definition.
~c
~m]
A D ~ ~=V~AD(X,[)
depend on the logical
;D =are
1970 , 1972 , since they deal exclusively with
(but based on classical
To each f o r m u l a o f
assign a translation x, y
of
structure of
A
contained among the free variables
If
A
is prime,
then
For the o t h e r c l a u s e s , l e t
or
~ [ ! - H A m]
or
in the same language.
~HRO-]~c
The types of
only ; the free variables of
The definition is by induction on the logical d(i)
logic).
A , AD
of
is quantifier-free.
complexity of
A.
AD ~ AD ~ A.
AD ~ Z~__V~AD(~,~) ,
V~[*~]D
~ ~2
BD ~ X u V ~ B D ( ~ , ~ )
d(ii)
(A&~) ~ ~ ~
d(iii)
(~v~) ~
d(iv)
(~zAz) D
_ ~zx= V~(~z~z)D
~(v)
(VzAz) D
~ v~(v~A~) D ~ ~=W~AD(X~,~,~)o
d(vi)
For the sake of clarity,
.
V~[*D(~'~) &BD(~'~)]'
~ ~xVyAD(~,~,z ) = =
we describe the construction
of
(A -~ B) D
in a number of steps :
(A~B) D ~ (~=V~A D
(a)
~ u V V B D )D
(b)
(c) (d) (e)
[~!~(*D(~'~ Note that with classical prime
logic and
AC,
A , A D ~ A , and for conjunctions,
fications
(A&B)De--~AD&B
D,
(f)
) " ~D(~'~))] AD ~ A
for all
disjunctions,
(AvB)De-eADvB
D,
A.
In fact, for
extensional
quanti-
(~zA)De-~ ~zA D , even
232
intuitionistically ; AC
(VzA)De-gVzA D
with
and classical logic as an inspection
refined result
see lemma 3 ° 5 . 7
equivalent 3.5.3.
B
to
quantifier-free, ~xB
A ~ ~xVyB,
stance of an implication,
B
quantifier-free,
(Kreisel 195~) mA
AD ~ A , (which is
as in the above theories).
~A
~ A ~ I=0 , as a simplified in-
we find
m ~Y=V~~ AD(~,Y~ )
This should be compared with the so-called no-counterexample
Assume
by
The motivation for the particular choice of interpreta-
If we first consider
(mA) D
D ~ B D)
For a more
(mm~=B) ~ ~ (~VxmB) D ~ ~_mmB
if prime formulae are stable,
Motivation.
tion is twofold.
(A ~ B ) D ~ ( A
below.
Note also that (i) for formulae and (ii) for
AC , and
of (a) - (f) shows.
of a statement
to be brought,
of arithmetic
interpretation
A °
first into prenex normal form,
then into
~V-form
by means of Skolem functions : ~X'If2 .°. Vxlx 2 ... B ( X l , f 1 ( 4 , ~mA
then becomes
(classically)
V f l f 2 ° . ° ~XlX 2 . . . . So t h e i m p o s s i b i l i t y functionals
~(xl,f1(xt),
Secondly,
interpretation
the purpose of the Dialectica ~w
or
~_~w
of
translation
into
order of shifting quantifiers
tion (lines
is
demonstrated
by
~mA
(of. also
in Godel 1958 , end of page 285 and top of page 286).
arbitrary formula of particular
mA)
f1(F1(fl,f2 .... )), F2(fl,f 2 .... ) . . . . ) ,
to the Dialectica
explanation
(i.eo
) .
such that
m~(Fl(fl,f2,.-.),
Godel's
x 2 ....
of a oounterexample
FI, F2, ...
which corresponds
x2,f2(xl,x2) .... )
is, to transform an
~V- form.
One may ask why the
to the front in the case of implica-
(a) - (e) in 3.5.2) has been chosen : any other order of shifting
to the front, using the classical laws
and the intuitionistic (2)
(~A~B)
combined with
AC
ular transformation applications logical
laws
~-~ V ~ ( A ~ B ) ,
( A ~ VxB) ~-~ V x ( A ~ B )
would also have resulted in an chosen in the Dia!ectica
of (I) needed are as weak as possible
complexity)
~V- form.
translation
so that the transformation
close as possible to the intended constructive
(i.e.
But the partic-
is such that the A
of minimal
may be expected to remain as interpretation.
This i m p r e s s i o n
is confirmed by the fact that even
asks for n o n - r e c u r s i v e
realizations
except the D i a l e c t i c a
~ V - transform.
The other p o s s i b i l i t i e s
(A) yields with
~
of shifting quantifiers
for suitable
to the front are
For
A m B ~Tzxy
~(~,~)]o
~[A(~,!~) z
A~A
~ V - t r a n s f o r m in all cases
AC
(3) cursive in
for its
9 a recursive
realization
and other arguments
of (3) asks for
Uz' Yz
re-
such that
Vxv(~T(z,x,YzX ) "~T(Z,UzX,V))
o
This y i e l d s
~T(z,x,Yj) and this w o u l d make
--
~Vv ~T(z,u,v)
~uVv ~Tzuv
recursively
decidable,
h e n c e a contra-
d i c t i o n follows. (B) y i e l d s w i t h
(4)
~!
AC
~y[A(~,~)
W e apply &gain to quested to find
-
B(~,z)].
A ~ B ~Tzxy Yz' Uz
; for a recursive r e a l i z a t i o n we are re-
recursive
Wv(~T(z,x,Y
in their arguments
xv) - - ~ T ( z , u
and
z ,
such that
,v)) o
Hence
W~(T(~,u,v)
-
T(~,~,Y~)),
i.e.
Vv(T(z,u,v) -- W
T(~,~,Y~v))
=
~v T(z,u,v) -- W ~w T(~,~,w)
=
(~T(~,u,v)
~Vx
This would make a complete
~w ~(~,~,W))o n ° - predicate z
w h i c h is impossible. (C) y i e l d s with
AC
w h i c h is a special
case of (4)°
equivalent
to a
Z oI _ predicate,
234
3.5.4
Theorem (Soundness).
Then,
for
A(~)
Let
H
HA~,
containing at most
~
Z-
,
~-~,
HRO-.
free
~ A(~) = ~f- ~ ~A~(~,~) for a suitable
sequence
t
of closed terms of
H o
=
Proof.
~e apply induction on the length of deductions
basis we take Godel's
system.
For definiteness,
fication to he given first for
in
H
as our logical
we shall suppose the veri-
H ~ I~- H ~ ~ ; afterwards we Comment on the
minor changes needed for the other systems considered.
PL 2).
Assume
(1)
AD(~ ~, ~, ~)
(2)
A~(~, ~ 2 ~ '
We have to construct
~) ~ ~ ( ~ ' T4
~' ~)"
such that
sD(~4 ~, v, ~).
Apply (~) to ¢ = ~2S[[' Take
x =
then AD(21s, ~2z~, ~).
~I~ ' then
AD(~t~, ~2~(~1~) v, ~)
and by PL 2
BD(~3~(~I~), Z, ~)" Take now f o r
T4 :
k~oT3~(TI~) .
In the remaining cases we shall not consider additional FL 3).
Assume
(3)
AD(~, ~ I ~ ) ~ BD(~2~, ~)
(4)
BD(~, ~3~) ~ c~(~4~,
We wish to find
T5, T 6
u
in (4)
~2~'
~).
such that
AD(~, ~ 5 ~ ) ~ % ( ~ 6 ~ , Take for
free parameters.
~) .
and take for
~
in (3)
A~(~, ~1~(~(~2~)~)) ~ ~ ( ~ '
~(~)~)' ~ ( ~ [ , ~3(~[)~) ~ %(~4(T~[), ~).
Then
A~(~, ~I~(~(~)~)) ~ C~(T4(~)' ~)"
~3~;
then
235
So we may take
T 5 ~ kx_w.(T1~(T3(T2~)w) , T 6 ~ ~_.T4(T2~ ) .
PL 7), PL 8).
~hen
Let
CD ~ ~ p V q CD(p,q) .
(A~--C) D ~ ( ~
D
~z(AD&B~) -- ~ ~ %)
and
D
(A ~ (~--C)) D ~ ( ~ A
which is equivalent PL 9)-
D-
to
~h~(~D(~,
(A & B ~
~ q ) ~ CD(P~ , q)))
C) D . Hence the two induction steps are obvious.
(I:0 ~ A) D ~ (I:0 ~ ~ V ~ A D ( ~ ) f
ly satisfied~
by any
t
~ ~=V_y(1=O " AD(X,~) )
can be trivial-
of suitable type.
=
~L~o). (AVA--A) ~ ~ (~o
, V~,[(Oo.A~(~,~))
~ (z°/O~A~(~',~'))]
D
m -~x .... ,~ )) ~ ~r,x,, := = v~°~,~"{[(~°:O-A~(~, ~ o
Z= x " V T-" A
(~%
,, ~,,)}.
--A~(~', ~'z°~,~"))l--~(~"°
Take for
~, ~'
term sequences
T, T' =
take for
X"
a
T"
,~,,)) a
such that
T ~ T' ~ kzxx'y".y"
:
=
=
= =
=
and
=
such that
=
T"z°xx ~ = =
x
= =
{
if
z° = 0
if
z° / 0 °
=
x'
[~ - Aa~] ~:~ [ ~ A ~
NOW let
TXD
T', T", T =
=
-- ~,~" V~,~"(A~(~,~,) ~(~",~"))]
be as in 1.6.14, and take for
X', X", ~
term sequences
such that
:
y, T' m kx.x,
T" ~ kx.x,
i~ ~A~{'/O
Txy'y" = I =
. . . . . .
~"
if
~A~,=O.
I% is in the verification of this axiom schema only, that the decidability of prime formulae plays an essential rSleo
(~/o - ~ ( ~ , ~ ) ) ] ~ (z~/o - ~ ( ~ , Take for
~z~,~[ ~ , ~ [ A , ( ~ ,
:
Xxy,voy,, ===
=
:
}~o0, =
fact arbitrary). Similarly for the other ha:f of P11). P12) is also routine. P13).
Assume
-- {(Z~=O - A~(~'~, ~'))
~))}].
Y, Z, X~, U: :
~,~)
kx_.x, kx.x --
--
=
~
respectively
(U =
is in
2}6 [CvA
CvB] D - [ ~z~ o
v~ [(~I=o - cD) & (~/0~AD) ]
~z~=p,=~v~, z [(~2=o-cD(~,~,))
& (~2/O-BD(U,Z))]]
=-
V °
Now t ~ e for Q
1 )o
Q, Y__, Z 2, ~', ~-
~o.,Z.T~Z,
Xz~=~.~,
Let
~D(~,~y) To
X~5'=~.q',
interpret
- AD(~,~ ~, ~, ~)
B -~ V z A z
we
have
to f i n d
T", T'" =
such
that
~D(~,~,,~=y~) -~ AD(~,,,~z , Y=, ~) . ~ake
~" --- ~2~z.T~,
~).
T'" --- X~z.~'zu.
(V=~s-,At) D -= (~Vz~D(Xz,~,z)
-. ~' V~' ~D(~',~',t))D --
-- ~zzx, vxy, (AD(X(ZX~,), YXy,, Z_X~,) ~ ~ D ( X , X , j , , Take f o r
Z, Y, X' :
Similarly
for
Q3,
~__y'.t,
~ == X y , . y=, ,
t))
kX.Xt . = =
Q4,
No~z~g!c~_5~i~m~o
The "defining axioms" for the various functional
are quantifier-free,
and therefore unproblematic.
also quantifier-free
(or purely universal,
closure) ; the rule Let
The equality axioms are
if one considers
so we only have to verify the induction axiom, BO, Vy(By--B(Sy))
(By) D m X u V v B
constants
their universal
or equivalently,
~ VxBx.
(u,v,y,z)
and suppose:
BD(~o~'~' 0, ~)
(5)
[ BD(~, ~ly~,,
y,~) - BD(~2y~, ~,, Sy, ~).
Now we define by recursion
t
such that
tO = T z
~(sy) = ~2y~(~y)° Then
BD(tO, ~, O, ~) 1.7.10
Now apply the induction lemma~-with
T ----def k y v ' T l y z ( t y ) v "
Then
Q(y,v)
BD(__ty, v , y , z)
=def B(~y, __v,y, z) ,
follows,
and with
and with
T' -= ~ z . t ,
BD(-T'z-Y, Z, Y, z) o The preceding verification
of the soundness theorem is directly applicable
237 to
I - H A w o For
HA w , we must note that the verifications
(kx__.t[x]_)~t ~t[t']_ metamathematical A fortiori,
for certain terms
t, =t' ,
where
sense indicated in Io6o15, remark
the verification
is valid for
and conclusion in an application
=
only require that
is interpreted
in the
(ii).
~-H~
~,
of the extensionality
noting that premiss rule are both purely
universal. For
HRO
3.5.5.
, the soundness theorem is also immediate.
Remarks.
(1). Similarly,
the extensionality
rule
E X T - R'
for the slightly in 1.6.12,
strengthened
version of
the soundness theorem applies.
(ii). If the deduction theorem holds for the system considered,
the soundness
theorem also extends to deductions under hypothesis : if At,
..., A n
~B,
then AID(~I'~I~I"''~n~) .... , AnD(~n,~n~1"''~n~) for some term sequence
b BD(~I°''~n'~)
T ° =
(iii). If we would have been satisfied with the weaker soundness
we could simplify the treatment of the induction need the induction lemma and its tedious proof,
schema,
theorem
inasmuch we do not
since instead of using the
induction lemma at the final step of the verification,
we no~e that
BD(~O, ~, O, ~) Vy ( ~ ~ ( ~ y , j , and therefore,
Vy (iv).
If
Dialectica
applying the induction
schema
~ BD(~y, ~, y, ~).
AD
holds for a certain model
interpretable,
-M-Dialectica Dialectica
y, ~) - v7 BD(~(Sy), 7, Sy, ~)) ,
interpretable.
interpretable
is), by a reasoning
~
of
H~Aw,
The
M-Dialectica
interpretable
formulae are closed under deduction
3°5.6°
Remark.
as is shown by the following counterexample,
N-HA w
does net have a Dialectica
correspondence)°
~e can derive by predicate
type
in
equality)
A
N-
and
H, H-~-
(provided
similar to that needed for (ii) above°
itself,
0
we may call
and if this can be proved in a formal theory
N-HA w
Vy1~Vu°~(u= 0 +_~yl = z I)
interpretation
into
due to W.Ao Howard
logic (and the decidability
(in of
238
If we treat
y
1
= z
I
as a prime formula,
[vylmvu°m(u:O<--ey=z)]
D =
IVy I ~u ° ~ ( ~ . O ~ y = ~ ) ]
D :
=- ZTJ2 Vy I -7 m(Uy:O ~
(~y
vy I shows
(by note
translation
becomes
(ii) in 3°5.2)
y = z ) ] I).
On the other hand, for no continuous
one easily
the D i a l e c t i c a
U2
o e-~ y = z) ]D
~yl -7 (Uy = 0 e--> y = }O~°o0) ]D
Since all the closed type 2 terms of (2.3.~0(D), it follows that
N-HA ~
N-HA
for any given continuous
~
represent
continuous
does not permit a D i a l e o t i e a
[[2 .
funotionals
interpretation
into itself. We also see from this counterexample lectica t r a n s l a t i o n
of any f o r m u l a of some extension
provable in the same theory implies decidable
in a w e a k sense, ~my
= z
V my
for it follows
In Luekhard%
should be
that
about the D i a l e c t i c a
of h i g L t r type equality
interpretability (but c o n j e c t u r e
is logically weaker).
1973 , page 55, the example of
quantifier-free)
N-H~A w
= z .
even implies d e c i d a b i l i t y
that the a s s u m p t i o n
of
that the Dia-
that equality b e t w e e n h i g h e r types is
We do not know w h e t h e r the a s s u m p t i o n perhaps
that T~he a s s u m p t i o n
is used to i l l u s t r a t e
~(~xAx
V m~xAx)
the need for d e c i d a b i l i t y
(Ax for prime
formulae. 3 . 5 . 7 - 3.5.11. A_xiomatization of D i a l e c t i c a interpretability. 3.5°7°
zP,
Lemmao
Let
denote the schemata
( Vx~x~ %By) ~ ~y( VxAx- By)
O
=
=
for M'
IPo, }Z'
=
A
quantifier-free,
_x_, y
with
arbitrary
x
with arbitrary
types,
- l ~ ~xAx -~ ~ A x =
for
=
=
A
=
quantifier-free,
types.
=
Then for
H ~ HA w,
I-HA w
+IP o+~{, +AC f f A ~ P ° P rOOfo Assuming
We establish H' ffA ~
r e a d i l y see that in (Vx~A) D -= Vx~A,
the lemma by induction
AD ,
H' ~ bB
H' ,
~-eB D
(A&B)D~-CA&B,
(~x~A) D -= ~x~A ;
U s i n g the n o t a t i o n
the t r a n s i t i o n
complexity
of
A o
(AvB)D~-~AVB,
so it remains to consider implication.
in 3.5.2 we see that the transition from
is justified by the induction hypothesis, istic logic,
on the logical
H' -= H + IP °' + M' + AC , we
for
to (c) by
the t r a n s i t i o n
IP~ ,
(A-~B) D
to (a)
%o (b) by intuition-
the t r a n s i t i o n
to (d) again by
239
intuitionistic VyAD~B D
and the transition from (d) to (e) by
is equivalent
and thus by
hence
logic,
M'
B D V (mBD&mV=YAD) ,
to
BD V ( ~ B D & ~ _ m A D )
;
hence
i.e.
~' ,
since
BD V ( ~ B D & ~ m ~ Y m A D ) ,
~=[B D V ( m B D & m A D ) ]
, and
~=(AD--BD) .
The transition from (e) to (f) is justified by 3.5.8.
Lemma°
~-~®,
The axiom of choice
AC
ACo
<~],
relative to
for
~
,
~a~-,
AC
w°~"
~(x,~) ~ ~(~)~ Vx~ A(x, ~x)
is Dialectica interpretable in [Vx ¢ ~ • A(x,y)] D and
qf-H.
[ ~ z ( ~ 7 ~ ~A(x, zx)] ~
Proof.
renaming bound variables), interpreting an instance
hence interpreting B~B
of
are identical
an instance of
AC
reduces to
PLI o
Lemma° Each instance o f the schemata N' and IP~ in HA ~ w , HRO-) is Dialec%ica interpretable in q f - H o
3.5.9.
(modulo
~H]
(for
H m l-H~w~
Proof.
(i)o
( m ~ __~Ax)D --- ~xAx
tion of an instance of
N'
(by note
(ii) in ~°5o2);
reduces to interpreting
so the interpreta-
an instance
C-~C
of
PLI. (ii).
Similarly for
IP' = O
3.5.10o (i).
Theorem° For H =- H~Am, I^-HA w, H + N w + IP w + AC ~ A + - ~ A D
(ii).
H+M
HRO-,
~ - ~ o
O
Here
N w,
®+IPo ~+AcbAz=qf-~bAD(L~,~)o IP ~
are the schemata
N, IP
O
extended to all finite types : O
M~
Vx(A v ~A) ~ ~
~Po
~ ( A V ~A) ~ (V~-- ~ B ) -- ~ ( V ~ - - B) o
Proof.
~ a --
(i) and (ii) hold for
i~[, and
IP'
in place of
M w, IP ~
0
3.5.7-3.5.9 sufficient
and 3.5.5
to show
(ii).
M~
and
IP ~
to be derivable in
O
-- ~ .
Vx(A V ~ A )
Dialectica interpretation,
then
~
Mw
V~(AV-~A) & ~ - ~ By lemma 5.5.7,
it is therefore
H + ~ ' + IP' + AC .
O
Consider any instance of
is equivalent i.e. to
in
H+M'
[ V=x ( A V -TA)] D o
+ IP' + A C Let
to its own
Zy VZAD(X,Y, z) ~ A D ,
[Vx(A v ~A)] D = (V~[~ V~ AD(~,y,~ ) v ~ZVv~A= = D(x'vZ'v)]°D).=. . .
-= mr~z v~v ( [ ~ o - , , ~ ( = , ~ x , z ) ] a [~/o-~ ~ A (x,,,,z~v)]) == === -
=
= =
=
From this we derive the existence
For ~et
~, X, Z
reSpo, by
0
To establish the theorem,
satisfy
D
of a
(~(~v~))O
U ~
~
=
------
such that i.e.
"
U x= = 0 <--->Ax ° =
~4o
u~ /
(~) If
U ~ = 0,
0 - ~A~(~,
then
Conversely, if
~_V~AD(~, ~, U x % 0 , then
which contradicts (2). Now relative to
Hence
for
~H]
~__V~AD(~,
~,
~) ,
so
"~A.
becomes equivalent to
the instance of
M' , and therefore derivable.
0
If
~+ F
is an extension of
(H ~ ~-~..~m, HR0-, HA ~ ~ , W~E - H A w)
H+F
AD([, ~, Z ~ ) ,
would imply
IP~o
3.5.11. Corollar#.
in
Ax.
o) - D ( ~ = o)
which is an instance of for
z) , i.e.
V~AD(~, ~, ~)
~ + ~ ' + IP ° + A C
~=D(~= Similarly
[, ~ )
holds
~
by a set of axioms
F
so that the soundness theorem
in the form
~+rbA
~+r~A ~
=
then
~+ r+M®+ IP~+AC ~ A ~-~ A D
~+r+Mm+=p~+Ac bA ~ ~+ r b AD" Remark.
In systems
H
extending
language, each assertion of ~=V~AD(~,~) , AD be
~ m (H~RO-, L - ~
for
quantifier-free.
AE~H']
in
m)
but with the same
can be brought into a normal form If the theorems of
HV , ~ _ D i a l e c t i c a interpretable,
form for assertions of AD
~
~'~H,
H
can be shown to
the result is a kind of normal
~' , obtained by interpreting the quantifiers in Mo
Taking
M m ICF,
~' ~EL,
H ~
this yields
(practically) the result of Vesley 1972. 3.5.12- 3.5.15.
The interpretability of the extensionality axiom°
3.5.12. Theorem° In
~CF(I~) ,
extensionality axiom ( 2 . 7 ° 2 ) Proof.
?~! satisfying
EL,
the
is Dialectica interpretable.
We have to show that
[Vz(~)~ x ~y o ( x
(I)
for any universe
is valid in
=e y "
z x =e
sZ)]D
ECF.
Let
~,
be sequences of variables such that
0 .
Then (I) can be stated as
[v~(V~(x~=
x~, z~
become terms of type
y~) ~ v~(~x~ = ~yz)] °
(2) Since we are working in a model with extensionality, we may make use of the reductions of the type structure (1.8.5- 1.8.8) to reduce (2) to
24t
(3)
m Vzx~(x(~xyv)
where
=y(u~xy~) - ~ x v = zy~)
z 6 (j+l)(~)j , x,y ( j+1, v 6 ¢, U ( ((j+l)(¢)O)(j+1)(j+1)(¢)j
We construct the desired
U
in our model as follows.
WI
Let
y 6
Let
~ 6 WI
o
1
(j+I)(¢)o'
1
I
~,~ 6 W. ~, 6 £ W~ , and let ¢ 6 W, ,. be an ~+ lkO)J enumerating function of the recursively dense basis for ~. (cf. 2.6.~9). represent
p - functor),
0¢
say.
Now we construct a
depending continuously
on
~, ~, y, 6 ,
1° )
If
(YI ~)(6) = (~1 ~)(6) '
2° )
If
(?] ~)(8) / (y I 8)(6) , there are initial
such t h a t
(~1~)(8),
must represent elements
(~lp)(6)
initial
Let
tensional in ~,~,7,6, 3.5.15 . Lemma°
Let
(given by some
~= ~ ; segments
can be ~omputed from them;
segments of different f ~ c t i o n a l s ,
of the recursively
different values.
we put
~
as follows :
~ so
F
dense basis of be
W!
to which
(¢)minz[~((~)z)/~((¢)z)]
AITA~A*~5.~
~x, ~x, ~x, ~x so
~x, ~x
hence there are ~, ~
assign
o ~ is obviously ex-
gives us the required
U.
be any formula in the negative fragment of
Then
(~)C where
QF - A C
is
~___~_'A(~,{) ~ Z~VxA(~,zx)
Q F - AC Proof.
+ QF-AC b F ~-~ FD
(A
quantifier-free).
We show by induction on the complexity of
F,
that
FD
takes the
form:
~v~*(~, and simultaneously
The assertions
{),
are both obvious for for
F~ G .
~ v~ F (x~,~,{) ~
(repeated use of
QF-AC,
F
prime.
Assume the assertions
vz F D
Vz[ ~_ F * ( x , z , ~ ) *--> ~
vz F
induction hypothesis).
Let
F0 ~ ~ V~F*(xz,~) , CD ~ ~ = V ~ G * ( ~ , v ) .
(h)
(FAG) D ~ a ~ V ~ ( F * ( ~ , ~ )
& G*(~,~))
~-~ e ~ V [ F * ( ~ , ~ ) & ~ u V T ~ * ( ~ y , v )
6-~ F D & G D .
~z[v~
F*(~,~)
to
Then
(VzF) D -z ~= Vz~ F*(Xz_y_,z,~) ~ ~ - ~ Vz
FD(~, ~ ) ,
that
have been established
(a)
~*(~y, y)
- ~c*(~,~)]
(classical
logic)
242
~[~V~F*(~,~) ~--~(F D 3 • 5.14.
~)
-- ~ V y G * ( ~ , ~ ) ] ~
proof.
F
w ~ -HA~ w + M w + IP ° + AC
Corollary.
negative formulae.
(QF-AC)
(~-- ~).
is conservative
over
~
W. rot.
(~reisd.~
Assume
a negative
arithmetical
of the e x t e n s i o n a l i t y
formula.
axiom~
say
+ M w + IP~ + A C
~
Then there are finitely many instances
FI, ..°, F n
~ F I -- ( %
such that
-- ...(L
" F)'")
and therefore D . FD FD + FI + "° + n + "
H~ Note also
(~w)c
+ QF - AC ~- F <--* F~
(m~)s
+ QF-AC
hence
If we not interpret it follows 3.5.15.
that
of
shown
~E-HA
(I) ( F[t]
the
by
ECF(£) ,
w
tylo..y
of type
of variables
of
~ that
hereditarily
the
majorizable
extensionality
of
E - H A wo
axiom
n =
sy1°..y
n
0 ,
tYlo..Yn,
~
Fit]
is not
is f o r m u l a t e d
(2.8.6)
Dialeciica for the variant as:
~ F[s]
sYl...y n
of type O,
net occurring free in any a s s u m p t i o n depends),the
funetionals
As a consequence,
the rule of e x t e n s i o n a l i t y
tYl°.°y n = sY1°°°y n
2.6.20 and 1.10.12
of the extensionali.ty axiOmo
the
by a functional where
then by 3o5o12,
.
model
in Howard
interpretable of
H~ ~
HA ~ F
The n o n - i n t e r p r e t a b i l i t y
By means it is
+ F DI + ° o. + F nD ~ F °
deduction
yl,°..,yn
a sequence
on w h i c h the d e d u c t i o n of
t h e o r e m does not hold.
For then
(I) w o u l d imply
VY1...yn(tYl...y and by the d e d u c t i o n it follows
n ~ sYl...yn)
theorem,
taking
that X=e y
-~ z ~ = z N •
w h i c h is the e x t e n s i o n a l i t y
axiom°
~ x, y
Fit] = ~[s] for
t, s , and
zx
for
F[x] ,
243
3.5.16.
Theorem
(i)o
CT, CT
are H R 0 - - D i a l e c t i c a o D i a l e c t i c a interpretable.
(ii)o
C-N-
continuity
~ H A m] ) o
The n e g a t i o n
Proof.
able,
so
(i).
[CT] D
is D i a l e c t i c a
EL- ICF-Dialectica
is
(~.r.t. ~ w ] (iii).
is
FAN
E~L - I C F -
and
is not
of
interpretable
HR0-
(Wor.t.
E~L- E C F - D i a l e c t i c a
IP
is
interpretable
HR0--Dialectica
I-HA w-Dialectica
is o b v i o u s l y
interpretable,
hence also
CT
interpret-
interpretable.
HRO-- Dialectica
interpretable~
and
AC
.
o for the m o d u l u s - o f - c o n t i n u i t y
MC
hence also
). of an instance
IP
(ii). The a x i o m
interpretable,
functional
(2°6°~)
is q u a n t i f i e r - f r e e : (I)
y ( ~ m j y ) = ~(~mcXY)
Therefore
(1) is identical
existence
of
~mc
ICF-Dialectica
in
-- x y = xz
with its own D i a l e c t i c a
ICF
(2.6.5)
interpretable.
EL- ICF-Dialectica The existence
Since
9 ~ kxlkz.sg(xlz) . of
M + I P + CT °
w h i c h interprets -I~ w
FAN
Then the
(2.6.4)
C-N,
it is C-N
variant
A, a variant . N. - H A. w . into
is
can also be expressed
an i n t e r p r e t a t i o n
and
EL, ECF-Dialectica
similarly.
(3.2.27) ; this directly y i e l d s the (i)o
of the D i a l e e t i o a of the D i a l e c t i c a
interpretations translation
N - H A w ~ as follows.
A ^ of the f o r m
A^ may contain b o u n d e d universal
~LxVyA^(~,~)
guantifiers,
is described
To each formula is assigned,
but no u n b o u n d e d
quantification
Vx ° < t
A
of
where
q u a n t i f i e r s orB.
In the d e f i n i t i o n of the t r a n s l a t i o n by i n d u c t i o n on the logical b o u n d e d universal
E~L,
= z(~y)
E L~ ICF-
is obtained
in c o m b i n a t i o n w i t h
The D i l l e r - N a h m
In D i l l e r - N a h m
MUC
~ z(~x)
is inconsistent
desired conclusion, 3.5.17.
Since the
E~L,
form
( ~ ) ( ~ u c ~) = ( ~ ) ( % J )
(iii)o
in
AC + (I) implies
of the f a n - f u n c t i o n a l
interpretability
translation.
can be established
interpretable.
in a q u a n t i f i e r - f r e e
where
(y,z6 I, x C 2) o
complexity,
is counted as a separate logical
operator. The i n d u c t i v e Dialectica
clauses for prime formulae,
translation.
(s~ c) ~ ~ ~ For
w
empty it follows
= ^
F u r t h e r we put
Let
&,
B ^ ~ ~vVwB~(v,w)= = = ,
V, V, E
are as for the
C ^ ~ ~ oy V z=C=^ ( y ,_ z )_
v~((vx < x~) s~(~,~v~) - c~(~,~)). that
Then
244
One easily verifies
((w< t)c)^~ Note that if we require to the Dialectica
(Vx(x< t ~ c))~° Xvz = I ,
the resulting
translation
is equivalent
interpretation.
The clause for implication
may be conceived as being obtained by replacing
in
V=wB^(v,w)
by the equivalent
assertion
and then use the same transformations the Dialectica Extending
translation qf-N-~
as for the implication
in the ease of
(3.5.2). by the addition of bounded universal
as a new "propositional"
operator,
quantification
with axioms
(vx< O)A (Vx< St)A -~ (Vx< t)A (Vx<St)A-~
[x/t]A
(Vx< t)A
[x/t]A -~ (Vx< St)A
&
I
t in
free for
(let us denote this system for the time being as N-~m
~A
for a suitable
x
A, H)
Diller and Nahm obtain
= H ~A^(t,x)
sequence of terms
t
of
N-~o
The crucial point in the proof is to show how to interpret A ^ = ~L~:=VyA^(x,y_) o
Let us suppose
A ~ A&A
.
Then ^
(A -~ A & A ) ^ - [~=I VYIA^(x--1'{1) -~ Z~--2--xsVY--2Y3(A^(-x-2'Y--2)&A^(=x3'Y--3))] =-
~Z_Ix2xsv~1~2y3[ (w < x~y2~3)A^(~I,X~y2y3)
-=
-
-. A ^ ( X 2 x ~ , { 2 ) & a ~ ( 5 3 ~ ~ , ~ 3 ) ] o Now t a k e
X -~ ~ = 1 { 2 ~ 3 . 2 ,
Y2(Sz)__xlY=2__Y3 = ~3 o In all other cases is
then It
is
formulae,
as for also
X
X 2 ~ X 3 -= ~ 1 . = ~ 1 , c a n be t a k e n
the Diatectica easy to verify
and by eases
t o be i d e n t i c a l
I,
~10~1[2{3 and t h e
= {2'
treatment
interpretation. that
for
theories
H
with
decidable
prime
H~A D ~ ~ ~A ^ .
An advantage of the present variant is, that in the construction interpretations special rSle.
for the provable formulae, In principle,
of the
the prime formulae do not play a
any set of formulae
closed under propositional
245
operations formulae,
and bounded universal
quantificaticn,
and containing
the prime
can replace the prime formulae in the definition of the translation,
and the soundness theorem might be established in the same manner.
~v~ 3.5o18o Shoenfield's fragment
variant°
By combining the translation
(§ Io10) with the Dialectica interpretation
restricting
tica interpretation
for classical
arithmetic
Shoenfield described a variant, m, V, V
fragment a formula
follows (Shoenfield
~et
equivalently,
one obtains a Dialec-
and systems
(~w)c
assigning to any formula
A S ~ Vx~
into the\
As(x,y) , A S
etc.
A
in the
quantifier
free, as
1967 , § 8.3) :
As ~ V~As(~,y),
Bs ~ V~BS(~,~).
(i)
AS ~ A ~ AS
(ii)
(mB)S ~
(iv)
(vws) s ~ v ~ _ S s ( W , ~ , Z )
for
W~_
A
quantifier
free
~3s(x,Y~ ) o
As compared to the translation
D,
for a formula in the fragment considered,
AS @-~A
requires,
~'
as defined in 3.5.7), namely in case (ii).
(M'
(or:
one's attention to the negative fragment)
'
besides intuitionistic
logic, only
QF-AC
together with
(Cf. lemma 3.5.13,
and
theorem 3o5.10(i).) 3.5.19- 3o5.21. Extending 3.5.19 . For the system DNS
Vx ~ m m A
added,
the Dialectica interpretation
EL,
lectica interpretation
Spector
1962.
DNS
~ m m Vx~A
which is equivalent
the defining
with the schema
to stronger systems.
to full classical
analysis
by means of bar-recursive
schemata of
N-H~ w
and
BR~
functionals,
functionals
for all
Spector gave the interpretation
free system of bar-recursive
(cf. Io10.9) a Dia(closed under
a ) was first given in
in an extensional
containing
the rule
quantifier-
EXT - R'
(1.6.12). A more elegant presentation
is given in Howard
with a strong rule of extensionality
treatment given there does not automatically Howard obtains other results besides:
r
~
[Vx (°)~ ~
apply to an intensional
The general
P(~Y) a
~ v ~ ( P ~ ~ P ( ~ . ~)) a
[ v~(P~~ Q~) & V ~( Vy a Q ( ~ , )
1968, also for a system
(contrary to what is said there,
Q~)] ~ Q< >
the version).
schema of bar induction
24-6
( ~, ~
variables for finite
concatenation, BwR~ ( ~ E - ~
-
sequences of objects of type
course-of-values
w + BR )
etc.),
then it is established
(translation into the negative fragment,
~ , *
is interpreted that the
denoting
in '- translation
1.10.2) of
VnVy° ~ J A ( n , y , z ) ~ ~(o)~ VnA(n, ~m,u(Sn)) can be derived in
~_~w
with types for finite Let
R u l e - BI v
for suitable
T 1 6 ((0)~)0,
(WE-H~ ~
to
T2, T 3 ~ , u
BI v , and
be given closed terms, let a variable
of type
R u l e - BR v
the
x
be a variable for
~ . Then there is a constant
{ T1[x ] < lth(x)
-- tx = T1x
~I[~] Z lt~(~) -- tx : ~ ( ~ u . t ( x ~ Howard BI
is derivable in
BR
is derived from
A direct functional
Detailed 1973.
))x°
shows
quantifier-free
w + R u l e - B l ( ( o ) ~ ) ~ + AC + E X T - R '
R u l e - BR v
for suitable
interpretation
systems with
expositions
Luckhardt
WE-~
Rule- BR
Rule - B I
are also given in Girard
1972 and Luckhardt
by a process of relativization
(similar to the interpretation
of
1962o
arithmetic
H
is a theory similar to
together with a generalized
a definition of a complete qf - H
a corresponding
In Howard
3°5°20.
1972 the Dialectica
he handles
the
to extensional
for systems
1.9o6)o H,
IDB w , i.e. a theory based on
inductive
definition
(classically :
fragment.
interpretation
for a theory
V*
of this
See also Zucker's discussion in § 6.8.
Church's thesis and bar recursiono
In Kreisel
1971
the bar-recursive
(pp.
126- 127) a proof of Godel is cited showing that
functionals
negation of Church's
thesis.
satisfy the Dialectica (A more roundabout
HA c , hence by § 1.10, in
interpretation
of the
proof is already implicit
in Spector 1962 ; see also Kreisel 1971 , page 126). holds in
1971,
His prin-
H~- set) extended to all finite types, and
quantifier-free
type is given in detail.
1970,
HAS in Ma~ w i ~ W o u ~ , c f .
It is also possible to give a Dialectica interpretation qf - H , where
in
is given.
pursues in detail the approach of Spector
axiom of extensionality functionals
;
v;
of systems with
cipal aim is to give a consistency proof ; in this context,
in
also
such that
Rule-BR
a) b) c)
~
to definition by bar recursion :
sequences of type t
+ E X T - R'
be the rule corresponding
rule corresponding Let
+ BI
sequences).
ZyVz[TxxyvmTxxz]
247
~
~v Vz[Txxy V ~Txxz]
is provable~
since
(i.e.
with
DNS)
T
is decidable.
A ~ ~
~VxZyVz[Txxy but this contradicts 1962,
DNS
following
But then, applying
Vz[Txxy V ~ T x x z ]
the non-recursiveness
is Dialectica
interpretable
satisfied by the bar-recursive
In Girard
In Girard
of
~yTxxy o
Since~ by S~ector
by bar-recursive
functionals,
it
functionals. 1970 , 1973 (chapter IX).
1971 , an extension of the Dialectica
HAS,
VxA
of the negation of Church's thesis is
A proof is also found in Luckhardt 3.5.21.
~ ~
V ~Txxz] ,
that the interpretation
described for
Vx ~ A
we find
using Girard's
interpretation
system of functionals
is
described in 1.9.27~
1972, this definition is still further extended to cover the
theory of finite types
(of species)°
For technical details we refer the reader to Girard's papers : here we restrict ourselves
to a heuristic
lectica interpretation. like quantifiers ed.
motivation for the extension
of the Dia-
In Girard's presentation no sequences
of adjacent
are used ; adjacent quantifiers
are automatically
contract-
Also for equations between terms: o
o
[t1=t2 ]D def ~ Vy [ t i = % ] , x~ y
not occurring free in
R~S
, AD
is of the form
As regards
tl~ t 2 . ~xVyAD,
species variables,
unary species variables.
Therefore, AD
variable
~
of type
To each species variable
(~)(~)(0)0,
for conjunctions,
in an obvious way (contracting A D ~ ~xVYAD '
BD ~ ~ u V V B D ,
of
Z
~Z' ~Z
we suppose to be ( ~' ~
for short) and
Then
[z~°] D ~def m=~Vy~(XzXyU°= o) The interpretation
A
let us for simplicity restrict attention to
assigned in a one-to-one manner two variables a
for any formula
quantifier free.
°
disjunctions
and implications
is adapted
adjacent like quantifiers), e.go if then
[A~B] D ~ ~ ' V y ' [ % ( D ' x ' , D ' y ' )
~.D(D"x',D"y')],
etc. etc. Now we shall discuss the choice of definition for
[vZA(Z)] °
~et
[A(Z)]D ~ g The
types
[~ZA(Z)] D
a, •
Intuitively,
will
VY~ %(=' Y' =Z' ~) in
general
contain
[~ZA(Z)] D corresponds to
~,
~
xz
~ (~)(~)(0)0o
and
248
~
~xz ~x Vy A]~(x, y, Xz, ~) .
(This is of course not a formula quantifications
over types° )
of our language,
Equivalently,
since we did not have
for a variable
x, 6 ~ X(~)(~)(0)0 ~- p[~,~] ~ { ~ Z~' VYAD(D'x',y, O"x',u=) which
is replaced
(1)
by
Zo~ZI8 Zx' "CfAD(O'x' , EXT(EXTY~)Ig)x', D"x', u)
with
Y ~ V ~ V ~ ((p[~,~])~);
give
~, 8, x' 6 p[~,~]
Using
EXT(EXTY~)~)
amounts
the rules for Girard's
(~)
l~o~Zl~pr°~,~q,o~(IZl~Pr~',!~],~ x 'J ) LL
if of type
to specification constants, ]
of
an
(~[~,~])~° Now to x, ~ ~=~s 0[~,~]
we see that
6 [~[~
p[~,~]
and
ST.(STay)
for
~ (~p[o,,s])o
v ~ (p[~,~])o
and STc~(ST ~ X x ' t [ x ' ] ) ( I ~ z s p [ ~ , ~ ] , ~ ( I z ~ p [ ~ , ~ ] , ~ x ' ) ) If we use AD
Xw.A D
as function
(3)
as an abbreviation of
w,
for (the characteristic
we find that
in (3).)
(I) implies
We take
Similarly,
(3) as
[VZA(z)]D
(3), substitute [~ZA(z)JD
function
of)
(I) implies
~ K V Y {STo~(ST~(XW.AD(D'w , E X T ( E X T Y ~ ) ~ ) w ,
(To see that
= t[x'] .
D"w, u))(X)
the left hand
= O} .
side of (2) for
X
o
corresponds
heuristically
to
v ~ v ~ Vx z ~x Vy iD(X , y, Xz, u) or equivalently
vo,v~ ~XVy, AD(x(D,y,), o"y,, D,y,, ~)
where :% ~ ( ~ , ) ( ~ ) ( 0 ) 0 , If we wish to give an X' E Vc~V~ p[c~,~] •
Then
EXT~(~XT
~x,)
and therefore
(5)
X,
uniformly
~ i~,~]
in
X ~ ((~)(~)(0)0)~,
~- p [ ~ , ~ ] .
~, ~ we can do so by finding
,
we can put
~ w v~
(4) which
y' ~ (~)(~)(O)0X~,
in turn may be replaced
~{ST
by the stronger
~(ST ~(~,w. AD(EXT~(EXT~X)(D'w), O"w', D,w, ~=))(Y) = O} .
an
249
(To see that (5) implies Y
(4), substitute
in (5).) We take (5) as our definition
I~sp[~,~]~(Is~[~,~B,~y' of
[VZA(Z)] D .
)
for
250
§ 6.
Applications:
consistency and conservative extension results.
:====::~=::=~==~====:===:==:~==::=====~:=:==~:==~===~=======
3.6.1.
Contents of the section.
The present section utilizes ~-, m ~ - realizability and the Dialectica interpretation to obtain conservative extension results.
For the case of
arithmetic, the results are given in extenso ; for analysis, only some of the more typical ones have been lifted out, since the treatment is very similar to the case for arithmetic,
so it may be left to the reader to
formulate further applications when he needs them.
As one of the more inter-
esting applications we point to the consistency of
AC
AC~
for
3.6.2.
extensions of
3.6.3°
HA m , HR._O, a n d
.Definition. . . . or
A--B
E-~
+ CT , HRO
are conservative
~.
I-
[_~w
form
~ - .~m + CT,
Immediate, by the fact that
model for
of
HRO , and of
HEO . Theorem (summary).
Proof.
for
A
~EO
Fo(FI,F2)
E-H~A w
or
~)
HR0
for
can be shown (in
~)
to be a
E-IJ!~A*o
is the class of formulae (in the language such that in all their subformulae of the
is an almost negative formula (negative formula, purely
universal formula) preceded by existential quantifiers. Let
Fn, Fan , Fpr
stand for the classes of negative, almost negative and
prenex formulae respectively. Remarks. (ii).
(i)o
F 2 ~ F I ! F° .
Since intuitionistically
(~xlo..XnA--B) ¢--~ VXl...Xn(A-- B) , we might
have omitted (modulo logical equivalence) "preceded by existential quantifiefs" in the definition of
Fo, FI, F2 .
(iii). Alternatively, we might have defined (a)
Prime formulae are in
F
Fo, El, F 2
inductively :
(F I F2) O
(b)
A , B 6 Fo ~ A & B ,
(c)
If
A
B 6 F ° (F I, F2) 3.6.4. where sion of 3.6.5 o
Convention. F
AVB,
VxA, ~xA6 Fo (FI' F2)
is almost negative (negative, purely universal), then
ZX1°o.XnA~B 6 F
(F I F2)
We use the expression "}] is conservative over
is a class of formulae, as an abbreviation for : " H ~'
which is conservative w.r.t, formulae of
Lemmao
(i) A~r O = ~ W ~ x ( x ~ A ) ~ A (ii) A c t 1 = ~ - ~ - ~ ( ~ A ) ~ A (iii) A~r 2 = ~ ~AD(~,~ ).A
F"
~, nr,,,
is an exten-
25~
Proof.
(i)o
We use the inductive definition of
F
(3.6.3, Remark (iii))
o and establish (i) by induction over the definition of
F
. O
(a)
For prime formulae (i) is immediate.
(b)
Assume (induction hypothesis)
then
~xVy(2{x}(y)
8c { x } ( y )
ray)
by our induction hypothesis.
,
~
~Zx(xrAy)
hence
-~ Ay °
Vy~x(x~Ay)
,
Let
therefore
~ x ( J 2 x r A ( J l x ) ) , so
thesis (c)
~yAy °
Assume
A
A VB o
to be almost negative ; then B
(by 3.2.11)
*--~ ~z(J2zrB(jlz)) ~--~ Z y ( y r ~ x B x ) .
Z~A -~ , % ( z r ~ ) .
follows that
hypothesis, we find (ii). for
~-~B
~x(xr ZyAy) , we
~ x ~ v ( x r A y ) , and thus by the induction hypo-
Similarly for
since for almost negative
~x1.o°XnA e-e ~v(ymr ~Xloo°XnA ) ,
$_xBx~
~x~y(y~Bx)
Therefore, if
Using
Zy(yr(ZxA-~B)) , it
~ ~-~z(z r ~ ) -~ ~
as
our
negative (3.4.4
(iii)o
The p r o o f
is
a g a i n more o r l e s s
similar
For prime formulae (iii) is obvious.
(b)
Assume
~__VyAD(x,y, z) -* Az
i.e.
xm~A
= A
(i)).
(a)
be g i v e n ,
induction
°
The proof runs parallel to the proof of (i), now using i
VyAy,
A &B °
Similarly for
Utilizing the same induction hypothesis, and assuming have
~x(xr VFAy) ,
io the proof
of
(induction hypothesis).
~X=Vzy=AD(Xz,y , z) ; t h e n
(i),
(ii)
Suppose
Vz ~j__Vy A D ( X , y . z)
:
(VzAz)D
hence
VzAz ,
etc. etc. (c)
Assume
Note that
Z~_ V_y BD(X,y ) -*B (~zAz) D ~ WzAz
(~=Az-~B) D ,
where
Assume ~Iso
Am,
ice.
(induction hypothesis)°
for purely universal
A z - = V=wC(z,w) . i.e.
VwC(z,w) ;
hypothesis,
Remark.
(i)
then
A
(3.5.2).
~Vyz(C(z,~=yz) V~C(z,__Wyz) ,
(iii) follows for
~y the proof
Theorem
(ii) a~
~zAz -~ B
hence
as
(Conservative extensions)°
(ii)
( ~ + z P ~ + A c ) n r I = ~ n r 1, ~-~®, E-~®, ~RO-
(iii)
I-HA~+IP~+AC+
(iv)
H+IPoW+AC+~
w
A o
AC
~-~,
~-~,
~mO-,
is as in 3.5.8.
nF o
for
~ ~ ~-~%
CT is conservative over is conservative over
H ~ I - H A w, ~ E - ~ ,
V__YBD(XZ,y ) , B ; hence,
and eliminating our second
(iii) now ho~d for
( ~ + ~ O T o) n r 0 = ~~ n r O ( ~ + ~ + ~ C T o) n r o = ( ~ + ~ )
for
~zAz-~B, = =
Now let
"* BD(XZ, y=) ) .
~x=Vy=BD(X,y=) and therefore by the induction hypothesis
eliminating our third hypothesis,
3°6.6°
Then
~ ~ nF 1 HNF 2
HRO-, ~eo, hence conservative over
nr
2 o
252
(V)
I - H~A w +
Proof.
I P Oe + A C + M
(i).
~ + CT
is conservative over
HA @F 2 ~
"
By the characterization theorem for realizability (3.2.18),
~(x~A).
~+ECT ° ~A ~ By lemma 3.6.5 (i)
~ ~x(x~A) ~ A
for
Act
~
Therefore
O
( H A + E C T c ) O F ° = HA 0 F ~
O
The second assertion is proved in the same manner (using 3.2.22 (i)). (ii).
Similarly, using the characterization theorem for modified realizabil-
ity (3.4.8), and lemma 3.6.5 (ii). (iii). Combine the characterization theorem for modified realizability with
the fact that HR0
C~
is
is a model for
(iv).
~m0-m~-realizable
in
~
(3.4.12 (ii)), ~ d that
I - H~A m , and use lemma 3.6.5 (ii).
Similar to (i), (ii) and (iii), using the characterization theorem
for the Dialectica interpretation (3.5.10) and lemma 3.6.5 (iii). (v).
Using the reasoning of (iv) together with the fact that
Dialectica interpretable in Remark.
CT
is
HRO-
HA.
The statements of the theorem in an obvious manner extend to ex-
tensions of the systems mentioned ( H~, ~ + M
I-HA*+zP~+AC+CT
in (iii),
~
in (i),
H
in (ii),
i~ (iv), I - H A ~ + I P ~ + A C + M ® + C T
in (v))
for which the appropriate soundness theorem is provable (3.2.19, 3.4.8,
3.5.11)o 3.6.7. (i)
Corollaries (for F~+ECTo,
HA+IP+ ~
(ii)
~+ECT over
o,
3.6.8.
are conservative over
HART
o
~.~+ I P + CTo,
HA@ F
(iii) H A + E C T O + M
~). CT
~
H A + IP o ÷ M +
CT °
. n
are conservative
pr is conservative over
(HA+M) nFan ,
Corollary of 3.6°5 (i) or 3.6.6 (i).
If
(HA+ M) n F p r .
KLS I (= KLS
relative to
V ~ set of a~! total recursive functions, see 2.6.15) is not derivable in HA,
then
Proof.
KLS I
KLS I
is also not derivable in
~+ECT
is expressible as a formula of
F
° . .
o
3.6.9.
Theorem (repeated from 3.5.14).
E-HA w+M+ ~
over
IP ~ + A C
is conservative
O
HA@ F ° n
3.6.10- 3.6.16. Axioms of choice for 3.6.10o Axioms of choice.
HRO, HEO .
For definiteness, we list the principal forms
which concern us here :
AC
Vx~ Kv~A(x,y) ~ ~z(a)~ VxaA(x,~x)
253
AC
~,T
~
is similar to
AC
but with
O,T
AC = O {AC~, T I ~'~ G =T} ; similarly T Tm
QF-AC~,
TI,...~T m
instead of
--~yT
AC~ .
Vx ~yl I ..o gy A(x,y 1 . . . . . ym) ~z (°)T1 I
(A
~yr
"'"
az(~)Tm Vx~A(x,zt x, .. m
"'
ZmX)
quantifier free).
~F-~C = U { ~ F - A C o , ~ I , . . . , % l o , ~ I 3.6.11.
Lemma.
x~ Va,
. . . . '~m:=T' m a r b i t r a : }
x~Wa,
Ia(x'Y)
(defined i n 2.4.8,
2.4.11) are
equivalent to almost negative formulae. Proof°
By induction over the type structure.
assertion is immediate. Then
x G V(a)T,
For
~ =0
Assume the lemma to hold for
XEW(~)T,
I(~)T(X,y )
the truth of the
o, T o
may be re-written as
Vy~V~[~uTxyu & Vv(~yv ~ Uv~ v ) ] , Vy~ W[~u~xyu & Vv(T~yv ~ ~v~ W ) ] &
respectively.
With the induction hypothesis for
o, T the lemma for
(~)r
follows. 3.6.12. Theorem. ably in Proof.
for formulae of
QF-ACa, °
I-HA w --
holds for
HR0
(prov-
HA). For simplicity we restrict ourselves to an instance of
without parameters.
Assume
formula of
( [B]HR0
~-~.
e~ in the obvious
way.)
~et
variables corresponding to VuG V
gv A*(u,v)
[Vx ~ ~yOA(x,Y)]HR 0 , A
a quantifier-free
is the interpretation of
A*(~,v) ~ [A(~,Y)IHR0' x, y
QF - A C
B
where
in
HR0 , defin-
~, v
are the
under the interpretation, then
(by our hypothesis)
and
b ~ ~v
- A*(u,v) W A ' ( u , v )
or equivalently
~ p V~v(uG V o ~ a<(z=O ~ A*(u,v)) a (z/O ~ m A * ( u , v ) ) ] ) . NOW apply the preceding lemma, and the closure of
~
under
proved in 3.7.2 (i)), then for a certain numeral
~p
u~ v o ~ ~ w ( T ( : , j ( ~ , v ) , ~ )
~ (uw=0 ~ , A * ( ~ , v ) ) ,
therefore
I- v ~ v a avw(T(:,j(u,v),,~) ~ uw=o)
ECR °
(to be
2%
Let
z = Au. J l m i n w [T(~,j(u,JlW),J2w ) & U(J2w ) = O] o
Then
vu~v~(~I~t(u) ~ A*(u, fzf(u))) so zcV(~)o,
and thus ~z~V(~)o vucv
Remark.
it is open whether
results
2.6°20, 2.6°21~
QF - A C
QF-AC
A*(~, Izl(u))°
holds generally for
HRO °
By the
HE0.
holds (classically) for
3.6o13. Theorem. (i)
~-I~+ECT
(ii)
[AC]smO
( ECT °
in
(i,eo
~-HA w+AC+
the class
of
~
CT
are conservative over
I~0F °
only)o
[A]HRO,
A ( { ( I - H A w) )
° ; so it is consistent (relative to
HA)
is derivable to assume
AC
HRO o
The first assertion is a consequence of the second statement ; for
interpreting
~-~
in
HA+ECT °
provable in
servative over for
and
w.r.t, the language of
~+ECT
for Proof.
o+AC,
HRO
in
H~A+ECT °
~HA+ECTo , hence over
~-HA m+AC+
gives, because
by assertion (ii), that
CT , since
CT
So it remains to prove (ii).
HA n~
holds in
Fo
[AC]HRO
I - H~A w + E C T o + A C -(3°6.6 (i))°
is
is con-
(Similarly
HRO o)
We shall derive
[AC]HR0
in
~HA+ECTo "
Assume
vx~v x EV
is
wi th
ECT
~y~V A(x,y).
equivalent
to an a l m o s t n e g a t i v e
f o r m u l a by 3.6o119
therefore
o
~u v~c v ~v[Tu~v & A(~,uv) & Uv~ vT], hence
3.6.14. Theorem. (i)
[AC~]HEO
(the set of interpretations in
in the language of
E - H A w ) is derivable in
consistent (relative to (ii)
E-H~W+AC~ HA O F
~)
+ E C T ° , and
( ECT
o
Proof.
HEO
to assume
E_p~W+AC~
+ CT
ACI
AC~
H~A+ ECT ° , so it is for
HEO .
a r e conservative over
with respect to the language of o
HA
only)
~
(ii) is obtained from (i) in the same manner as in 3.6.13.
So it remains to prove (i). ~yEW~
Assume
(I)
VxE~
(2)
io(~,~,) ~ A(~,y) ~ A(~,,y,) ~ 4(Y,Y')
x6~
of instances of
A(x,y)
is equivalent to an almost negative formula (3.6.11)~ so by
ECT °
255
there is a
u
such that
Vx6 W Assume
~v(Tuxv&A(x,Uv) & U v 6 W q ) .
Ia(x,x~ ) , then there are
~u~,~,,
T~v,
By (2)
4(~v,uv,).
A(~,:v),
v, v'
such that
A(~',:v'),
Uv~W,
:v,~W T
~herefore u~ W(=)~, and thus
~u ~ w(o)~ vx ~ ~
~v(T~v~A(x,Uv))
3.6.15. Remarks. (i)
ACt, O
is false for
HEO ; for consider
Vx I ~yO Vz o ~vO(Tyzv & Uv = xz) which holds for
HEO ; ACI, c
(ii)
By the characterization
system
HR0-
(3.4.8),
HEO
(in the language of
consistency model of
of
AC
(cf. remark (i) under 2.4o11)o
theorem for modified realizability,
it is consistent relative to
objects of finite type are hereditarily AC, IP ~
~w 2 vx1~ ° ~v°(T(~,z,v) & ~v: xz),
would imply
which obviously does not hold for
for
~_~w).
recursive
~
for the
to assume that all
operations and satisfy
Note that this does not imply the
HRO , only the consistency
of
AC
for some sub-
HR0 °
(iii) It is open whether implies
CT °
CT
[AC]HRO
does not imply
implies
ECTo ; it is obvious that it
ACo, ° : CTN
satisfies
CT , but not
ACO~O (2.5.3 ( i v ) ) . 3.6.16.
n,C
Theorem.
The following
schema
Vx~[Ax -- ~ ( B ( x , y ) & A J ) ] -- W~[Ax ~-~z(°)~(zO= ~ &Vy ° ~(zy, ~(y+l)))]
can also be shown to be consistent Proof.
for
The argument is more complicated
consider an instance of
(1)
[ Vx~ lax ~
RDC
HR0
relative
than for
with parameters.
to
HA.
AC.
For simplicity we
Assume
B~y]]~ o ,
Joe.
(2)
vx ~ v~[A*x - ~ ~ v°(B*(x,y) &A'y)
where
A*x ~ B*(x,y)
HA+ECT
, A*x
are obtained by interpreting
is equivalent
to
(2) is equivalent
to
~zA'(x,z) , A'
A,
B
in
HR0.
almost negative
O
(i)), hence
VxVz(x~ V By
ECT
o
&A'(x,z)
-~ :yz'(y~ V ~ & B*(x,y) & A'(y,z')) o
In
(3.2.18
256
~uvVxz[x~V aA,(x,~)-~lu}(~,~) a~Ivl(x,~) (lu}(x,~) ~ v= a B*(x, {~}(x,~)) a A,(lul(x,z), {vl(x,~)))]. NOW define
~(x,z,y),
~'(x,z,y)
by simultaneous reoursion on
y
as
follows (recursion theorem) :
~(x,z,0)
=
~(~,z,Sy)
~ {ul(~(x,~,y),
~,(x,~,y))
~,(~,~,sy)
~ {v}(~(x,~,y),
~,(x,~,y)).
By induction over for all
y
y
whenever
x,
~,(x,z,0) ~
one then proves
~(x,z,y),
~'(x,z,y)
to be defined
x6 V a & A'(x,z) , and moreover
VxW[x~ v ~ A,(x,~) ~ Vy(~(~,z,y) ~ v ~ ~(~(~,~,y), ~(x,~,y+ I))] which implies Vx[A*x ~ ~ z 6 V ( ~ ) ~ [ l z } ( O ) = x
& VyB*(lzl(y),
Izl(y+1))]].
Q.e.d. 3o6.17-3.6.20.
Extensions to analysis.
3.6.17. Exploiting the analogy.
We shall not attempt to give an exhaustive
list of results for analysis which can be obtained analogously to the results in 3.6.2- 3.6.16 for arithmetic, but restrict ourselves to some of the more interesting and striking applications.
Once the analogy is clear,
and the proof ideas of the preceding subsections are understood, the reader will have no difficulty in formulating and proving other applications to analysis for himself. Roughly, among the formal systems analogy ; E _ ~ W
E~L
takes the place of
we usually have to be satisfied in proving results for As regards the models, The analogues of
ECF, ICF
ECTo, CTo, CT
continuity" in 3.3. 9 ),
C-N
correspond to
are respectively
N-H~ w
HE0, HR0 GC
3.6.18. Theorem
(Examples).
(i)
n r
(EL+GC+M I)
=EL
n r
r
(ET.°+ ~I ) n r ~
0
(ii)
H ~ EL,
:EP~
z2x1= z2u I) .
Here 0
lVI1
is
V~[A V m A ] & -n~{Rc& ~ 2c~.A. H + IP I + A C o , I
is conservative over
EL+FAN,
E~L+WC-N , where
(-~A--, ~,B) -, ~ ( ~ A ~
~-~w,
instead. respectively.
(1.9.19) and the assertion that type-two
Vz2Vx I ~yOVu1(~lyO = ~ l y ° ~
N1
in the
(the "generalized
objects are continuous :
(E~+CC)
~
remains the same ; there is no direct analogue to
B).
H O F1 , for
257
(iii)
E L + IP~ +ACo, I + M
Proof°
(i)° Completely
(ii). First note that ECF
or
ICF
in
E~L).
is conservative
is conservative
Now let
EL + W C - N
(iii)oLet
over
FE F I N~(E~L) o = ~
+ IP w + AC
~
b
F
(by
E~L+FAN , using 3.4.16 and taking
the previous argument, For
E~LnF 2
similar to 3.6.6 (i), using H~Aw
EL + IP + ACo, I ~ F
Similarly for
over
instead of
I
E~L
where
_ realizability. (by use of the model
Then ~ F
3.6.6
(i~))
E-~W
+ FAN + MUC
in
~Wo
we must use 3.4.1 7 (ii)o
F6 F 2 .
Then (3.6.6 (iv))
_
~I
~
M~ W F
etc. etc. 3.6.19. Theorem. (i) QF-ACa, °
(ii) (iii) Proof.
holds for
[AC]IcF (AC ~.r.t. [AC~]ECF (AC~ w . r . t . Entirely
ICF
(provable in
EL)°
~ ( N - I ~ w) ) i s d e r i v a b l e i n ~(N-~) ) i s d e r i v a b l e in
similar to the proof of 3.6.12,
EL + G C . EL + G C .
3.6.13 (ii), 3.6.14 (i).
3.6.20. Remarks.
(i).
A consistency proof for
[AC]IcF
i s a l s o c o n t a i n e d in Vesley 1972
(implicitly). (ii). The results of Kleene 1965 are special cases of results of the type of
3.6.1~ (i).
258
§ 7.
Applications : proof-theoretic closure properties. =
3.7.1.
:
:
=
~
=
:
=
:
:
:
:
=
=
=
=
=
=
:
=
=
:
=
=
=
=
=
:
=
:
=
=
=
:
=
:
=
=
=
=
=
:
=
=
:
=
=
~
=
:
Contents of the section.
This section is devoted to establishing proof-theoretic closure properties by means of functional and realizability interpretations. the principal closure properties considered.
ED'
~Ax
If
x
~ ~t(~At)
(~Ax
closed).
is a numerical variable, and the system is complete w.r.t, closed
equations between terms of type type
We briefly discuss
The first is
0
0
(which implies that each closed term of
can be shown to be equal to a numeral), then
ED
~Zx°Ax = Zu(~A~)
(A[
ED'
implies
ED :
closed).
If the system considered contains enough arithmetic to prove ~((x=O
~A)
DP
& ( x % 0 ~ B)) ~ - ~ A v B ,
~AVB
= hA
or
WB
ED
(AVB
in turn implies closed),
Among the rules corresponding to the schema
IPR'
DP:
IP
we consider
~ A ~ x O B x = ~xO(A--Bx)
( x not
free in
A,
A
negative)
~A--~x~Bx
(x= not free in
A,
A
negative).
and IPR 'w
= ~xa(A~Bx)
Earlier~ in § 3.1, we showed how to establish for certain systems the stronger rules : ~ ~A~x°Bx
IPR
~ ~ZxO(~A~Bx)
(x
not free in
A)
(x
not free in
A )
and iPR w For
~A-~ ZxCBx = ~ x S ( - T A - ~ B x ) IPR,
see also 4o2o137 4°4°4°
Corresponding to
ACR
AC
~Vx~TA(x,y)
we consider
the rule
~ ~.%(~)Tv~A(~:,ZX) .
Of course it is possible to formulate other rules corresponding to RDC . Corresponding to CR
and
CT, CTo, ECT o
~VtI ~ ( ~ - Vy°({~l(y) ~ fly)
we consider the rules:
DC
and
259
ffV~(A~ - ~ y )
sc~ °
(A
= ff
~uW(A~
av(T~
~
& ~(~,uv)))
almost negative)°
In the case of applications
to analysis,
CRo, ECHo
are replaced by certain
continuity rules ; see 3.7.9. Subsections systems.
2 - 8 are devoted to applications
In 3.7.9 we discuss,
briefly,
to arithmetic
and related
analogous
applications
to systems
over
with respect to closed
stronger than arithmetic. 3.7.2. Theorem. (i)
Assume
H~+ F
O
~I - formulae,
(1)
to be conservative and let
F
be a set of closed formulae
FE r ~ ~ a ~ ( ~ + r ~ q F ) Then
Let
HA+ F
satisfies
F
H be
ED, DP, CR , ECR
° O
H~A~, N-HA w, ~ - I~ *, WE-HAAw, E-HA w or
be a collection of closed formulae
(2)
such that
o O
(ii)
~
HR0- , and l e t
such that
F~ r ~.=~s(H+_ r b s m ~ F ) . Then
H+ F
satisfies
Proof.
Ad (i).
Assume
I~+ F ~xAx
i.e.
By
IPR'
3.2.4 (iii)
,
~Ax
t~+ f b J2~A(JI~)
closed.
Then for some
~,
HA+ Z ~
q ZxAx ,
& A(Jl~) "
Since we Can find a numeral
ECR
ED', ACR,
~'
is obtained as follows.
such that
}~ ~ jl n = n' , it follows
that
Assume
O
~+
r~
and assume that
Vx(Ax~xBxy) A, B
(A
almost negative)
do not contain free variables
more general case with additional contraction of variables°)
parameters
besides
x, y .
(The
can be reduced to this case by
Then far some numeral
m :
SO
~+
r ~ vxVu[uqAx ~ Ax ~ :t~(aTt&~(x,jlt))
t ~ II~l(~)l(u)° ~y 3.2.11 (i) u q A x ~ A x and also u q A~ ~ ' ~ (x) ~ ~A(~) q Ax, hence " A
where
:+ where
for
&B(x,jlt) ~ A
almost negative,
r ~ w [ A x ~ ,t, :B(:,jlt,) ~
t' : {Iml(x) l(~A(X)) o We can find a numeral
HA ~ : = AX°Jlt,
hence
:
such that
26O
The proof of (ii) is similar to the proof of (i),
m qq- realizability taking
~ - realizability ; since in this case we deal with arbitrary terms, a condition such as conservativeness w.r.t, closed Elo _ formulae is the place of
not necessary. 3.7-3. r
Remark.
For closure under
ECRo, ACE
the conditions
(I), (2) on
in 3.7.2 respectively may be weakened to
F~ r ~ + r ~
~(y~F)
and
3.7.4. (i) (ii)
Theorem
(Corollaries of 3.7.2).
~ + F, F ~ E C T ° or M c F c E C T o U M satisfies ED, DP, CR ° and ECR o. H + F, F c _ I p w U A C whore H may be HA w, I - H ~ w, W E - ~ w, E-~H~A w, N-HA w
(iii) I~ Proof.
or
HRO-,
satisfies
is closed under
ED', ACR, IPR'W.
IPR' o
(i). We use 3.7.2 (i), 3.2.4, 3.2.15, 3.2.21.
The conservativeness of instances of
o formulae follows, for the addition El-
w.r.t, closed
ECT
, from 3.6.6 (i) and for the addition of instances of O
~
by 3.6.6 (iv).
(ii). Similarly, from 3.7.2 (ii) and 3.4.7.
(iii)
From
3.7-5.
3.7.2
Theorem.
H+AC+IPW+N
m
( i i ) , using For
H
satisfies
HRO-
-= HA ~ w , I-HA
w , WE-HA
ED' , ACR;
for
O
WE-HAW: Proof.
(~)
H+AC+IPm+~ For
w
satisfies
H' m H + A C + I P 2 + M
w
w , I-
HA w
+
iE
H = HA w ,
I-HA
~
--
~
, HRO-
+ IE
~
:
, O
DP°
we have (3.5.10)
~, ~A = ~(~, b ~ %(~,~))
and also (2)
~, ~ A ~ - ~ V y % ( x , y ) .
The soundness theorem for
~-H~ w
automatically extends to
1 - H A m + IE ~
. 0
Therefore if
~' ~ ~x~ Ax ~
( ~x~Ax ~
closed)
then H' ~ V~ AD(t a, ~, y) for suitable closed ~' ~ A(t~).
t~, ~ .
Since
Axa ~--~ ~ z V ~ A D ( X ~ =
_
'
~, ~),
it follows that
291
If any closed type possible in H' ~ A V B ,
0
term can be shown to be equal to a numeral,
HA ~ w, I-HA~+IEo AVB
, W~E~-~ w,
closed, we find
then
~,s, t
DP
holds.
as is
For then if
such that
H' ~- V=yv[(~=O -~ AD(_t,y) ) & (~/0 -* BD(S,V)) ] , which in view of (1), (2) implies
~,pA 3.7.6.
or ~,ffB.
Lemma.
Then
~
N-HAWcH,
Let
is closed under
Proof.
J-~H] = ~ N -
CR.
Consider any closed term
= [t I]
such that
w].
tI
of
Then there is a numeral
N- ~w
([tl], I ) r e p r e s e n t s
tI
in
HRO ; by 2.4.13
[tly]HRo ~ {[t,] }(y), and by 2.4.14
- ~ = b tly ~ l[ tl] t(y) • 3.7.7.
Theorem°
H + F,
£ c IP w U AC
for
H ~ N
-
HA ~ , E - H A w,
--
~
is closed under
CR
~
, and O
under the rule
ECRI
~yoB(~,y)) = ff~u°Vx ° ~v°[Tuxv & (*~B(x,Uv))]
~ vxO(ix ~ (A
Proof.
(a)
negative). By the preceding lemma,
~+ £
more, by 3.7.4 (ii),
H+~
ED
hence closure under
holds for
(b)
~+ £
~+F,
is closed under H~
is closed under
IPR' , and
is closed under
is closed under ACR CR °
and
ED I
CR ; furthertells us that
follows.
CR ° , and as a result
ECR I
follows.
3.7.8.
CorollarZ.
Proof.
Immediate,
3.7.9-
Extension of the precedin~ .me.t.h.ods to systems of anal.Tsis. I I ~r - and ~ - r e a l i z a b i l i t y to obtain properties such as
since
N-HA w
ECR I .
is a conservative extension of
~.
The use of ED,
DP,
and the following forms of Churchls z~ale
( ~A~
closed) and
t- Vx ~yA(x,y) = ~ ( t for certain extensions of The extensions of
EL
EL
vx ~ T~xy & , ( ~ , ~ y ) ) are discussed in detail in Kleene 1969, 5.9.
considered are the same as in 3.3.4.
In addition to the applications given by Kleene, we can also use
q
-
262
realizability to show closure under the following generalized continuity rule
GCR :
where
A
is almost negative.
for the extensions of A variant of might be called
m~
For example, we can obtain closure under
GCR
E~L mentioned in 3.3.4. realizability specialized for
ICF
(by analogy this
m q - realizability), or in other words, a
q - v a r i a n t of
Kleene's special realizability, has not been investigated in the literature, but may be expected to yield similar results for extensions of include
IP
EL
which
(not only w.rot, numerical, but also w.r°t, function variables).
V esle[ 1972 , after formalization, may be expected to yield results similar to those listed in 3.7.5 for extensions of IP °
and
M,
E~L
containing forms of
since Vesley 1972 amounts to carrying through a Dialectica
translation within the language of analysis by interpreting the objects of finite type in
ICF o
For systems in the language of such as
HA+T!(~)
~,
but proof-theoretically stronger,
for example, the results obtained by
readily extend (in view of 3.2°23), so in 3.7.4 (i), or
TI(<) O ~ c F c ECT
U TI(<) U M O
~ - realizability
TI(~)c F c E C T o U T I ( ~ )
is also permissible, etc. etc.
263
§ 8.
Markov's schema and Narkov's ruleo = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
3.8.1.
Contents of the section.
In this section some miscellaneous
information concerning Markovts rule
and Markovts schema is brought together. O
rules
MR, MRpR , ~RpR
M, MpR , ~ R
and the corresponding
have already been introduced in I°11.5.
In theories of finite type, such as
N - H ~ w,
the generalization
M~
(defined in 5.5.10) and the corresponding rule MR w
~Vx~(A v ~A),
~ x ~ A
= ~x~A
play a r61e. It is useful to note that due to H. Luckhardt) : M, N ~
M, M ~
can be formulated as rules (a remark
are (relative to
HA, ~ - H A ~
respectively)
equivalent to Vx°(A V ~ A ) -- 7 ~ ~x°A = Vx°(A V 7 A ) ~ ~x°A and VxU(A V ~A) -~ -7-7~x~A = Vx~(A V : A ) -* Sx~A respectively.
This can be seen by taking for
Ax
in the rule
-~SyBy-*Bx.
Then certainly ffVx(Axv ~Ax) ~ ~ A x , since
~-7-TSXAxo
On the other hand,
Vx(A V-TA) -* ~zr~A is
Vx(-l'7 ~yBy-~Bx) V "-I('7-7~yBy-~Bx)) -~ ~x(-7~ SyBy-~Bx) , which implies Vx[ ( ~
S~vBy-~ Bx) V -7 (-7-7~vBy-~ Bx) ] & --77 ~yBy -~ ~xBx , hence
Vy(By V 7 B y ) & - ~ y B y - ~ As regards
MpR,
~yBy° we note that this is equivalent to the specific single
instance
If we make use of the fact that in Ix I :zT(JlX'J2x'z) }
and
it is provable that
Ix I :zT(x,x,z) I are both complete
therefore recursively isomorphic MpR
~
O
ZI
-
sets, and
(cf. Rogers 1967, § 7.2), then we see that
is also equivalent to
~e interpret 3.8.2.
-7~R
Theorem°
as the negation of the universal There exists a formula
that for no negative formula
B
~
~:A
17 A, *-* B o
A
closure of this formula.
almost negative,
such
In other words, negations
of almost negative formulae are not always equivalent to negative formulae.
264
Proof.
A ~ Vx[~
Take
~ y T x x y -- ~ T x x y ]
negative, then a f o r t i o r i : ~ o that
H~Ac b ~ B ,
(1)
~
~ut ~
(s~A)
ff ~ A
and therefore,
~ ~
.
since
B
r e l a t i v e to
3.8.3.
Theorem
bA '
,
B
it follows
~.~ b ~ B .
Thus
(Kreisel).
The system
.~+ ~%R
(by ~.4.4 ( n ) ) ,
so
is consistent, and
MRPR °
We establish first
(~+
(1) The system
+ C%)
~%R
HRO- + IP w + AC
3.6.6 hi)).
c~ o
hol~s in
(3.2.27), therefore
HR0-
~c
is negative,
hence < >mr~A
i~0,
is consistent, contradicting (1).
closed under
Since
A .
HA+ ~A
Proof.
H~A b ~ A @ - ~ B
Assume
+__> ~ B °
~
Now assume
~
+ ~MpR
n rl°
is conservative over HR~O- w.r.t.
FI
(cf.
~ 0 - + AC,
and m ~ R holds in ~ + I P + C T ° c HR~- + IP ~ + AC. On the other hand, + CTo --
+ ~R
is conservative over
HA ~ ~
n r1 ~ ~
~,
and thus (I) follows.
~ ~yAxy,
A
~yAxy , and by the closure of
primitive reeursive.
HA
under
~PR
'
Then by (I)
H~ ~ ZxAxy , so
+ ~%R P ~yA~y. 3.8.4.
Theorem°
HA c H C I~ c . Then the following two assertions are
Let
equivalent : (i)
In
H
every almost negative formula is provably equivalent to a
negative formula (ii)
%R
Proof.
is derivable in (ii) = (i)
~.
is immediate.
Assume (i) and consider
F ~ [ m ~ ~zTxyz-~ ZzTxyz] and assume
H b F e-~B ,
BE r ° Then
HAC ~ F <--~B,
n
therefore 5 •8.5. (i)
H~ ~ B ,
and thus
HAC ~ B ;
This proves (ii)o
Theorem° The following rule H ~ Ax ~ V o A x ~, H ~ -7-~x~Ax -~ m ~ x ~ A
holds for
~ ~I!,+r,
F ~ IPW+AC . H~ ~
MpRO
(iii)
NR w
holds
holds in
Proof.
(i)
~, ~ m ®,
~ - H A ®,
The rule also holds for
(ii) MR
and so
~
H bF °
in
I-
~
H~
=
m~, (with
E- H~w, HR0-, w ~ - ~ ® , ~=0)o
RAw, HR0-, WE-HA w, HAm
~. Assume
A
to contain
a
x , y
T
free, then
265
~k Then by term
T
Vx y ~z
ACR, ED'
tA6 (~)(~)0
Or
L(~=o~A) a (z/~--A)].
for
H
(3.7.4 (ii)), if
V mA
H kA
such that
Vx y [(tAxy=O-~A ) & (tAxy/O-~ m A ) ] Then
we find a closed
.
H ~ A ~-9 tAxY= 0 , and a s s u m t ~ H ~ m ~ x S A
-~ ~xaA , we have
H ~ -~-7~x~(tAxy= 0) -~ Zx a(tAxy= 0) . By closure under ability;
IPR'W;
ACR
(or more directly,
soundness for
~-realiz-
3.7.4 (ii))
of.
H t- ~[Z(T)=VYT( mm~x=(tA xy= 0)-* tA(zy)y= O) ; since H ff tA(zy)y= 0 V tA(zy)y / 0 , we have
~ Vx~(mtAxY = O) V ~:~ (tAxY : 0) hence Vx ° ~A(~,y)
v e~=A(x,y).
To obtain the result for the fact that (ii)
use the result just obtained together with
is a conservative
extension of
HA o
See 1.11.5.
(iii) Let only.
I-HA w
H~,
ffVxa(A v ~A) , ff ~ m
~x~A . Assume
As under (i) we may construct a term
A
tA
to contain
x, y
T
free
such that
ff tAxY = 0 *-~ A(x,y) ° Then
ff ~
~x a Axy
is equivalent to
~Vy T~
translation of this is (equivalent to) Note (ii)).
Therefore we can find a closed term
~VyT(tA(sy)y=
0)
(3.5.4), and therefore
extension of
I-HA w
(by 3.5.2,
such that
. is a conservative
.~.
Corollary.
In
H~
and
~c
the same g~delnumbers
to represent total recursive functions Proof.
s 6 (T)~
~xSA(x,y)
For arithmetic we must again use the fact that
3.8.6.
~x (tAxY= 0) . The Dialectica
Zz(~)T Vy T tA(zy)y = 0
Assume
H A c b ~uT([,x,u) ; then
can be proved
(Kreisel 1958). ~.~ b ~ V u mT(~,x,u) , henoe
~ffXuT(~,x,u)° 5.8.7.
N, ~IR for systems stronger than arithmetic.
The underivability
of
N
in systems of intuitionistic
analysis with
function variables was shown in Kleene and Vesley 1965, § 10, 11, by means of "special realizability"
(i.e.
m r - realizability relative to
ICF)o
266
Otherwise there are few systematic investigations in the literature ; e.g. Scarpellini
1971, chapter IX.
sion of results on
~, NR
see
Another line of approach for the exten-
and its variants is given by the present section :
scan around for systems to which the methods of proof of the present section apply° For theories with species variables, chapter IV,
~.2.|~
, and for
Further remarks on in 1 . 1 1 . 5 .
M, N[R
see for closure under
~PR'
NR, Girard 1972 , V I , ~ . and its variants are found in this volume,
267
§ 9.
Applications
3.9.1o
of
p - realizability°
Contents of the section°
In this section we consider a notion of
p - realizability
1972), which does not quite fall under the schema of § 5.2, but the treatment of which is rather similar Our principal Beeson
interest
1972 ; namely
in ~
p - realizability ~KLS
o
found in
The result has a certain intrinsic
settling the old problem about the intuitionistic
See
%he
definition in 3.9.9), but our principal in
~ppllcahiO. o F
of
(cf. remarks in 3.~ ~).
is in an application,
est,
provability
inter-
of
KLS I
reason for including it
here is that it provides an example ofWYealizability not accessible
(due to Beeson
~ p - realizability
techniques up till now
by other methods°
At the end of the section we have summarized
proofs)
(without
some other
1972.
related results from Beeson
(Postscript : a new simplified proof is in ~eeson 9 esonB B. s e e 3 13 Beeson 1972 discusses m~tters in somewhat g r e ~ T ' - g e n @ r ~ l l t y , ' O ~ t our
presentation we have restricted
ourselves
to the essentials needed for the
application. 3.9.2.
Definition
placing the clauses (iii)
of
~ - realizability°
(iii),
(iv),
x ~ A ( x I .... ) VB(xI'°'')
A definition
is obtained by re-
(v) in 5.2.2 by
~def [Jl x = O ~ J2x~p~A(Xl .... ) & P r ( ' ~ A ( X l
(JlX/0 ~ J2x~B(x~ .... ) ~ P r ( ~ B ( ~ (iv)
xpA(xI'°°')'B(xI'°'')
))
~ ~A(Y'Xl .... ) ~ e f J2x~A(Jlx'~1 .... ) & P r ( ~ Y - - J ~ A ( ' ~ ' ~ I ' ° " P )
and replacing in the other clauses Here
.... P)
~def V u ( u ~ A ( x l .... ) & P r ( r ~ A ( ~ 1 ' ° ' ° ) ~ ) "
- ~lx}(u) ~ {x}(u)~ ~(~ I . . . .
(v)
.... )~)
Pr(rA ~)
abbreviates
r
~-ProofD~
by
p.
( y , r A ~) ,
and in
is supposed to be a list containing all the variables Note that the definition is more general * than in § 3°2~ since to the predicate P(A,y) ~ y
P(A)
a free variable not in
A(x1,°.°) free in
~ p - realizability
there now corresponds
" xl,o..
A. as used
a predicate
A o
Let us introduce as an abbreviation :
tP~-~pA(Xl. . . . ) where if Perhaps
~def
t~A(Xl ....
t m t[X1,o..,Xn] , ~
P(A(x I ....
))
) & Pr(r~A([1
indicates
)u) '
the numeral giving the value of
~ ~r(r~(y~A(~ I ....
but it certainly would have complicated
....
))~)
might also have worked,
the treatment.
"
268
We collect in a lemma some properties needed in the proof of the soundness theorem. 3.9°3.
Lemma.
~
indicates provability in
(A
~.~ o
(i)
~pr(rA ~) - Pr(~Pr~A ~)
(ii)
~ P r ( ~ A ~ B ~) & P~(mA=) -- Pr(rB :)
close:)
(iii)
~ x p p A(Y 1 . . . . ,Yn ) ~ P r ( ~ p ~ p A ( y I . . . . . yn )~)
(iv)
# x p A ( t , ~ . . . . . yn ) ~ [ ~ / t ] ( x 2 A(y~,~ . . . . ,~n )) A(~ .... ) & ~(A(~ . . . . ) ~ S ( z ~ . . . . )) -- ~l~l(~) ~ l ~ l ( ~ ) p p S ( z ~ . . . . )
(vi)
~xp~A & y p p
(A, B
closed)
B ~ t[x,y]~ C = (t
(vii) (viii)
ffwp~ w ~ - . :{~l(x) & :bl(~)p~sx ~t ~A = ~tppSo
Proof.
(i). Immediate,
since
Pr
is a
primitive recursive)
( VxBx
closed)
E oI - predicate.
Cf. § 1.5. 40.
(ii)o Immediate.
( i i i ) o Assume x p A ( y t . . . . ) & P r ( r [ p A ( y 1 . . . . ) ' ) o Then by ( i ) P r ( r P r ( ~ p A ( ~ 1 . . . . )~)~) , hence : p r ( r ~ p A ( ~ t .... )') & P r ( r P r ( ' x p A ( ~ l , . . o ) ~ ) ~ ) ,
i.e.
Pr(r: pp A ( y l , . . . ) ~ ) .
(iv). Proof by induction on the complexity of
(v). Then
A.
Let x p ~ A ( z 1 . . . . ), y ~ A ( z 1 . . . . ) ~ S ( z 1 . . . . ) o ' { y } ( x ) ~ t y t ( x ) ~ A ( ~ . . . . ) ~ S ( z ~ . . . . ) . Also P r ( ~ p A ( ~ l , . . . ) ~ ) ,
e r ( ~ p A ( ~ I . . . . ) - S ( ~ I . . . . ) ~ ) , hence PrCfy~--FHT) p S ( ~ l , . . . ) ~ ) , and thus {y t ( ~ p ~ B(~ I . . . . ) " ( v i ) . Let W ~ p ~ A ( ~ , . . . ) ~ y p ~ S ( ~ , . . . ) - t [ ~ , y ] B C ( s , . . . ) , and assume x p p A ( ~ , . . . ) & y p p ~ ( ~ . . . . ) . We have W P r C ~ A ( f f , . . . ) &~pps(~,...) -~ t [ ~ , y ] p C ( ~ .... )~)
with (iii),
therefore
pp C~) o
Pr(~
Pr(rt[~,y]
(vii).
~ O(~,.°.)~), ~et ~pp VxSxo
hence
Pr(r~w~)
(viii). Suppose 3.9.4.
P r ( r ~ p ~ A ( ~ .... )~),Pr(CyppB(~,...)~) ,
Also
pr(rt[~,y]
=t[x~,y I) ,
therefore
fix,y] ~ C o Then Vx(, I ~ l ( x ) & {wl(=) p S~) & P r T ~
~Bx~), ~tp~pAo
and thus Then
V~Sx ~) ,
lw}(x) p ppBx°
~Pr('-tpA~),
Theorem (Soundness theorem)° For closed
hence
~tppA.
A
~ A = _~(~-:~A). Proof.
The proof is rather along the lines of the proof of 3.2.4, but we
have to appeal repeatedly to 3.9.3 ; we establish, by induction on the length of a deduction of
A,
269
where
A*
is the universal closure of
A o We select four typical
We may take G~del's system for the verification. examples of basis and induction step. PL 2).
Assume (induction hypothesis)
(3.9.3 ( v i i i ) ) hence ff~imJ(x) & { m } ( x ) ~ A x ~ Bx. Also ~ p ~ V x A ~l~l(x) ~ I ~ l ( x ) ~ , and therefore PL 7). Suppose b ~ V x ( A ~ c ) . Then A~^v~w. {~ }(~, ~ (~,w)) ~ V~(A~ (s~ c) ) (3.9.3 (vi))° Q 3). Consider Vy(A(t[y]) ~ ~xAx) . Let ~ = a ~ A u . j ( t [ y ] , u ) . (with a t a c i t Assume uppA(t[y]), then ~[~l(u), and use of 3.9.3 ( i v ) ) .
Then
I{~}(y)}(u)~m~Ax
Induction.
such that
Take
We then easily show
BpA(O)
For prenex
f~}(u,O) ~ j~u, t~}(u,S~) ~ t j 2 u } ( x , t ~ } ( ~ ) ) .
& V x ( A x ~ A ( S x ) ) ~ VyAy,
5.9.5.
Lemma.
Proof.
By induction on the construction of
similarly to 3.2.4.
A , HA ~ x p A--A . A . If
A
is quantifier free,
the result is immediate. Assume
A m VyBy o Then x~A
~ Vy(~IxJ(y) & [ x / ( y ) ~ B N ) VyBy
Assume
(by induction hypothesis).
A ~ ~yBy . Then
x pA ~ j 2 ~ B ( j l x ) B(JlX ) ~yBy
~ Pr(~j-~
(by induction hypothesis)
.
H o2
formulae
3.9.6.
Lemma.
For
Proof.
Let
be of the form
A
~(jl~)')
A
Vy~zP(y,z,u)
(P
prime).
Assume
Vy~z P(y,z,u) o Obviously,
~= m i n P ( y , ~, u) ~ o p P(~,w,~)o Note also
P(y, w, u) ~ Pr(rO p P(y, w, ~)i) o
Let t ~ ~.j(minP(y,~,u), O). Then j21tl(y) =0, Itl(y)~P(y,~,u).
:Itl(y), JIItl(Y) = minzP(Y,Z,U),
270
Refinement. taining
If
A
O
is a
n2
formula,
(only) the free variables ~ A ~
3.9.7.
Corollary.
ing (only)
of
then there is a A ,
p - term
tA
con-
such that
~tA & t A r A . O
For
~2
formulae
the free variables
of
A ,
A , there is a
p - term
tA
contain-
such that
KA ~ A ~-~It A & t A r A
•- ~ 3.9 • 8.
Lemma.
Proof.
Let
besides
~(x ~A) E oI
For
.
formulae
A = ~yPy,
Py
A : HA ~ ~A ~ ~(xppA)
.
prime (and possibly containing other variables
y).
First note that
0~Py
HA ~ O p P y
(I)
~ Py,
hence
(§ 1.5)
-~ P r ( r O p P y ~) , i.e.
~ ~ o~Py- O p p F y
NOW suppose
(2)
(in
HA)
that for some
y, l°y.
Then
OpPy,
so by (I)
op2Py
Hence
(3)
j(o,y) pA.
Also by (2) and 3.9.3 (iii) (4)
Pr(rj(O,~)p A D
Thus, from (3) and (4) 3.9.9.
~x(xp~pA) .
Some definitions.
Let us use
Kreisel-Lacombe-Shoenfield Tpt(y)
Pr(~p~P~ ~) , and hence
= "y
KLS
theorem for
to denote the assertion of the n V ~ objects of type n in ECF(~) .
is total and provably total",
i.e.
w ~v Tyxv & P r ( ~ v x ~w T ( ~ , ~ , v ) ~ ) Def(z)
~ "z
Ext(z)
m "z
is defined on
Tpt-indices",
i.e.
Vy(Tpt(y) -~ ~uTzyu) is extensional
Vyy'vv'(Tpt(y)
on
Tpt-indices",
i.e.
& Tpt(y') & y~y' & Tzyv & Tzyv' -~ U v = U v ' )
where y ~y'
~ Vzz'w'(Tyzu
~od(n,z)
-= " n
& Tyz'v' -~ U v = U v ' )
.
is a modulus of continuity for the functional
respect
to
Tpt-indices",
z
with
i.e.
Vyy'vv'(Tpt(y) & Tpt(y') & {yl(n) = {y'i(n) & Tzyv & Tzy'v' ~Uv=Uv').
271
Cont(z) = "z
is continuous w.r.t.
~n ~ o d ( n , z )
Tpt-indices",
i.e.
.
3o9.10o Lemma. ~.~ ~ ~x(xppy~Wl)~-¢ Tpt(y) ° Proof.
Suppose (in
HA)
y~w~ ~ P r ( ~ ) ,
xppy6~
i.e.
I . Then by 3.9.7 and 3.9.3 (ii),
Tpt(y).
For the converse implication, assume By 3.9.7, for some
Tpt(y) , i.e.
yC W I & pr(r~E~?
t
b y c w t ~ : < y ] ~ t [ y ] ~ (y~wl) ° Hence p r ( r y E w I --' ' t [ ~ ] & t [ y ] p (~EWl)") , and thus
;t[y] ~ t [ y ] ~ ( y ~ v q ) Therefore
~ ( = P2 y c Wl)).
3.9.11. Lemma.
but
Assume
mCont(~) .
Proof.
~ Pr(~t[y] ~ t [ y ] ~ ( y ~ w ~ ) ~)
Then
~
is such that
KLS I
is not provable in
~
Our first aim is to show that there is an
° ~
such that
We may write u E W 2 =- Pu & Qu , where Pu
-= V y ( y E w I ~
~v~uyv)
,
Qu --- Vyy'vv'[Q1(y,y' ) & Tuyv & Tuy'v' -, U v = U v ' ] where Q1(y,y' ) ---y E W 1 & y' 6 W & y ~ y ' Let
~I = AYAv'j(minvT~yv' O) . Then
by lemma 3.9.10, since
Def(~) .
Also, if
~2 =- AFJiv'Aw'Aw'Ax'O'
(3)
b~2 P Q~
then
(by 3.9.10, and 3.9.5 applied to Put
Next we show that Assume
HA ~ K L S I , b~x(xp
(4)
y~y'
, and
Zxt(~) ).
~ -= j(n1,~2) , then (I) follows from (2) and (3). KLS1 then
is not provable in
H~
as follows.
HA ~ ~ x ( x p K L S 1 ) , and then by 3.9.3
V z 6 W 2 Cont(z)),
~- ~ x ( x ~ Cont(~)) ,
and so by (I) which implies for certain
n
).
272
(5)
~(x p r~d(.,~)).
We will show that (5) implies in fact
Mod(n, ~) .
So suppose, for certain
y, yl, v, v v
(6)
Tpt(y) & T~t(y') & {y}(n) = {y' }(n) & T~yv & T~y'v' .
By le~nna 3o9.10, applied to the first two conjuuots of (6)
~ ( x p~ (y ~ w11) ~ ~ ( x p~ (y, ~ w I I). Also, by lemma 3.9.8. to the last three conjuncts of (6) (since
: {y'}(n)
{y}(n) =
Z o1- formula)
is e~pressible by a
Zx(xp2 {y}(n) = {y' }(n)]& ~{x(xp~p T~yv) & Zx(xpp T~y'v') . Applying all this %0 (5), we find true, i.e.
{~}(y) = {~}(y') . Hence
p - realizable, hence
Cont(~) , giving us
We may construct a numeral
{~}(y)
to be
a
contradiction°
H-A t-/ZKLSlo
3.9.12. Theorem. Pro.of.
Uv = Uv'
such that
0
if
0
if
Zj(Proof(j,r~EWl~ ) & ~ i j ( { y } ( k ) = 0 ) ) Rn(Q(y,n) & m q ' ( y , n ) ) ;
if
~(Q(y,n)
where
~
Q,(y,~)) ,
Q(y,n) ~ Vj < n[mProof(j, r y E ~ )
& {y}(n)%0 & {y}(j):0] Q'(y,n) ~ Vj < n
;
&
, and where
V w < n ( P r o o f ( j , r ~ 6 ~ 1 " ) -- Sk< {yJ(n)Twnk) ;
undefined otherwise. We now proceed to show Assume
Tpt(y) . Then
chosen minimal.
H~ b Def(~) o P r o o f ( j , r ~ E w Il )
for some such
j ; let this
j
be
Then
Vk !j({y}(k) = O) v m V k ij({y}(k) =0)o In the first case, Since there is no
{z}(y) = O. j' < n
In the second case, let
second or third alternative in the definition of n
{[I(Y)
applies with the
as indicated. Next we show
Let
= minu(Iyl(u)/O).
Proof(j',ryEW11- ) , i% follows that the
such that
Tpt(y)
HA ~
& Tpt(y')
EX%([) o & y~y'
.
Then
{~l(y)
and
{E}(y')
are
defined.
Assume e.g. I~l(y) = 0 and {~l(y') = I. Let a~ain n = minuIY'l(u)/O. Since y ~ y ' , also n = m i ~ l y ~ ( u ) / O . We have { ~ } ( y ' ) = 1 , therefore Q'(y',n),
hence also
Q'(y,n).
But then, since
{~l(y)=O,
it cannot
be computed according to the second or third clause in the definition of I~l(y) , so it must be computed according to the first clause, hence there
275
is a
j
such that Proof(jJycW1~
Then
j < n ; also
y < j,
y< n
on the usual
we have
) & ~!J({Y}(k) (since
we have obtained Finally
it occurs
definitions
~ k < {y}(n)(Tynk)
.
Take a canonical
index
Therefore,
for a standard Therefore
to show that
m ° = I + sup{k I XW<no
in a proof with godelnumber
for "Proof").
Since,
a contradiction.
we proceed
=O) .
ZJ<no(Pr°°f(J/wE
y >n
Uk > k ,
= Izl(Y') . n o , define
For any given
°
Q'(y,n) ,
godelnumbering,
Izl(y)
~Cont(~)
since
j ,
W~) & TWnok)} "
such that O
{yl(x) Then
= 0
if
Tpt(y) . We note
to compute j >y>n
{~}(y) ,
X/no,
that
since
O , and (b) if
tion of
the definition
of
~!j({y}(k)
k< m o ,
{~}(y)
On the other hand,
= moo
(a) the first
clause
= 0 ) ~ j < n o,
such that
= O .
Hence
but
W~)
Twn o k,
must have been used,
{~}(Ax°O)
cannot have been applied
Proof(j/~6
j < no, W < n o ,
there is a
m°
Iyl(no)
,
Proof(j,ry6W1~
so the third clause
and therefore n
{~}(y)
in
= 1.
cannot be a modulus
of
o
continuity
for
~
at
Now the assertion 3.9.11.
Ax.O
w.r.to
total and provably
total indices.
of the theorem follows with 3.9.11.
Definitions.
Let
KLS
denote
KLS
with
E(V)
~ Wn;
and let
n
KLSR n
denote
KLSH n
~ ~
~m
the rule corresponding E
Wn+1
V~
to
KLS n :
=
n Vvc~l(Iul(lml(u)) = {vl(Iml(u))~ {~l(u) = {~l(v)).
Let
x_~y -=def Vz(:I~l(z)~:Iyl(z) ~ Ix}(z) = {yl(z)) x~y
-def x C y
Ext'(z)
& y~x
=def
z
is extensional
on partial
indices",
Consis(z)
-=defVUvw[Tzuw & uc__v-~ Sy(Tzvy & U y = U w ) ]
Pras(x,z)
= "x -def
is a partial
reoursive
associate
i.e.
for
z",
i e.
Vnm(fn_~ f m & "{xl(n) -- {~l(n) = l~l(m))
Vy('Izl(y) where
f
is a uniformly
constructed
~(fn<_y ~ {~l(n)= {~l(y))) godelnumber
for the finite
n
function
whose graph is coded by the number
Now the M y h i l l - Shepherdson MS
theorem
(Ext'(z) -~ Consis(z))
n °
can be stated as :
~ ((Consis(z)-~
)
then by the defini-
~xPras(x,z))
@
partial
274
3.9.12. Some ether results from Beeson 1972o The following results, among others, are established in Beeson 1972 :
(i)
m + ~ R WMs
(ii)
MS, KLS
are
H R O - m r - realizable, hence
(iii)
H~ ~ N S
(by
p - realizability)
(iv)
}~
is closed under
Similar results hold if <
~
~+MS+KLS
fl ~ R
KLSRI, KLSR 2 . is replaced by
~
+ axioms of the form
TI(<) ,
a primitive recursive well ordering.
3.9o13.
Postscript added in proof.
HA ~ M S ,
HA ~ K L S I
ability.
In Beeson B, a simplified treatment of
is given, omitting "realizing numbers" from
The resulting "realizability"
r I C - relation (§ 3.1).
p - realiz-
can be seen as a variant of Kleene's
Chapter IV NORmaLIZATION
THEOREMS FOR SYSTEMS OF NATURAL DEDUCTION
§ I. The strong normalization = = = = = = = = = = = = = = = = = = = = = = = = = = 4.1.1.
theorem for
HA
= = = = = = = = = =
In this section we shall discuss normalization
theerems for the natural deduction There are different
treatments
Prawitz A and (implicitly,
system for
available
HA,
and normal form
as described in 1.3.6.
in the literature : Jervell
as a special case) Martin-Lof
1971 contains a method for predicate
1971;
1971,
Prawitz
logic which is easily adapted to
arithmetic. Jervell
1971 has been inspired by the method of Sanchis
1967 for establish-
ing normal form theorems for the terms of theories of objects of finite type (such as
~-~;
cf. our remarks in 2.2.35).
Technically Martin-L~f
the methods of Prawitz
for terms in
~_~w
tion of validity and strong validity to the definition of computability have originated
(Prawitz
defini-
1971, Appendix A) is similar
used by Martin-Lof
1971, but seems to
of the logical
constants
(cf. Prawitz
1971, II 2.2.2,
1935, § 5)-
Cut-elimination
theorems for calculi of sequents for first-order
metic originate with Gentzen,
especially Gentzen
The methods of Gentzen
to systems of intuitionistic tuitionistic
of com-
(cf. 2.2.5) ; Prawitz's
in the attempt to clarify Gentzen's ideas about an operation-
al interpretation A 1.3.1, Gentzen
1971,
1971 are similar,
1971 has been inspired directly by Tait's introduction
putability predicates
metic).
1971 and Martin-Lof
arith-
1938 (for classical
arith-
1938 have been further explored and extended
arithmetic
in Scarpellini
analysis and other extensions of
H~
1969, A and to in-
in Scarpellini
A, 1970 ,
1922.
For some further general
comments
Here we restrict ourselves
see also Prawitz
closely follow the presentation
in Prawitz
and extended to the case of first-order
Prawitz also deals with permutative theorem.
arithmetic.
reductions,
complication
(This adaptation 1971 in two respects :
and proves a strong
It is the first addition
absolutely necessary for some of our applications) additional
We rather
1971, Appendix A, but adapted
yields a stronger result than stated in Martin-Lof
normalization
1971, III.
to systems of natural deduction.
(convenient,
although not
which is the cause of
in the treatment as compared to Martin-L~f
1971.
276
For sections
fl- 3 the p r e r e q u i s i t e s
sections 4 - 5, also
§ 1.4,
Special n o t a t i o n a l by a lower case 4.1.2.
Zo, ZI, indicate
convention
conventions
for arbitrary
that
for this chapter :
successor is indicated
about p r o o f - t r e e s
H, n,, N", NI' "'"
...
§ 1.1 - 1.5 ; for
s.
Notational
We use
are to be found in
1.5 are needed.
N
or
Z
for a r b i t r a r y deductions,
finite
sequences
and
of deductions.
Z, Z t, Z", ..., If we wish to
possess a set of open a s s u m p t i o n s
of the form
A ,
we w r i t e [A] H
[A]
[A] Z
~
does not n e c e s s a r i l y
refer to all a s s u m p t i o n s
w i s h to refer to different indexing
(e.g.
[A]~,
sets of a s s u m p t i o n s
[A]~)
to indicate
of the form
A ; if we
of the same form, we may use
the d i f f e r e n c e
and use n o t a t i o n s
such as
[A]~ [A]8 etc.
~ [A]~ [A]~ Z
In case of a single occurrence
we may omit the square brackets around
A. If we wish to indicate that a d e d u c t i o n N has a c o n c l u s i o n A , we n Z A" We write ~ to indicate a d e r i v a t i o n obtained by an a p p l i c a t i o n
write
of a rule to the c o n c l u s i o n s If [A] H,
N or
or
A
[A] Z
is a d e r i v a t i o n
for the d e r i v a t i o n
for every open a s s u m p t i o n Z
Z
as premisses,
with conclusion
with c o n c l u s i o n
A.
A , we write
obtained by inserting
of the set indicated by
N
[A].
in
[A] 9'
or
[A] Z
Similarly for
Z
[i],
[~].
H
Z~ This concept
understood If
of
B
indicates
of substitution
of d e d u c t i o n s
for open a s s u m p t i o n s
must be
as follows. is an open a s s u m p t i o n a set of
n
(occurrence)
occurrences
in
A ' and
of open a s s u m p t i o n s
[A]
in
N,
in
[~]
then all
n
f?*
occurrences
of
B
in
u n d e r the s u b s t i t u t i o n
~']
d e r i v i n g from the same occurrence
are in the same a s s u m p t i o n - c l a s s
of
B
in
N'
(i.e. must be dis-
charged or remain open simultaneously). More precisely, coded by finite
assume
the n o d e s
(positions)
sequences of natural numbers,
in a d e d u c t i o n
such that
0
tree to be
corresponds
to
277
the conclusion,
and if
they are l a b e l l e d classes
n
may be s p e c i f i e d
an a s s u m p t i o n
class
In * m
I m B
In * m
I n E ~, m E B I ,
~n = In * m
in
Similarly
for
Z(a)
t
for
4.1.3.
there
B e l o w we
finite
shall
assume
indicated
below would
(A) R e m o ~ a ~
A redundant
simplicity a
noting
introduce
A reduction ~'
are of four contractions,
notation,
of
~
types:
Z(t)
has
classes
sequence ~(a), refer
obtained
of
Z(a)
has
to the
by s u b s t i t u t i n g
[a/t]H,
to satisfy
[a/t]Z.
the general
if a r e d u c t i o n we shall
removal
conditions
process
assume
the c o n d i t i o n
as de-
proper
parameters
on parameters.
of r e d u n d a n t
a
A parameter
is an i m p r o p e r
shall
always
from
parameters,
~
take
O
steps
- contraction
to
~
of
E.
which
a constant term,
by
does
term for
so the r e d u n d a n t
[a/O]~.
described
consists H"
~").
permutative
Let us call a
E
deduction
contracts
contractions,
contractions.
H
of
H
sub
(B) - (F) b e l o w
parameters.
by a "simpler"
~-
parameter
for this
by r e p l a c i n g
to a d e d u c t i o n
of a
in a d e d u c t i o n
by s u b s t i t u t i n g
the r e d u c t i o n
(~'
a
nor in the c o n c l u s i o n
can be removed
proper
B
steps.
if
H"
H(t),
we can use
condition
new redundant
contr.
assumption
if
processes
p ~ m ~ .
step applied
H' ;
if the n o t a t i o n
so as to satisfy
that
in
or finite
of derivations,
assumptions
is r e m o v e d
It is w o r t h
H'
we
a ;
this
of ~ d u ~ a ~
parameter
discharged
of an argument,
1.3.6 ;
of r e d u c t i o n
in the open
application
Let in [A] , [A] stand for H' H in [A] then consist of
a derivation
are of two k i n d s :
is said to be redundant,
deduction
it,
Assumption
.
1.1.7,
violate
processes
and a p p l i c a t i o n
Z [~] Z'
all d e d u c t i o n s in
The r e d u c t i o n
we w r i t e
above
n E ~.
parameter
of the r e d u c t i o n
automatically
do not
immediately
..., n * < m - I>.
a set of distinct
to denote
sequence
to be renamed
for
the nodes
Z [~],
As an a l t e r n a t i v e
on p a r a m e t e r s
it ;
is
one for each
in the course
resp.
parameter
nodes
has not been
the n o t a t i o n s
Description
not occur
~;
~
w i t h an i m p r o p e r
a.
scribed
if
may be used
been introduced derivation,
m
n * O,
as sets of top nodes.
H' ,
I m E ~I ,
derivations
as
a node of n', n E ~I U Im I m a node of HI . The a s s u m p t i o n H' A gives rise to a c o r r e s p o n d i n g a s s u m p t i o n class
been d i s c h a r g e d
H(a),
to right
of top nodes
class
in
is a node w i t h
f r o m left
with
a sub-
the same conclusion;
The possible contractions,
a reduction
- reduction.
in r e p l a c i n g
contractions induction
step o b t a i n e d
by
278
(B) P~op~r ~ e d u c t ~ s . with c o n c l u s i o n
If in a deduction
A , is i m m e d i a t e l y
rule with major premiss both a p p l i c a t i o n s
A ,
of rules.
to the f o l l o w i n g possible ~)
HI
Hr
A1
Ar
stands for "right",
hi A. i
A I
~)
is a
&r-
or a
[A]
92 contr.
B A ~
3)
~I - contraction.
- contraction
HI
B
[A]
~2
~1
A
B
V - contraction n(a) Aa
contr.
~(t) At
Vx Ax At 4)
V.- contraction
(i = r,l)
I
A
5)
H
[All
[Ar]
Ai
HI
Hr
AI V Ar
C
C
V- contraction
is a
V
r
-
1
contr.
or a
[Ai ] g. 1
V I - contraction°
~- contraction n
[Aa]
At
H'(a)
~x Ax
C C
E -
according
contractions :
contr.
&- contraction
I - rule,
the deduction may be simplified by c a n c e l l i n g
AI & Ar
A
of an
This gives rise to proper reductions,
(i = r,l ; r
&i - c o n t r a c t i o n
an a p p l i c a t i o n
followed by an a p p l i c a t i o n of an
[At] contr.
H'(t) C
I
for "left").
279
(C) Permutative reductions.
Permutative reductions make further
applications of proper contractions possible by changing the order of application of certain rules. 6)
V E -contraction [A I ]
[A 2 ]
H
HI
H2
[A I ]
A I VA 2
C
C
H1 C
C
[A 2 ] Z
~2 C
Z
contr.
C
is major premiss of an elimination rule).
A 1 VA 2
A
A
A (the lowest occurrence of 7)
Z E -contraction [Aa] ~x Ax
[Aa] H'
C C
Z
(the lowest occurrenne of
contr.
C
C
ZxAx
A
acts as major premiss in an elimination
rule). (D) Induction reductions. term in applications of
8)
IND.
Induction reductions simplify the induction The corresponding constructions are
[Aa] n
n(a)
o
A(sa)
A~o
contr,
A0
9)
n
o A0
[Aa]
no AO
[Aa]
n(a) A(sa)
contr.
no
n(a)
A0
A(sa)
[At]
A(st)
i(st) (E)/k- reductions.
The effect of
)k- reductions is to lower the logical
complexity of the conclusions in applications of )~I" contractions are
The corresponding
280
10)
#k&-
contraction
2<
contr.
A 1 &A 2
11)
2k
~.
AI
A2
A I & A2
#kV- contraction Z
,Q
Z
2~
AI
contr.
A I VA 2
A I VA 2
12)
~-
contraction n
Jk
H contr.
Jk
A2 A I ~A
2
A I -~ A 2
2kV- contraction H n
contr.
,,k
AO
Vx Ax 14)
Vx Ax
/k~_ contraction H
2k ~k
contr.
-AO
~x Ax (F) ! ~ ! i ~ removal
~5)
~x A x ~i~![fi~![[~"
of r e d u n d a n t
applications
Immediate of
simplifications
VE , ZE .
consist
The c o n t r a c t i o n s
in the are
YEs -contraction H
AI V A2
H1
H2
C
C
contr,
i C
C if in the d e r i v a t i o n
on the left
side no a s s u m p t i o n s
are closed
in
281
H. i
16)
by the a p p l i c a t i o n
of
VE
shown.
Z Es - c o n t r a c t i o n n' ~xAx
C
contr.
C if in the d e r i v a t i o n by the a p p l i c a t i o n 4.1.4.
C
on the left hand no a s s u m p t i o n s
of
~E
are closed in
H'
shown.
Definitions
A thread in a proof tree is a sequence of formula o c c u r r e n c e s such that
AI
each
i ( n,
i ,
is a top formula, Ai+ I
A segment in a d e d u c t i o n (i.e.
Ai+ I
An
the end formula,
is i m m e d i a t e l y
immediately
H
below
and such that for
Ai .
is a sequence
below
At, ..., A n ,
At, ..., A n
A i ) formula occurrences
of c o n s e c u t i v e in a thread of
H
such that I)
AI
2)
A i , for each
is not the c o n c l u s i o n
3)
An
i < n,
of an a p p l i c a t i o n
is not a m i n o r premiss of an a p p l i c a t i o n
F r o m the d e f i n i t i o n of the same formula.
it is obvious
individual
formula o c c u r r e n c e s a ~ A 1, ..., A
of a rule, if
An
some t e r m i n o l o g y
for
~ ;
premiss of an a p p l i c a t i o n
premiss of
~
of a rule, if
¢. if it is conclusion
of an i n t r o d u c t i o n
and m a j o r premiss of an e l i m i n a t i o n ; a maximal formula
(occurrence)
is
similarly.
A deduction ~)
VE , ~E .
that a segment consists of o c c u r r e n c e s
of
A segment is said to be maximal,
defined
of
VE , ~E .
to s e g m e n t s :
is said to be (minor, major) is (minor, major)
;
is said to be c o n c l u s i o n of an a p p l i c a t i o n
n is c o n c l u s i o n
AI
V E , ~E
of
This permits us to transfer
a segment
a
of
is a m i n o r premiss of an a p p l i c a t i o n
H
is said to be in normal form (w.r.t.
if no c o n t r a c t i o n
strictl [ normal
form,
of
~
is applicable
if it is in normal
to
~ °
H
a set of c o n t r a c t i o n s is said to be in
form and does not contain redundant
parameters. W e write
~ ~I H'
single c o n t r a c t i o n >
if
H'
is obtained from
to a s u b d e r i v a t i o n
is the transitive
of
r e l a t i o n generated
H ; by
H
by a p p l i c a t i o n
of a
H' I H, .
>I ;
H ~ H' ~def H > H,
or
H = n' , H ~ H, ~def H, ~ H o Let us call if for all
HI, H2, H 3, ...
i ,
said to terminate
Hi+1
a reduction
sequence
If the r e d u c t i o n
(startin 6 f r o m
sequence
is finite,
(in the last d e d u c t i o n of the sequence)
of the sequence is normal.
HI ) it is
if the last term
282
Similarly a pair finite
to 2.2.17, we define a reduction
, where sequences
(a)
~
(b)
if
n 6 T,
H"
such that
=
is a n o n - e m p t y
such that
assigns d e d u c t i o n s >
T
n * ~ 6 T
to the elements
=
of
T
tree of a d e d u c t i o n
set of natural numbers n E T , and
~
H
as
representing
a function w h i c h
such that
n
mn = H, , and
H~,
..., H'
is a complete
listing of the
n
H'
(without repetitions,
I being imposed),
of deductions ~(n*)
H"
then
some standard o r d e r i n g
n * 6 T
for
I < i < n , and
= ~!. 1
The length of a reduction 4.1.5.
Remarks on reductions,
A normal F
tree
normal
is the n u m b e r of elements
A normalization is a r e d u c t i o n
normal
deduction
of
If A
A from
is derivable
starting from
A strong n o r m a l i z a t i o n
H
from
F.
t h e o r e m is of the form : For every d e d u c t i o n
sequence
T.
form and normalization.
form theorem is a theorem of the type :
then there is a (strictly)
in
H ,
there
w h i c h terminates.
theorem is of the form : All reduction
sequences
are finite. Let
~p' ~c' ~ '
Ps
denote
the sets of contractions
according
to
I - 5 + 8 + 9, I - 9, 1 0 - 14, 1 5 - 16 r e s p e c t i v e l y ; we a b b r e v i a t e unions by etc.
s
R~ma~k I. maximal contains
A d e d u c t i o n which is normal w.r.t.
formulas
(segments).
atomic a p p l i c a t i o n s
~p(~c)
does not contain
A d e d u c t i o n w h i c h is normal w.r.t. (i.e. a p p l i c a t i o n s
~A
with atomic conclusion)
only. Remark II. There e x i s ~ a primitive
recursive
arbitrary d e d u c t i o n into a normal d e d u c t i o n conclusion,
by the i n t r o d u c t i o n
remark of Jervell
of redundant
1971, page 106).
procedure (w.r.t.
for t r a n s f o r m i n g
Pc~ )
parameters
(elaborating
a
For let in n A
A
be a maximal H A
formula.
We then transform
0=O A &
sa
(0 = O)
A &
=
(ca =
this d e r i v a t i o n
sa
ca) IND .
A & (b = b) A E'
an
with the same
as follows :
283
The two occurrences
of
A
are not maximal
anymore.
We may deal similarly with maximal
segments : if
last formula of a maximal
then the same
segment,
A
H A
in
occurs as the transformation
makes the segment non-maximal.
Normalizing with respect to
a primitive recursive process:
one needs
total number of occurrences applications
of logical
if
n
is the
of ~ I -
in the deduction considered.
Y
is required,
R~mark III.
Remark IV.
~
' when only normal form, not
a derivation w.r.t.
when contractions
Contractions
VE , SE .
~c~
becomes trivial.
If we first normalize
remains normal w°r.t.
of
is itself
~ - contractions
symbols in conclusions
Hence the normal form theorem w.r.t° strictly
n
~
from
For example,
~p
from
~
~cs
,
then it
are applied.
may introduce new redundant applications
consider
(~)
c&c
B
B~C
C
C
CVC
CVC
(3)
-(1) (2)
C
C&C ~ CVC
c v c
C&C ~ CVC
&I
(c&c~ cvc) & (c&c~ cvc) &E
C V (B~C)
C&C~
CVC
C&C~
CVC
-(2),-(3) C& C ~ C V C Contracting
the
&-introduction
tion with a redundant ~s
and
& - elimination marked yields a deduc-
V E -application.
On the other hand,
do not introduce maximal formulas or segments in
(but they may do so in
Remark V.
Certain contraction
defended for proper contractions, permutative
reductions.
would be sufficient
to the formal deduction.
For a discussion
we might try to establish
i
(O ~ i < n)
For example,
~c
The
see Kreisel
197 I,
1971, 3.5°6.
Hi+l
is then obviously false, however,
plificationso
the
This can be
and perhaps also for
the conjecture : H, H'
present the same proof idea, if there is a sequence
conjecture
~s ) "
to ensure identity of the underlying proof idea.
1 (c), pp. 114- 117, Prawitz
such that for all
from
Hence equality of normal forms with respect to
converse is a more dubious hypothesis.
~ore generally,
segment may
of contractions
steps may be conceived as preserving
intuitive proof idea corresponding
of
~ c - normal derivations
~ p - normal derivations : a maximal
reduce to a maximal formula by applications
contractions
the following
or
re-
H ~ Ho,HI, ..., H n ~ ~' , Hi <1 Hi+l"
This
if we permit immediate
two derivations
represent
slmintuitive-
284
ly distinct proofs,
and are both in normal A
(from Prawitz
1971, 5.5.6)
(I)
A -* A
(1
A-~A
-(I)
II =
B -~ (A'*A)
B -~ (A'*A)
(A'*A) "* (B'* (A'*A))
(A'*A) "* (B-- (A'*A)
Now consider the following derivation
H" :
A--A
A'*A
CVD
-(1)}=~'"
B-* (A-*A)
B-* (A'*A)
(A-*A) -* (B-* (A-*A))
(A-*A) -~ (B-* (A-*A)
(A-*A) -* (B-* (A-*A)) Now we may apply in two different ways a ing either
H
or
as intuitively H"
H' .
Y E s -contraction,
thereby obtain-
Our conjecture would then lead to regarding
the same proof,
as is illustrated
may he said to combine two different
itself a proof different from each of these). tions make normal forms non-unique,
by the sequence
proof ideas
H, H", H,
(and hence represents
Since immediate
simplifica-
the most plausible alternative
be, in the presence of a strong normalization
theorem,
H, H'
seems to
to reformulate
the
conjecture as follows: Con~ect~e
: Two deductions
H, H,
if the sets of normal deductions Remark VI.
represent
the same intuitive proof idea,
to which they reduce are identical.
If we would have failed to distinguish
and correspondingly non-uniqueness
between
&r-
and
between
&l - contractions,
of normal forms would have resulted,
&r E
and
&l E ,
another source of
as will become clear
from the following example : A
A-*B
C
B
C-*B B
B&B B
could then be reduced by a & - c o n t r a c t i o n s 4.1.6.
Isomorphism and homomorphism
References : Prawitz Martin-Lof
to either
A
A-*B B
or
C
C-~B B
between terms and deduction s
1977, IV, especially
IV 2.5 ; Howard A ; Girard 1972,
19ZI.
We consider a very special case to illustrate
the idea, namely pure in-
285
tuitionistic for
implication
simplicity
bles"
HA,
~,
this
H'~, ...
for each
as the only rules)
H
type also by
(in the
consisting
Assumptions
in the
We define
type
A.
A ,
a "deduction
inductively
to each f o r m u l a
We have
"deduction
and a b s t r a c t i o n
~(H)
A
varia-
with
~A'
~I,
~E
as follows.
only we assign
have a s s i g n e d
A ;
operators
deduction
term"
of an a s s u m p t i o n
same class will
further
a type
system for natural
we a s s o c i a t e
To a d e r i v a t i o n
to them.
Let us a s s i g n
we denote
To any d e d u c t i o n
HA .
logic.
a variable
the same v a r i a b l e
:
f[A]] ~nA-'[: ,
IAI N o w we
- contraction
see that
corresponds
I - conversion :
to
[A] n B is m a p p e d
A~B
A
onto
(XE A . ¥([))(y(H'))
B H
[A]
and
onto
[rIA/ Y(H')]Y(H)
.
H B This makes used
it u n d e r s t a n d a b l e
to prove
deductions. relations duction
reduction In fact,
between
relation
and
as
can be obtained.
VI,
-HAW; "IND"
VE,
SI,
as ~E
and
techniques
defined
This
of
theory
isomorphism,
VI
which
N - HA w ,
and
have also
of terms with
w i t h respect
idea is exploited
substitution
vE
of individual
been apply
to
reduction
to the re-
in Girard
1972o
may be interterms
respectively,
l - conversion. we have no direct
in this
sense we have
behaves
similarly
if we a s s i g n
same
a type for individuals,
X- abstraction
V- contractions To
a complete
the
form for terms
for a suitably
them,
If we introduce preted
that
to normal
to extend
to a special
to a d e r i v a t i o n
[Aa] H, Asa
H A0 At
analogue
in the
the results instance
~- version
of
f r o m § 2.2.
of the r e c u r s i o n
operator :
28~
the deduction term
RIND(E)(V~I')t , then the
IND-reductions
correspond
to RIND(~H)(YH' ) 0
cony.
YH
RiND(~)(,n')(et) cony. (~,)(Hi~D(~)(~n')t). It should be noted that distinct assumption classes correspond distinct deduction variables(an "unspecified"
deduction of the assumption).
variable
assigned to all assumptions
HA
terms of a
Having the same deduction
of the form
isomorphism)
assumption
classes corresponds
in Prawitz
1965 (I, § 4, pages 29- 31).
Historical note.
classes.
calculus only).
for an
The method of
The earliest publication where the analogy between types
seems to be C u r r 2 - F e y s
on the other hand, was
1958, 9E (for implicational
Their remarks concern not the equivalence
calculus and natural
deduction,
but between
4. I. 7. Strong normalization
implicational
simplifications
ization theorem relative
yields,
logic.
theorem relative
afterwards,
~c%
we also obtain a normal-
~cAs"
We might also have proved the strong normalization first normalizing w.r.t.
k-
for
Below we shall establish a strong normalization Carrying out immediate
between
the pure theory of combinators
and a Hilbert type system fc~ intuitionistic
~c
in
available.
to the second method for defining deductions
and terms on one hand, and formulas and deductions explicitly noted,
A
correlation with a theory of terms ( a f o r t i o r i
we must distinguish
assumption
relative
A , corresponds
k - calculus of having only one variable of type
Hence, for a natural
to
open assumption may be viewed as an
~c
only ;
~/~, then using the strong normalization
theorem
by 4.1.5 Remark I I I a
relative
normalization
theorem relative
~c~" For applications,
we only need a normalization
theorem relative
~c%;
so if the reader wishes, he may delete everything in the proof below referring to
/k- contractions.
4.1.8. Notational
conventions
a) "Rule" denotes a function which assigns to a deduction of
H .
E - rule, b)
So ~I
Con (H)
implications c)
Rapp (9)
Rapp (9)
Rule (9) or
H
the final rule
takes as value one of the basic rules, an
I- or
IND.
is the conclusion of C ~D,
n
Premiss ( C ~ D )
(the formula derived by
is the final application
has a proper parameter
a,
~) .
For
= C . (instance) we put
of
Rule (~)
Param (H) = a ;
in
H.
Param(H)
If may
287
be fixed arbitrarily elsewhere. d)
If
Rule(H)
E {~I, VE,
~E, INDI , Ass(H)
class of assumptions discharged by e)
If
Rule (H) = ~ I
ends with
At,
~Ax
then
Term (H)
~ or if
denotes the (index of the)
Rapp (n) . denotes the term
Rule (9) = IND,
then
t
such that
Term (n)
9
is the in-
duction term. f)
"Subst" maps a triple
t , and a deduction proof
[a/t]H
9
(a, t, 9) , consisting of a parameter
containing
"Sub" maps a quadruple
duction
9",
The value of h)
Let
t
A
Subst (a, t, 9)
(A, 9, U', ~) ,
may be fixed arbitrarily.
where
PRD(H)
Con (~) ~o~ " H
in
onto a de-
H, r e v e r y open
belonging to the assumption class (indexed by)
Sub (A, 9, n,, ~)
Rapp (9) .
are the deductions of the first, second . . . .
of
Rapp (H)
respectively.
If
Rule (~)
is an
E - rule,
~.
may be fixed arbitrarily on other arguments.
denote the set of deductions of the premisses of
Prd1(H), Prdl(H), ...
a term
for a throughout the proof ;
which is obtained by substituting
assumption of the form
a,
as an improper parameter, on the
obtained by substituting
for other triples the value of g)
a
Prd1(9 )
premiss
will be assumed to refer to the de-
duction of the major premiss. i)
Norm (H) :
9
is strictly normal.
If we assume the discussion to be formalized, we may introduce certain code numbers for the various rules, and identifying the other syntactical objects with their godel numbers, on a standard godel numbering, all functions and predicates described in (a) - (i)
then become primitive re-
Cursive. 4.1.9. Definition of strong validity. Let us write
SV
as an abbreviation for the predicate of strong validity
to be defined below°
The definition of
SV(~)
("9
primarily given by induction on the complexity of 9
with
Con (9)
is strongly valid") is Con (H) ;
of fixed complexity, the definition of
for derivations
$V(H)
takes the
form of a (non-iterated) generalized inductive definition. A deduction
9
is said to be strongly valid (sv(H)
following clauses applies: (i) Rule(H) 6 { & l , Vl,
(ii)
R u l e (n) = - - I ,
~I} ,
(iii)
R u l e (9) = V I , P a r a m
(iv)
Rule (9)
is not an
~9'(H'
(b)
If
~,
P r e m i s s (Con ( n ) ) = A ,
~H'(SV(n') and Oon (9') = A
(a)
and
=
(if) = a , I-rule,
= SV(H'))
holds) if one of the
~ PRD(H)[sv(n')].
then
SV(Sub (A, U', H, Ass (9))) .
Vt ( S V ( S u b s t ( a , t ,
Prdl(ff))))
o
and or
91
is normal.
Rule (9) = V E , then the reduction tree of
Prd1(9 )
is
288
SV(Prd2(H)) , SV(Prd3(H)) ,
finite,
Con Prdl(H ) = A 1 V A 2 , the sub-derivation
and if
then for each
9' ~ Prdl(~ ) : if
above an endsegment
of
Rule (H) = SE ,
finite each
, H"
SV(Prd2(H)) , and if
above an endsegment
SV(Sub(At,
H", Subst
clause
(Param(9),
(iv)(c)
Ass (9))).
tree of
Prd1(H)
Con Prd1(H ) = ~xAx ,
which is a sub-derivation
immediately
More pictorially,
then the reduction
requires:
with
of a
is
9', with
Con (H") : Ai , ~he. SV(S.~(Ai, n", P r d i + l ( ~ ) , (c) If
H"
is
then for
Con (~") = At ,
H' ~ Prd1~ ,
t, Prd2(H)))) . If
9
is of the form
[Aa] n1
H2(a)
~xAx
C C
[Aa] then HI
H2
strongly valid,
reduces to a
H'
9"
containing
H',
an endsegment of
the reduction
tree of
~I
is finite,
as a sub-derivation
At
and if
immediately
above
then
[At] H2(t) C should be strongly valid. Similarly for clause
(iv)(b).
A variant of the definition is obtained by reading clause (iv)(a) as follows:
(a)' Each reduction
sequence from
H
either terminates
or passes
through a deduction which is strongly valid by clause (i), (ii) or (iii). 4.1.10.
Lemma.
Assume
Rule (n)
H' A
and let 5o
to be an
~, or
H, HI, H2, ...
H
is of the form
'
be a reduction for all
H
H! i
or
so
H" A
Rule (Hi) = Rule (H) i A
I - rule,
i ,
sequence starting from Hi
H ;
and
being of the form
H'~ i
A
the requirement that the reduction additional requirement as compared itate formalization in § ~.
tree of Prd1(H) is finite is an to Prawitz 1971, and serves to facil-
2s9
respectively.
Then
H,, N~, H~, ...
and
H", 91" ~2" "'"
are reduction
sequences after omission of repititions. Proof.
Trivial, by inspection of the various cases.
4.1.11. Lemma. and let
~1 (a) h n 2 ( a ) ,
Let
[A(a)]
a
not a p r o p e r
parameter in
indicate a set of open hypotheses of the form
~l(a) ,
A(a) in
El(a) , then H
[A(t)] > ~1(t ) H A(t)"
for each derivation Proof.
Trivial, by inspection of cases.
4.1.12. Lemma. I f Proof.
[A(t)] n2(t )
9 1 h H2 '
SV(nl) '
We show by induction over
SV
SV(~I) = ~]2 (HI h U2 = SV(H2)) " (a) If Rule (91) = & I , then HI, H 2
A respectively,
and
9~ ~ H~, SV(~),
(b)
Rule (91) = --I.
(c)
Rule (HI) = V I •
or
(91) = Vl, (91)
SV(HI)
fulfilled.
are of the form (lemma 4.1.10)
H~ ~_ 9~, SV(n~)
SV(H~), SV(n~)
hence
imply by the in-
SV(~2) .
Similarly, using 4.1.10 and clause (ii) for Use lemma 4.1.10 and clause (i) for ZI .
is not an
Similarly to ease I-rule.
holds because
~I
Then there is a
the assertion for
SV(~2). w.r.t. ~1 that
A
duction hypothesis
(d) Rule (e) Rule
then
HI
Either
SV .
(a), (b). ~I
is normal, and then
is not normal and (iv)(a), 9, , SV(n')
SV.
~I = ~2 '
(b), (c) are
n2 ~ ~'
such that
follows from the assertion for
9'
4.1.13. Theorem.
Each strongly valid deduction has a finite reduction tree.
Proof.
R(~) m [H
We prove
the
SV
deductions.
(a)
If
Rule (H)
is an
has a finite reduction tree]
I-rule,
it follows by lemma 4.1.10 that the in-
duction hypotheses for the deductions from (b) ~'
If
Rule (9)
is not an
(~'
implies
by induction ever
PRD(n)
I- rule, then either
is our induction hypothesis,
imply H
R(9).
is normal, or which immediately
R(9) .
4.1.14. Remark.
The preceding theorem is classically equivalent to : each
reduction sequence terminates istically,
(by an appeal to Konigts lemma).
Intuition-
the stronger conclusion follows by an appeal to the fan theorem
29o
from the weaker assertion.
The weaker assertion might have been proved
directly, by applying induction to starting from
H
R'(H) ~ [all reduction sequences
terminate].
The advantage in proving the stronger assertion is in fact that it is directly expressible in
HA,
whereas the weaker assertion must be ex-
pressed in a conservative extension of
~
containing function variables.
4.1.15. Strong validity under substitution. We define:
a deduction
H
is strongly valid under substitution,
if
for each substitution of terms for parameters, and each replacement of open hypotheses in ing deduction is
H
by
SV.
SV
deductions of these hypotheses,
the result-
(It is not assumed that all open hypotheses are
replaced : an open hypothesis is a strongly valid de~uc~io~ Our next aim is to prove each deduction
to be
SV
of itself.)
under substitution.
To do this, we need the following lemma. 4.1.16. Lemma.
A deduction
~
for which
Rule (H)
is not an
I-rule,
is strongly valid if the following conditions are satisfied : (I)
The reduction tree of any
(If)
If
Rule (9) E I&E, ~ E ,
E E PRD(E)
is finite.
VE, ~ i I , then the elements of
PRD(H)
are strongly valid ; (III) If
Rule (9) = IND ,
i.e.
H
is of the form
[Aa]
n,
m'(a)
AO
Asa At
then
AO
is
SV,
and for each
SV IAt I '
~tw
[At] ~"(t)
is strongly valid.
Ast
(iv)
If
Rule (9) E {VY~
~EI,
then conditions(b),
(c) in clause (iv) of
the definition of strong validity are fulfilled. Proof.
To a deduction
value
(~, B, Y, 6, e),
~
satisfying condition where
=
logical complexity of
=
length of reduction tree of O
Con (~) , Prd1(H ) , if
length of
8 =
sum of length of reduction trees of
Prd1(H) , if
Rule (H)
complexity of induction term of 0
is an
Rule (H)
E - rule,
otherwise ;
y =
e =
(I) we assign an induction
otherwise.
is an
E-rule,
0
otherwise ;
PRD(H) ;
Rapp (H)
if
Rule (H) = IND ,
291
We assume the induction values to be orSer~ lexicographically, i.e.
(~o' ~1' e2' ~3' ~4 ) < (~o' e~' m~' ~3' ~4) m
~i < 5 [ Z j < i ( . j = ~ )
% < ~].
~nd
Now the proof proceeds by induction on the induction value of deductions satisfying the conditions of the lemma.
We have for a deduction which
satisfies I - IV, only to verify (iv)(a) in the definition of strong validity in order to ensure that the deduction is strongly valid, since (iv)(b), (iv)(c) hold by IV. If
H
is normal, we are finished.
that each
H,
Case (a).
H'
If
H
is not normal, we must show
such that H, < H is SV. I is obtained from H by replacing a proper subtree of
by a contraction.
So if
H1
"" "
H
H
is
Hn
A
then
H'
is n
n A
where for some
i
(I < i < n) --
induction value of
~'
premiss, and then H,
j /i
H
(~', B', Y', 6,, ¢').
(~, 8, y, 6, e)
~ = ~,
= ~t . 0
be
(m,, ~,, y,, 6,, e,) <
~i ~I H~ , and for
--
8' < ~,
or
Obviously,
since either ~ = ~,
Let the
0
Hi
is the major
~ = 8', ~ = ~'
and
6' < 6.
obviously satisfies condition I of the lemma ; II and III are satisfied
because of lemma 4.1.12. For example, if
H =
Rule (H) = IND ,
H1 i
n2 [Aa] Asa
A0
'
then
Rule (H,) = IND ,
H' =
n~ I
AO
At [Aa] [Aa] H2(a) >1 H~(a) , Asa
H~ [Aa] Asa
,
and if
where
At
then by condition III for
H,
and lemma 4.1.1Z
~5 [At]
Asa Ast
is
SV for each SV H5 At" (IV) is also satisfied for
~'
For assume
[B I] [B~] B I V B2
ZI
Z2
A
A
H
to be
I[Bj] ,
[Bj]
and let
A
A
s92 then H! is strongly valid (condition II for H, lemma 4.1.12), and if 1 E H" B I V B 2 --• ~o' go containing Bj as subderivation immediately above an endsegment of
H
O'
then
H"
[I'~
rill
[Bj] _~ [Bj], _,l
"
A If
H,
-
z
[Bj] Z' _~l A
hence
A
BI ZV' B 2 ~IB I ZVB 2 ,
and
is
H:l reduces to
derivation immediately above an endsegment of for
9'
follows from
Similarly if Case_ (b).
IV ~or
H'
B
~o
by lemma 4.1.12.
with
Ho ' then
N" Bj
as a sub-
~c --< ~ ,
and IV
~.
Rule (H) = ZE . is a proper reduction of
Then the major premiss of an introduction. When
SV,
B
H,
and case (a) does not apply.
of the last inference of
H
is the conclusion
is a conjunction, implication or universal quantification,
we
may apply, by condition II of the lemma, clauses (i)- (iii) in the definition of strong validity, and conclude that
E"
g'" A2 [AI] AI-~A2
= let
I AI
H
,
then
H'
is strongly valid.
9' = I
For example,
~" [At] ~,,, A2
A2
Since
, AI
[At] 9"' A2
are strongly valid by (II), the strong validity of
AI"~A 2 is immediate by clause (ii) in the definition of strong validity. H'
~2
If B is a disjunction or existential formula, the strong validity of is immediate by (IV).
(c).
Rule(n) ~ IND,
and case (a) does not apply.
[Aa] =
n1
n2(a)
A0
Asa
H' =Inl AO
AO
then the strong validity is immediate by (III).
,
Suppose
9,
293
[Aa]
[Aa] Suppose
HI = I~ [At]E2(t)AsaH2(a)
H = I HIAO Ast AsaH2(a) '
A(st)
[Aa] Now
H" =
I gl ~2~) AO
Asa
is strongly valid, since it has a lower induction
At value
(~", ~", y", 6", ¢") :
Then by condition III
H'
~ = ~", ~ = ~", Y = V", 8 = 8",
e
=
e " + "1 .
is strongly valid.
2e~ (d). Rule(n) = ~ ,
A single example
and case (a) does not apply.
illustrates this case.
I ri,, Let
H =
[ £I,,
A1Jk ,&A2
II"
H, = I A ~
~
,
A , Jk satisfy the conditions of the lemma and have a lower induction AI A2 value, hence are SV ; therefore ~' is SV. Case (e).
H,
is obtained by a permutative reduction from
does not apply.
We discuss the case of an
~E-reduction,
N,
and case (a)
the case of
VE-
reductions being similar.
Let
H =
I
Hl(a)
~o
Xx Bx
for
~'
IV
=
(j
~1(a)
o
Q
Bx
l
-----K----
A
A
The lowest occurrence of
Condition ~ 9o
17'
C 0
wish to verify that
tree of
'
for
C
9'
shown is major premiss in an elimination.
~'
implies condition
I
for
is finite because of condition I for
implies that
We
satisfies conditions I - IV of the lemma.
[~I a) C
~' , since the reduction 3,
and condition
IV
is strongly valid, hence by 4.1.3 the reZ
A
duction t~ee of this deduction is finite. Conditions II, III are vacuously fulfilled for sho~m that condition IV is fulfilled.
H' , so it remains to be
For this we have to show that
294
H2 =
i
[Ba]
[Bt]
Nl(a ) C
and
Hl(t )
H~ =
Z
C
A
are strongly valid, under the assumption that
nBt4
such that
is the sub-deduction of
We note that the induction value than
E
A
(~, 6, Y, 6, ¢) , since
(~5' 83' ~5' 63' ¢3 )
of
H5
~xBx~° reduces to a
(~2' ~2' ~2' 62' e2)
~2 = ~' ~2 ~ B, Y2 ( ~ "
H3
U5
immediately above an endsegment.
is also lower than
of
H2
is lower
The induction value
(~, 8, ~, 6,¢) , since
~2 = ~' 83 < 8 , as is seen by the following reasoning. Assume e.g.
[Ba] n
o
~x Bx
~3
nl(a )
H*
Bt
1
C
~xD I
ZxBx
[Ba]
~x Bx
H~ n
~xD
~xBx
n
hi(a)
~x Bx
C C
(for simplicity we assume
94
to have a single endsegment) ; then by a
number of permutative reductions, we obtain as a result a deduction
N~
containing a sub-deduction
nl(a) ~x Bx
which by a proper
C
~- reduction reduces to
It is also obvious that
H2, H 3
93 .
H 2 , since if
B 5 < B.
satisfy (I) of the lemma, since
satisfies this condition (cf. our argument for satisfied for
Therefore
83 < B ).
Rule (H2) E { VE, --E, & E } ,
Condition II is then
[Ba]
H6 =
i no
hi(a)
~x Bx
C
C
[Ba] is strongly valid, hence SV
because of (II) for
~1(a) n.
is
SV ; the derivations of
Z
are also
295 Condition
II is satisfied for
H3 ,
Z
since
is
SV , and
94
[Bt] n~(t) C is
SV
by clause
(iv)(C) in the definition of
Condition llI is vacuously
satisfied for
Condition IV is satisfied for For example,
let
H
H2, 9 3
SV,
applied to
H 3.
H2' ~5"
because it is satisfied for
9.
be of the form
[Ba] HI(a)
No ~x Bx
[Db]
~x Dx H7(b) A
~x Dx
and assume
[~a]
[~a]
nl(a)
n° ~xBx
Dtn4(a)
~x Dx
~x Bx
~x Dx
ZxDx Rx Dx
[~] Then by condition
IV on
H ,
hi(a )
is
SV,
etc. etc.
[Dtl
~7(t)
Therefore we may apply the lemma to in turn shows IV to hold for Therefore (e',
~'
8', ~',
satisfies 6,, ¢,)
H2' ~3
(induction hypothesis) ; this
9'.
condition
I, and thus has an induction value
for which obviously
~'~8,
y, < ~.
So
9,
is
strongly valid by the induction hypothesis.
4.1.17.
Theorem.
Proof.
The proof is by induction on the length of a deduction.
All deductions
I° ) The basis step for deductions 2 ° ) If
Rule (~)
clauses 3 ° ) If
is an
!-rule,
are strongly valid under substitution.
of length 1 is trivial. the induction
step is also trivial :
(i) - (ill) in the definition of strong validity.
Rule (N)
is not an
I - rule, the induction
step is also trivial :
we have to apply the preceding lemma. For this, it is sufficient tion
H :
to show that lemma 4.1.16 implies that a deriva-
296 H I
"'°
n
A Rule (~)
not an
l-rule,
is strongly valid under substitution
are strongly valid under substitution. from in
H H,
by substituting
"'''
if
H I, ..., H n
be any deduction obtained
strongly valid deduction of open hypotheses
Then
Hi,
H*
terms for parameters which are not proper parameters
and substituting
for these hypotheses.
Let
H*
of
H
is of the form
9" n
A*
HI'* "''' ~*n
are
so condition condition of
SV,
if we assume
II holds for
HI, "''' Hn
SV
under
substitution ;
III holds also because of the strong validity under substitution
~2 ' which yields that the substitution
Rule (~) = IND
I
[A*a] ,
in
H~(t)
A*sa
deduction
A*t ' if
is
SV.
A'st
~1 = ~L HI
~*.
'
~2 =
~xBx
. ~ ~ IB*x L ~*
HI =
*
SV
[A't]
H~(a)
Condition IV holds also for
Then
of an
and
H~ =
Assume
to be
~* , condition I hoZds because of theorem 4.1o13 ;
For assume
I
Rule (H)
to be
~E,
i.e.
[~a]
H2 (a)
"
A
.
H2 =
'
A* I [B*a]~(a)
to reduce to e.g.
H" I ~x DIx
B*t
(For simplicity we have
~x B*x
assumed a single endsegment).
• ~x B*x nll n
.-
~xDx
Sx B*x
n
~[X B * x
Since
n *I
nition of
is
SV
this is also
SV,
applied
B*t Sx B*x
is
n
$V,
hence by clause
(iv)(c)
times,
SV ; then by clause
(1),
H' B*t
is
SV.
in the
defi-
297
The strong validity under substitution of
H2
yields that
n, [B't]
is strongly valid.
H~t A*
Similarly,
if
Rule (~) = V E .
This completes the proof of (IV) for
9" ,
and thereby the proof of the theorem. 4.1.18. Remark.
The proof can be greatly simplified if we disregard permu-
tative reductions. value
Then in the proof of the lemma we may use as induction
(~, 6, ¢) , where
¢, 6
and
e
have the same meaning as before, and
we can leave out the discussion of the troublesome case (e). 4.1.19. Uniqueness of normal form of deductions. To establish uniqueness of normal form relative
~¢2~ ' we have essential-
ly the same methods available as in 2 . 2 . 2 7 - 54 for normal forms of terms in - HA w . Technically the easiest
(but less elementary) method runs parallel
to 2 . 2 . 3 2 - 33. 4.1.20. 9"
Proof. I°).
Lemma (Analogue of 2.2.51).
with
If
H ~'I lq', U P1 ~'' '
then there is a
H" ~ ~*.
H' )- H*,
We distinguish a number of cases and subcases. The contractions used to obtain
H', H"
i.e. were applied to disjoint subtrees of
respectively are disjoint,
H.
Both contractions may be
applied one after another, and their order is interchangeable ; the result is in both cases 20).
H*.
The contraction which transforms
H
into
of the tree involved in the contraction from
H
E"
is applied to a subtree
to
H' .
We have to dis-
tinguish a number of subcases. a)
The reduction from
H
to
E'
is a
&-reduction.
Then we have a
situation as follows : I
H =
I~
n2 B
,
~,
=
E
[ n3
A&B
n} The contraction used for the transition from tree of)
U1
A
or
[J2
A
H
to
~"
applies to (a sub-
In both cases the contractions may be applied one
29s
after another, b)
and are i n t e r c h a n g e a b l e ;
The r e d u c t i o n from
t
n1
H=
H
to
n2 n5 c c
A V B
H'
~'=
c
the result is
is a p e r m u t a t i v e
nI
H2 C Z
A V B
D
z
H* .
reduction.
Then e.g.
n3
C D
D
D
111
L
n1 In the r e d u c t i o n from
UI bl)
n2
AVB'
C
changeable, b2)
n3 ' C
H
to
~"
there are two possible
are reduced ;
as u n d e r
then the contractions
The r e d u c t i o n f r o m
U
to
H"
reduces
from
Z
has the same effect as first applying
c)
Z
to
are again inter-
(a).
First applying the contraction
t r a c t i o n of
cases :
Z"
twice.
to the
E
to
Z" .
Z" , then the
VE-contraction
rE-contraction,
and then con-
The result is in both cases
All other cases can be treated
in a very similar manner,
H*. see 2.2.25
(cf.
also our remarks u n d e r 4.1.6). 4.1.21.
Theorem.
The normal form of a d e d u c t i o n
in
~
relative
~c~
is
u n i q u e l y determined. Proof. Remark.
Completely
parallel
Of course,
its t r a n s c r i p t i o n
to the proof in 2.2.26.
the other method m e n t i o n e d
to the present
in 2.2.34 can be applied also ;
context is c o m p l e t e l y routine.
299
4.2.1.
Contents.
deductions in
In this section we investigate the structure of normal
~,
and give some applications of the normalization theorem.
Strictly positive parts (s.p.p.'s) are supposed to be defined as in 1.10.5. 4.2.2. Definition.
A path in a deduction
rences
such that
(i)
(ii)
At, ..., A n
H
is a sequence of formula occur-
AI
is an assumption not to be discharged by
AI
is the conclusion of an application of
VE,
~E ,
or
IND.
If
A. is not major premiss of an V E - , 2 E - a p p l i c a t i o n , i Ai+ 1 is the formula occurrence immediately below A i in
(iii)
A. I If
(iv)
An
is not minor premiss of an A. I
then
or
IND
if
i < n.
is major premiss of a (non-redundant) application o f Ai+ I
VE,
SE ,
is one of the assumptions discharged by the application ;
is the end formula (= conclusion)
an application of
~E
,
premiss of a redundant Remarks.
~E
then H ;
VE,
H , or a minor premiss of
VE,
I N D - application,
or a
~E-application.
(i) Every formula occurrence in
"or a premiss of a redundant
of
or a premiss of an
H
belongs to a path.
~E-application"
for deductions without redundant applications of
If we omit
in (iv), this is only true VE,
2E.
The formula
occurrences not belonging to a path on this alternative definition disappear if we apply immediate simplifications° (ii) For practical purposes,
the concept of spine defined below turns out to
be more convenient and useful. 4.2.3. Definition.
A spine
formula occurrences in (i)
An
For
H,
I ~ i < n
Remarks.
Ai+ I
Ai
is a sequence of
occurs immediately below
Ai
for
is either premiss of an introduction or ~ I -
or a basic rule, or
tion with conclusion A1
H
application,
(iii)
of a deduction
such that
is conclusion of
I~i (ii)
H
A I, ..., A n
A. 1
is major premiss of an elimina-
Ai+ I ;
is a top formula or the conclusion ef an application of (i)
A deduction may have more than one spine, due to
IND.
&I-appli-
cations. (ii)
If a spine does not pass through
(iii)
The concept of spine was suggested by Martin-Lof's use of "main branch"
VE,
E-applications,
it is a path.
(Martin-LSf 1971), but since it does not coincide with that concept, we have introduced a new term for it.
300
4.2.4.
Lemma.
~I' "''' an ; (a)
In a normal
deduction,
a path can be divided into segments
the segments may be divided into
an e l i m i n a t i o n
(I ~ i < m - I)
part
(E-part)
~1' "''' ~m-1 ' where each segment
is m a j o r premiss of an elimination,
and contains
~i
~i+1
as
a subformula ; (b)
a m i n i m u m part
one is premiss of (c)
an i n t r o d u c t i o n
the c o n c l u s i o n
~m' "''' em+k-1 '
A I
in w h i c h each segment except
the last
or a basic rule ; part
(I-part)
of an i n t r o d u c t i o n
~m+k' "''' ~n ' where each segment is and a subformula
of the i m m e d i a t e l y
pre-
ceding one. Proof.
Entirely
straightforward,
tion that an i n t r o d u c t i o n
deriving
a contradiction
would precede a A I - a p p l i c a t i o n
i n f e r e n c e or an elimination,
or that a A I -
from the assumpor a basic
a p p l i c a t i o n w i t h atomic conclu-
sion or a basic inference would precede an elimination. 4.2.5.
Lemma.
In a path in a normal
of the path is a s.p.p, Proof.
Straightforward,
E - rule is s.p.p, relation 4.2.6.
" A
Lemma.
deduction,
each formula in the
E - part
of the first f o r m u l a in the path. n o t i n g that the c o n c l u s i o n
of an a p p l i c a t i o n
of the major premiss of the application,
is s.p.p,
of
of the
and that the
B " is transitive.
In a normal deduction,
a spine
At, ..., A n
can be divided
into three parts. (a)
an e l i m i n a t i o n part
(~ ~ j < m - I) (b)
a m i n i m u m part
an i n t r o d u c t i o n
Remark.
Aj Aj+ I ,
or a basic rule, part ( I -part) ~m,k ..... A n , where each
is premiss of an i n t r o d u c t i o n Proof.
A I ..... Am_ ~ , where each
A m, ..., A m + k _ I , w h e r e each f o r m u l a except the last
one is premiss of 2k I (c)
( E -part)
is m a j o r premiss of an e l i m i n a t i o n with c o n c l u s i o n
with c o n c l u s i o n
A
(ruth< j < n)
A j+1"
Similar to the proof of 4.2.4. If in a spine
At, ..., A n
does not contain disjunctions does not pass through
VE-
or
AI
is a Harrop formula
or existential
formulae
ZE-applications,
(1.10.5),
as s.p.p.,
and therefore
i.e.
the spine
coincides
with a path. 4.2.7.
Lemma.
(i) In a normal deduction
be an a s s u m p t i o n
d i s c h a r g e d by a
VE -
the top formula of a spine cannot or
2 E - application.
(ii) The top f o r m u l a of a spine of a strictly normal d e r i v a t i o n c o n c l u s i o n not ending with an i n t r o d u c t i o n deduction,
or a basic axiom.
is an open a s s u m p t i o n
with closed of the
301
Proof.
(i)
Let
a ~ At, ..., A n
AI
would
be an a s s u m p t i o n
AI
would
occur
would
have
premiss
to pass
through
such a minor
of spine. ~ ~ At, ..., A n
be once
by a
VE-
-introduction,
or
that
~
be the c o n c l u s i o n
would
be of the f o r m
but
tion nor above
b
would
a minor
premiss
of
assumption
VI- ,
of
the d e d u c t i o n
Since
Let
hence
Corollary. If
E
dant
a basic
H
of
dant p a r a m e t e r s subdeduction Almost
dant v a r i a b l e s hence
H
deduction
A
and w i t h o u t
of
A~
by 2.8,
of
must
end w i t h
A VB,
or
B A V B
IND - applica-
AI
inference,
is an open
of
~xAx ,
of the theorem. an i n t r o d u c t i o n part of the
closed, then
~
without
redun-
contains
a
B.
numeral
closed, then
deduction
etc.
without
~
contains
H
without
reduna
~ °
an introduction,
etc.
of
I - rule.
also.
~IxAx
a normal
of
an atomic
If the c o n c l u s i o n is an
A VB
H A
then
be a p r o p e r
to the minimal
formula
is of the f o r m
A VB
for
does not occur
ending w i t h
open assumptions,
since
It is also
open assumptions,
the a s s u m p t i o n s
for a suitable
immediate A VB
applied
or a s u b d e d u c t i o n of
by
4.2.6,
IND - applica-
cannot
open assumptions,
deduction
(lemma
Therefore
a spine not
of
AI
I-rule.
rule
belonging
and w i t h o u t
is a normal
b
without
of an
satisfying
axiom
discharged
AI
since no
of a basic
ends w i t h an atomic
is a normal parameters
subdeduction
Proof.
deduction
the final
be an a s s u m p t i o n
term of the
and also
de-
axiom.
an a p p l i c a t i o n
be a d e d u c t i o n
(i)
If
normal
then
by the
IND - application,
so
if
hence
normal
an introduction).
parameter, At,
or a basic
atomic,
the spine
below
IND-applications.
are no open assumptions,
4.2.9.
(ii)
below
cannot
of an
the i n d u c t i o n
or an a p p l i c a t i o n
is not H
there
end w i t h
~ E - application,
or
ends w i t h
can only begin with spine,
ef an ~E-
~I
b
occurs
A strictly
formula,
application
Proof.
Bb ,
of the deduction,
Theorem.
a closed
does not
then be a r e d u n d a n t
VI- application
parameter
4.2.8.
AI
is excluded
of a strictly
occurs
would
that
which
AI
N ;
~ E - application,
n o r an a s s u m p t i o n
~-introduction
excluded
deduction
~rE- application,
or
again a spine
~ E - application,
since no
or
VEpremiss
and our h y p o t h e s i s
tion ;
VE -
of a
H , n o t e n d i n g w i t h an i n t r o d u c t i o n .
discharged
AI
of a normal
by an
a minor
(ii)
duction
discharged
above
definition Let
be a spine
i.e.
VlI
or
redunVrI ;
3o2
4.2.10.
Remark.
theorem
4.2.8,
as well
as 4.2.11
below,
do not require
respect
to p e r m u t a t i v e
reduction
rules,
but only w i t h respect
4.2.11.
Theorem.
F
(i)
Inspection
Let
shows
formal
deduction
F,
rule applied
any main path has l e n g t h (ii)
In a strictly formula
the proof
be a f~nite
In a strictly the final
that
normal
of lemma's
set of closed of a closed
is not
VE
Harrop
formula
or
4.2.6,
4.2.7
normalization
~E;
to
and
with
~Jk "
formulae. A
from assumptions
i.e.
the e n d s e g m e n t
of
1.
deduction
from assumptions
F
of a closed
the final
disjunctive
rule a p p l i e d
or e x i s t e n t i a l
is
VI ,
~I
respec-
tively. Proof.
(i) Let
from
F,
e.g.
VE .
premiss
~
coincides
is ruled
out
occurrence
4.2.5
fore
H
(ii)
Let
since
in
in
of a closed
H ~
An_ t
to be passes
VE
formula or
through
is of the f o r m
cannot
with
through
premiss
AI .
rE-
A
~E,
say
the major
C VD,
Hence or
If
~
and
of an element
of
F,
V E-
or
~-
since
the elements
there
such
formula
application
;
and hence
by
of
F
or e x i s t e n t i a l
is no
possibility
are
formulae
Ai ,
as
and there-
~ E - application.
would
(for
A m ~xB
application
the argument
is
or a basic
rule,
of ~ I
end with an elimination,
and as under
an a p p l i c a t i o n
The first
be the first
of a path,
A m B VC
E - part,
F. Ai
piece
disjunctive
end w i t h an a t o m i c
its
AtE
of a
But
contain
and let
is not atomic.
or
N o w let
an initial
follows. a
axiom
atomic.
of
do not
before,
pass
with
is a s.p.p, which
end with
be a s.p.p,
of
VE
which
(i) it would
or
~E ; but
is excluded.
a spine
follow
then
that
A = B V C
Therefore
H
would
must
end
V - introduction.
4.2.12.
Co r o l l a r 2.
A strictly F
4.2.13. ~ A ~
F
be a finite
deduction
a subdeduction
of of
set of closed
AVB, A
AVB
from
F
Harrop
closed,
formulae.
from assumptions
or a s u b d e d u c t i o n
of
B
F.
A strictly F
Let
normal
contains
from (ii)
is a basic
is major
n~as ~
i.e.
is not
which
cannot
A
with a
AI An_ I
coincides
Ai
coincide
cannot
~ E AI , -.., An
a contradiction
similar).
(i)
a
formulae,
s.p.p.'s,
would
since
then
deduction
applied
E - part.
(ii),
in
normal
rule
VE-application,
with its
At, ..., A i lemma
the final
Then a spine
of this
By lemma 4.2.7
Harrop
be a strictly
and assume
normal
contains
a subdeduction
Corollar2. ~xBx
~
deduction
(The ~
of of
I P - rule,
~x ( ~ A -- Bx)
~xAx , A~
from
without (x
~xAx F,
closed,
from a s s u m p t i o n s
for a suitable
numeral
n .
parameters)
not free
in
A,
~ A ~
~xBx
closed).
3O3
Proof.
Let
~
be a strictly normal deduction of
must end with an
~l-application
n"~ i
(4.2.8),
A ~ ~xBx ,
so
then
is of the form
[ ~ AZ] 5~xBx
--, A -~ ~xBx By 4.2.11
H"
must also end with an introduction,
~"
i.e.
is of the form
Bt ~x 3 x
We then obtain a derivation
°
of
~ x ( ~ A ~ Bx)
as follows:
[~A] Z' Bt A ~ Bt
4.2.14.
Theorem.
Let
P
be any basic constant of
(Y~rkov's rule
MRpR , cf.
P*y
denote any formula with a single free variable
n+ I
stituting numerals for all arguments Then
~ ~
VyP*y
~
1.11.5 , § 3.5.)
arguments
of
P
of our language, y
obtained by sub-
except one.
~ ~y ~ P*y.
Note that this implies the validity of
NR~R , Markov's rule for primitive
recursive predicates
applied to closed formulae,
recursive
A(x) , we can find a constant of our language
that Proof.
predicate
and let
P(x,O) *-~ Ax ; and
I VyAy ~-~-i VyP(y,O)
A closed strictly normal deduction of
since for any primitive
~
~ VyP*y
~ VyP*y
P
such
etc.
takes the following
form
H~
i [vy Z P'y] Vy P*y -~
We wish to show that the tree ~* , obtained by deleting all assumption . formulae of the form V y P y from H is again a proof tree, deriving from assumptions
of the form
P*~.
To see this, we show that every path
~
in
H
begins with
VyP*y,
P*~,
followed by atomio formulae only, or w~lha basic axiom, followed by atomic formulae only.
304
Since empty.
H
ends with
Hence,
must be an a s s u m p t i o n a disjunction VE -
or
formula
Since
~
applied since
only
if
t
~E,
redundant,
no
ring above
a minor
closed
all paths #k
(= spine)
of an
premiss
belonging
4.2.15.
a proof
an
below result
in the context
Definition.
Let
(i)
Prime
formulae
(ii)
A,B E ~
=
(iii)
B e ~
A~B
(iv)
A(a) 6 ~
4.2.16. HA+ ~
=
ED
Note
assumptions
P*~I'
a in
in an open assumption° part,
the first
P*t .
P*t ,
b
and
would P*t
is a
Obviously,
I N D - application,
H*
provides Since
~x ~ P*x
be
not occurP*t
or
P*~u" of
rule
is a numeral,
Therefore
part of the path.
"'''
t
the
hence
a deduction
P~
H*.
6 in 8 c a r p e l l i n i
a calculus
of sequents.
deduction
a class to
using
A,
We give
systems.
of formulae
~,
of
is decidable,
from
of Theorem
defined
and formulae
inductively
~IxP(x) ,
P(a)
by : prime
~ ; 6 4 ; E ~ ;
VxA(x) Let
~
E ~. be a set of closed
over
HA
w.r.t,
that a f o r m u l a
belonging
to
from
s.p.p.'s
to be a strictly
normal
~.
If ~
A~
from assumptions
belonging
to
So in the r e m a i n d e r to end with
are excluded,
formulae,
closed
~ ~
~,
such that
Z ~ - formulae.
Then
HA+
DP.
and then
rules
is
contain
through
is o b t a i n e d
belong
A&B
and
a ~ At, ..., A n
pass
is a m o d i f i c a t i o n
~-introduction,
~
below
a proof
an
assume
does not
a parameter,
As a consequence,
denote
and existential
N
b
3 - elimination.
of natural
is c o n s e r v a t i v e
S.pop.'s, Assume
=
Theorem.
satisfies Proof.
~
to
~
w i t h a path e n d i n g
VyP*y,
--- e l i m i n a t i o n
that we can construct
belong
VyP*y
beginning
with
snb,
to the m i n i m u m
through
The result
of
if not an axiom,
does not
coincides
~
occurring
f r o m a set of a s s u m p t i o n s
but S c a r p e l l i n i ' s
of
starts
of an
are main paths.
it follows
since
a spine
E - part and a minimal
so the path
VI- application
formula
and
a spine
be of the f o r m
path does not pass
;
l - part of a spine
and vice versa.
consists
would
VyP*y
Hence
be a main p a t h
must be
the
of the spine,
quantifier,
~ E - application.
~
formula,
the top f o r m u l a
or existential
Con (H) (= main path) Let
a minimal
as before,
~
does not
deduction H
Y,
,
let
of
E - part.
~
E
be
deriving
~ . H ,
and
A I- or basic
is not atomic). By lemma 4.2.7
from
it must
H' ,
of
(atomic
prime.
closed,
numeral
be a spine
E-rule
as
P(a)
XxAx
as a s u b d e d u c t i o n
of an
the c o n c l u s i o n with its
XxAx ,
for a suitable
of the argument,
since
of
disjunctions
XxP(x)
ends w i t h an introduction,
contains
an a p p l i c a t i o n
coincides
contain
are of the form
Hence~
(ii),
3o5
AI E ¥ ~ 1°).
~,
Let
of
second 2o).
AI
is a basic
axiom.
A. be the first major premiss of a V E i there is such an a p p l i c a t i o n o c c u r r i n g in
(assuming s.p.p,
or
AI ,
of the form
BVC
case is only p o s s i b l e A.
cannot
l
r e d u n d a n t : A.
if
nor above
of the f o l l o w i n g
~xB .
~E- application
The first
Then
Ai
is an
is excluded,
the
A I E T.
a parameter
does not occur
1
application,
contain
or
or a ).
free,
above
since
a minor
an a p p l i c a t i o n
of
such a p a r a m e t e r
premiss
VI
or
of a
IND.
would
VE-
or
be
~E-
Therefore
H
is
form
~"(a)
n' A. ~ i
~xBx
C
At+ I ~ C
30). no
The only a s s u m p t i o n s ~I-
or
and thus B~
for
IND- application
HA ~
~Bx
;
some numeral
N o w change
H
occurs
therefore ~,
H' ~xBx
open in
using
are a s s u m p t i o n s
below
Ai .
Hence
rules
and axioms
¥,
HA+ Y ~
we can find a d e r i v s t i o n basic
from
~*
since
~xBx ,
in
HA
of
only.
{-
into
n*
~I ~
Ai+ I ~tgJ
~x Ax 4o).
HI
minor
premiss
below
At+ I
So
NI
is again C
strictly in
only
~
E - rules
satisfies
the
applications
of
must f i n a l l y
terminate
an
end either w i t h of
a.
or
H A + Y,
hence
of
for a suitable
A~
in
HA.
4 . 2 . 1 7 . Corollary. Proof.
~+MpR
than
we must realize
introduced
assumptions
H.
by an
Hence
as
H.
HI
if we repeat
H*
V - elimination.
But then a spine
case,
or
~[xAx
In b o t h numeral
H~ + N p R
that the
I - rule,
~xAx
in w h i c h
is a closed
cases,
no
is a s.p.p,
contains
the process,
in a d e r i v a t i o n
an introduction,
In the l a t t e r
to see this,
since
are applied.
same general
E - rules
S- elimination
normal ;
could not have been
spine ~
passes of
H*
fewer it through must
of the top formula
Z oI - formula
derivable
we Can find a d e d u c t i o n
in
in
HA+
~. satisfies
is c o n s e r v a t i v e
over
the u n i v e r s a l closures of i n s t a n c e s c o n d i t i o n s of 4 . 2 . 1 G are satisfied.
of
ED, D P .
closed MpR
Z oI - formulae all b e l o n g
to
(see
3°6.7
(ii~;
~ ,
hence
the
3O6
4.2.18.
Theorem.
Let
H
be intuitionistic arithmetic with induction
restricted to quantifier-free formulae, and let
qf-HA
free part of (the natural deduction system of) vative over
qf-HA.
Proof.
n'
Let
be a closed deduction in
A ; then we can find
B
such that
qf-~
H
~.
be the quantifier-
Then
H~
of a quantifier-free formula
~A
~-~ B ,
B
a prime formula.
By the normalization theorem we can find a strictly normal, H
of
is conser-
closed deduction
B.
Now consider any spine or an atomic conclusion of
At, ..., A n
of
H ; AI
is either a basic axiom,
IND ; in both cases the spine consists exclusivs-
ly of atomic formulae, and especially, does not pass through an
Z-
or
V - elimination. Let us define a spine of order
0
as a spine ; a spine of order
n+ I
is defined as a spine, but its lowest formula is premiss of an application of
IND , whose conclusion is top formula of a spine of order We can easily establish, by induction en
for all
n
through a in
to consist of atomic formulae, V-
or
Z - elimination.
n,
n.
all spines of order
n,
and hence especially not passing
As a consequence,
H
is a deduction
qf - } ~ .
4.2.19. T h e o r e m .
HA
is conservative w.r.t, closed formulae over the logic-
free fragment without induction. Proof.
This theorem was established before by computability methods in
combination with models for
N-HA w
(2.5.6).
A proof via the normalization theorem is very similar to the argument under 4.2.14, but simpler : all spines in a closed deduction with closed conclusion must begin in a basic axiom, hence no
VE-
or
~ E - applications
occur, and all formulae in the deduction are atomic. 4.2.20.
Theorem
(Reflection principle for closed
Let
Proof N
~,
i.e. Proof~(x,y) =d~f Proof (~,y) & Norm (~).
Then, for closed
~yAy ~ProofN(x, r ZYA2)
Proof. ~
-- ~ y A y .
By formalizing our argument in 4.2.11, 4.2.12.
ProofN(x, V Z y A y u) -- Z y z ProofN(y , r A ~ ] ) ; HA ~ P r o o f N ( y , rA~])
where
Z I°- formulae)
denote the (standard) proof predicate for normal deductions in
Proof o
"
formalizing 4.2.19 yields
Proof (y, rA~1)
is the proof predicate for the system without logic and
induction ; finally, using the reflection principle for this system,
b Pr°Ofo(Y' [ A~) .
A~
(of. ~ t S ] .
3o7
§ 3.
Normalization for
4.5.1.
.HA+ IP,
with applications.
In this section we study a natural deduction system for
H A + IP,
which is obtained by adding to the natural deduction system for
~
the rule:
A -- ~ x B IP Sx ( N A--B) As follows from the results in 4.2.13, 4°4.5
, this rule is derivable
from null assumptions. We add a new contraction
(IP-contraction) :
[~A]
~I
IP
[~A]
H
n
Bt
Bt
contr.
~ I
~xBx
~ A ~ Bt
A -- ~ x B x
~ x ( ~ A--Bx)
~(~
A~Bx)
In Troelstra A, normalization for intuitionistic second order logic relative the reductions of
~C
(extended to second order logic)
+IP , + IP-
reductions is discussed ; the proof, an adaptation of the proof of strong normalization for intuitionistic
second order logic in Prawitz
1971, App. B
was only indicated. Prawit~ A contains a proof of a normalization theorem (including reductions,
IP-
but not including permutative reductions).
Here we prove strong normalization by extension of the method of § I (so ultimately by an extension of the method of Prawitz 1971, Appendix A). 4.3.2.
Theorem.
In
~+
IP , every deduction has a finite reduction tree.
Proof.
We only have to make some additions to the argument in § I.
definition of strong validity remains unchanged ; but we classify
The IP
among
the non-introduction rules. Lemma's 4.1.10, 4.1.11, 4.1.12 carry over unchanged,
just as theorem 4.1.13.
In lemma 4.1o16, we must read for II : (II)
If
Rule (H) ~ I&E, ~ E ,
VE, ~ ,
IPI,
then the elements of
are strongly valid. The proof of 4.1.15 has to be extended with an additional case : Case (f).
H'
is obtained by an
I P - contraction from
9.
PRD (n)
3o8
[9 A] H" =
Let
r
Bt
,
U'
[~ A] Bt
=
f Sx (~ A--Bx)
Zx Bx
_~_A
Bt
~x (~ A
-~ A -~ Bx
-~ Bx) NttT
~"
is
(by (If)),
SV
hence
by
SV ( i i ) ,
for each
for each
SV
strongly
valid
-~ A '
~tfl [-~A] Bt ~x Bx
~rw~ is
SV.
Therefore,
by
SV
(i) ,
deduction
A
'
[~A] Bt is
SV.
Therefore,
Now theorem 4.3.3. with :
"or
A.
~I
Lemma.
deduction
(ii)
A spine three
by a
of a normal
as in 4.2.3, of
is strongly
valid.
but
clause
extended
deductions
in
the top formula V E-
or
(ii)
IP"
is never
may be followed
deduction,
E-IP-part,
is either
followed
by an
by an elimination,
I P - application.
HA+ IP. of a spine
cannot
be an assump-
Z E - application. A1, ..., A n ,
A I ..... Ai_1 ,
major
premiss
I -application then Ak
Ak+ I
may be divided
atomic (c) An
Ai,
instances
I - part
premiss
of an
part
each
an a s s u m p t i o n
of an
Ak
into
IP- application
(k < i - I)
or premiss of the form
IP- application
..., A i + j _ I ,
of 2k I
Ai+j,
where
of an elimination,
discharging
is premiss
is premiss
(b) A m i n i m u m
The
H'
parts :
(a) A n
(iii)
again,
of an a p p l i c a t i o n
We consider
tion d i s c h a r g e d
spine
an i n t r o d u c t i o n
- application
In a normal
SV (i)
over unchanged.
We define
in a spine
an
SV (ii),
is premiss
1
that
although
(i)
by
carries
Definition.
We note
4.3.4.
4.1.17
and
of an ~ B ,
and
k < i- 2 ,
or
;
where
only basic
rules
and
are used ;
..., A n , where
each
Ak
(i + j ~ k < n)
is
of an introduction.
top f o r m u l a
of a spine
w i t h an i n t r o d u c t i o n
with
of a strictly closed
assumption
of the deduction,
discharged
by an
~
normal
conclusion
is a basic
or an a s s u m p t i o n
-introduction
followed
deduction
by
axiom,
of the form IP.
not
~ B
ending an open to be
309
Proof. (ii)
(i) As for lemma 4.2. 7 (i). Let
A1, ..., A n
be a spine ; let
A.
be the first formula not m a j o r
1
premiss of an elimination. (a) If
A.
is atomic,
let
i + j
be the maximal i n d e x such that
1
Ai,
..., A i + j
Ai+j+ I
are all atomic.
(if existing)
followed by an ductions
occur in the spine,
application described
A
nor an elimination.
they can
or an elimination.
which cannot be
If two successive
nevermore be followed
by an
intro-
IP-
So in this case the spine has the structure
in the lemma.
(b) If (b')
must be o b t a i n e d by an introduction
IP-application
Ai
is not atomic,
there are the f o l l o w i n g
possibilities:
is f o l l o w e d by a sequence of i n t r o d u c t i o n s
l
ending in
A
n
, or
i=n; (b")
A.
(b'")
A.
is followed by an
-~-introduction
is f o l l o w e d by an
IP-application.
and
IP,
1 l
If (b") (resp.
(b"'))
Ai+1,
..., An)
Ai+l,
..., An) , etc.
shown that
is the case, we can consider
and iterate our argument for
Ai+2,
Ai+ 2 . . . . . ..., A n
till we arrive at case (a) or (b').
At, ..., A n
can be split up into parts
An
(resp.
(resp. Then we have
~o - AI' "''' Ai
'
°I -= Ai o' " ' ' ' A i I' ~2 = Ai I' " ' ' ' A i 2' " ' ' ' ~ m ' such that ~k (k <°m) passes through a (possibly empty) sequence of eliminations followed by
-*I,
IP or by IP only, and ~ passes through a sequence of eliminations, m followed by a sequence of basic rules and atomic ~ I - a p p l i c a t i o n s , followed by a sequence of introductions.
Hence the spine is of the form as described.
(iii) Similar to the argument for 4.2.7 4.3.5-
Lemma.
Let
H
(ii), u s i n g
be a strictly normal d e d u c t i o n in
open assumptions,
with
I P - applications,
and ends w i t h an a p p l i c a t i o n
i n s t a n c e of ~ I
(i) and (ii) proved above.
Con (n)
closed.
Then a spine of
or an a p p l i c a t i o n of an
I - rule.
If the c o n c l u s i o n of the
the final rule applied is an
Proof.
be a spine of
Let
At, ..., A n AI, ..., A i
passes
s.p.p,
AI ,
AI
of
formulae
since
as a s.p.p.
application
or
D , and let
through eliminations
By 4.3.6
(iii) by an
i
be the maximal
only.
A
Then
Ai
so no
is possible.
as in a normal
index
is a
or existential
can be atomic o n l y ;
IP- application
a spin¢in this case has the same structure and for the remainder
I - rule.
does not contain d i s j u n c t i o n s
~I'~Tfollowed
without
does not contain
of a basic rule, an atomic
d e d u c t i o n is not atomic,
such that
H A + IP H
deduction
of the lemma we may argue as in 4.2.8.
IP-
Therefore in
HA ,
31o
4.3.6.
Theorem.
assumptions similarly, A VB
of
In ~xAx
IP
every strictly normal deduction without open
contains a subdeduction
of
A~,
if
~xAx
strictly normal deductions without open assumptions
closed,
contain either a subdeduction
Hence also ( A V B H~+IP
Preof.
~+
closed, ~AVB
~xCx
~ HA+IPbA
Immediate by 4.3.5.
of
A
~+IP~
A VB ,
or a subdeduction
closed) or
is closed ; of
B
of
B .
511
4.4.1.
Conventions°
normalization itself.
Below we shall describe the formalization
of the
theorem (for subsystems of bounded logical complexity)
in
HA
In describing the formalization we tacitly identify the various
syntactical categories and objects with arithmetic predicates and natural numbers;
Rule (~), Prd1(H),
recursive numbers, etc.
Norm (~)
Con (n) (N
etc. are now assumed to be primitive
is normal) a recursive predicate of
H
(This is automatically ensured for anyone of the standard g~del-
numberings.) We need some additional conventions and abbreviations. (a) Rule (~)
is a number,
Jl Rule (Z) / 0 (b) Red I (H', ~)
such that
JIRUle (H) = O
I - rules
for the other rules ;
is a primitive recursive relation, expressing that
obtained by a single contraction from
for the
(applied to a subdeduction of
n'
is
U)
H;
(c) Red (n', n) ~def ~m(Ith(m) > 0 & (m)o = n & (m)ith(m): I = n' & & Vi < ith(m) : 1[Red((m)i , (m)i+1)] ; (d) V1(n,m ) m
is a primitive recursive predicate expressing :
is a subdeduction o c c u r r i n g
and
n, m
immediately above an endsegment of
n ,
are (code numbers of) deductions ;
(e) If
H'
is a p r o p e r subdeduction f r o m
4.4.2.
Formal definition of strong valid it 2 for deductions with conclusion
~ ,
~' < ~ .
of.bounded complexity. Let us assume
SVd_ I , the predicate of strong validity for deductions
of formulae of logical complexity
< d
to be given, and consider the
following closure conditions on a set of deductions (i)
Rule (H) : & I & SVd_1(PrdIN ) & SVd_1(Prd2E) ,
(il)
Rule (n) = V I l & SVd_1(Prd1~)
X :
V
v Rule (n) : V l ~ SVd_1(Prd2n), (iii) Rule (n) : ~i & SVd_1(Prdln), (iv)
Rule (H) = --I & VH'[Con H' : Prem (ConH) &
(v)
& SVd_I(~') " SVd_I{Sub (Prem (Con H), H', Prd1(H), Ass (n))}] , Rule (~) = V I & Vx[Term(x) -- Subst(Param(n), x, Prd1(H)) ]
(vi) JIRUle (H) /
0 & Norm(H) ,
(vii) Jl Rule (n) / 0 & vn'(Red1(n', n) -- X~)) & &[Rule (H) = ~E -- ~ n { n / O
& Vff'(Red(H',
Prdlff ) --
-- ~i < Ith(n)(n'= (n)i) & Vi < ith(n)(Red((n)i , Prd1(n)) ) & & X(Prd2n ) & Vi < Ith(n) Vm i (n)i{V1((n)i,m)
&
& Rule (m) = ~ I -- X(Subst (Param (a), Term (m), P r d 2 (~))) }}] &
3~2
& [Rule(R) = V E - - Z n { n / O & Vn' (Red(n', Prd I H) --- ~
< Ith(n)(n' = (n)i) & Vi < ith(n)(Red((n)i , Prd1(~)) &
& X(~rd2n) & X(Prd3~) & & Vi< lib(n) Vm i ( n ) i { V 1 ( ( n ) i , m ) & Rule (m) = V I r
- X(Sub (Con (Prd1(m)), Prdln, Prd~,
--
Ass (n)))~ &
& Vi < lth(n) Vmi (n)i{V~((n)i,m) & Rule (m) : VI 1 -"
Now let
X(Suh (Con (Prd1(m)) , Prdln , Prd3n , Ass Z)) }] . A(X,n)
(H
a number variable ranging over proofs) be given as
(i) v (ii) v (iii) v ... v (vii). Now
SV d
must satisfy
(I)
A(SVd, ~) -- SVd(n )
(2)
V n (A(R,H)-- RH) -- V~ (SVd(H) -- RH)
for all arithmetical
R.
Inspection of (i) - (vii) shows that so as to become an element of the class by 1.4.5, SV d
A(X,H) F
can be slightly rewritten
introduced in § 1.4.
is arithmetically definable, and can be proved in
Hence, ~
to
satisfy (I), (~). Since the proof of the normalization theorem for deductions of formulae of bounded complexity uses only the methods of intuitionistic arithmetic, the arithmetic definability of 4.4.3.
Theorem.
In
such that for each
HA
implies:
we can find a primitive recursive function
~(z,y)
n
(i)
~
(ii)
H~ b V z [ZxProofn(Z,X ) -
~ P r o o f n ( z , r A ~) -- Zy(Proofn(~(z,y),rA~ ) & Norm ~(z,y)) ,
(iii) H~ ~ ( z , O )
(iv)
~
b vy(~
(v)
~
~Norm
Proof.
SV d
~y Norm ~(z,y)] ,
= z ,
~orm ~(z,y) -- Red1(~(z, Sy), ~(z,y))) , m(z,y) -- V y ' > y ( ~ ( z , y ) : ~ ( z , y , ) ) .
The proof is for the greater part already contained in the dis-
cussion of the preceding subsection. Let us suppose a standard order of carrying out reductions to be prescribed, and let
~(z,y)
be the primitive recursive function , such that
denotes the (godelnumber of the) result of the
yth
reduction step in the
prescribed order, applied to the deduction (with godelnumber) is no such reduction step, ~(z,y')
~(z,y) = ~(z,y')
~(z,y)
for the first
z ; if there y'
for which
is normal.
On this definition,
(iii), (iv), (v) obviously hold.
We n o , that for deductions of formulae of bounded complexity, we can prove in
HA
that every reduction sequence terminates (and a f o r t i o r i
our
313 standard reduction sequence according to the prescribed order) ; hence (iii) holds. (i) is then satisfied in view of the fact that if not contain formulae of logical complexity
Red (H', R) , and
> n , then
9'
R
does
satisfies the
same restriction. 4.4.4.
Remark.
Now it will be clear why there was a slight advantage in
strengthening Prawitz's definition of strong validity by requiring in clauses (iv) (b), (c) (4.1.9) the reduction tree of
Prd1(~)
to be finite instead of
only requiring the termination of all reduction sequences, i.e. the well foundedness of the reduction tree.
For the well foundedness requires for
its expression function symbols, the finiteness not ; possible to define
SV d
in the language of
HA,
and thereby it is
so that it becomes possible
to apply 1.4.5 directly, without any intermediate steps. 4.4.5.
Theorem.
IPR HA ~ y
not free in
Proof. a
(Cf.3 I 15.)
~ A -- ~ y B y
= ~
~y
(7 A ~ By) ,
for
A, B
arbitrary,
A.
Let us assume for simplicity
A, ~ y B
to contain a single parameter
free, and let
(I)
~
~ ~ Aa ~
By B(a,y) .
Then obviously, since (I) implies the existence of a definite proof from which, primitive recursively in
x,
proofs of
A~
~yB(~, y)
may be
obtained, ~Vx~z
Proofn(Z , r = ~
A~--
-- n ) ~y R(x,y)
Formalizing the reasoning in 4.2.13, and combining this with 4.4.2, we obtain: b Vx~uProofn(U, ~ = ~y ( ~ A ~ ~ B(~, y)]) , and with the help of the reflection principle : ~y
(~ Aa ~ Bay).
The case of more parameters is readily reduced to the case of a single free parameter. 4.4.6.
C~ Proof. (1)
Theorem.
~
HA
is closed under Church~s rule :
bVx~yA(x,y)
= ~
Assume, for simplicity, ~
~
Vx ~
V x ~ y A ( x , y)
A(x,y) .
Then there is a numeral ~
b~zVx~u [~zxu&i(x,U~)].
z
Proofn(~ , ~
A(x, y)~) ,
to be closed, and assume
3~4
and therefore a primitive recursive function (2) Let
~ Y
~
PrOOfn(~'(x),
~'
of
x,
such that
r~y A(~,y)1) .
be a primitive recursive function such that for every closed
(3)
~
k
~x Bx :
Proof (x, r~yBy1> & Norm (x)-~ Proof (Jl ~x, rB(j--~>1)
.
Then it follows that
H~ ~- Vx ~yz Proofn(y , r A ( x , z ) 7 ) , combining
(2), 4.4.2, (3).
Now using the partial reflection principle :
~- Vx Zyz Proofn(y , r i ( x , z ) 1) & A(x, z) and therefore,
since
ff
min u Proofn( j qU, rA(x, ~2u) q)
~z Vx ~u(T~u ~ A(~, ~u)).
is a recursive function,
315
4.5.1.
Introductory remarks.
Basic sources are Girard 1971 , Martin-Lof
1971 A, Prawitz 1971 (Appendix B), Girard 1972. In Prawitz 1970, Prawitz established a normal form theorem for intuitionistlc second order logic utilizing a Beth-type semantics different from Kripk~semantics in this case).
(inessentially
In Girard 1971 , a system of
terms with variable types (cf. also 1.9.27 in this volume) was described which made the extension of the Dialectica interpretation to second order logic with full impredicative comprehension possible.
In establishing
normalizability by defining a computability predicate,
analogous to the
treatment for the first order case (cf. § 4.1), we encounter the following difficulty : the induction on the type structure in the definition does not work, because what complexity is to be assigned to a variable type ? complicated type is substituted for a type variable, plexity is introduced.
If a
suddenly a high com-
The difficulty was overcome by Girard by the intro-
duction of to some extent arbitrary predicates as computability predicates ("candidats de r@ductibilit@")
to be assigned to certain occurrences of
terms ; computability of other terms was then defined relative such assignments.
It turns out that computability predicates relative such assignments
are themselves examples of such "candidats de r@ductibilit@" ; thus, by an essential application of impredicative comprehension on the meta level, it becomes possible to prove normalizabil~ty for all terms in Girard's system. The ideas of Girard 1971 were applied in Martin-Lof 1971 to second order intuitionistic arithmetic.
1971A
and Prawitz
Martin-LSf has also extend-
ed the method to simple type theory, as did Girard in Girard 1972.
In
Girard 1972 the isomorphism between terms and deductions (cf. our remarks in 4.1.6) has been exploited to the full. A simple treatment, using Girard's idea, in the context of Schutte's
,~
~97~.
system for intuitionistic type theory is in O s s w a l d ~ 9 ~ T r o e l s t r a
A discusses
various extensions of these results to other systems (also with applications; special attention has been given to the addition of
IP , and the so called
"uniformity principle" UP
VX ~x A(X,x)
~x vx A(X,x) Intuitionistic analysis formalized with sequence variables only, formalized as a calculus of sequents, has been extensively investigated by the methods of "pure proof theory" in Scarpellini
1971, 1972 ; they will not be discuss-
ed in this section. The present section is not intended as a self-contained treatment, but contains some supplementary discussions.
3~6
4.5.2.
The system
~2(S) o
As the basis for our discussion we take the system of minimal (= intuitionistic) second order logic a constant
0
(zero),
=
~2(S)
....
S ;
(equality) and a function constant
The individual variables are means of meta variables
over the basic system
Vo, vl, v2, ...
2
•
(usuany
s
contains
(successor).
(usually indicated by
x, y, z, ... ) and variables for
p = o ,
S
p - arN relations
indicated by meta variables
X p, YP, Z p, ... ). First and second order quantifiers are distinguished by indices : 51, 32, Vl, V2 ; the other logical operators are
&, ~, V, )~.
ity, we shall restrict our attention to the language based on
For simplic~, VI, V 2 ,
regarding the other operators as defined symbols by the second order definitions (cf. Prawitz 1965 V, § I). A & B ~def
VX [(A -- ((B ~ X ) ~ X)]
A V B ~def
VX [(A ~ X) ~ ( ( B ~ X ) ~ X)~
ix A
~def
VX [Vx(A ~ X ) ~ X]
~Y A
~def
VX [ V Y ( A ~ X ) -- X]
~" ~ e ~ (X
VX X
a zero-place predicate variable).
There are two variants of the system, with the
X- operator as a primitive,
and without. In the system with
k
as a primitive, second order terms are defined
thus : 1°)
A
p - place relation constant or parameter is a
I- place second
order term ; 20 )
If
A
is a formula,
kxl... XpA
is a
p-place
second order term.
In this case, besides the first order rules we add V2I)
A(P n)
V2E)
VX n A(X n)
Un A(Xn) I)
A(Tn)
At1"''tn
~E)
{Xx1"''x n Ax1"''Xnltl"''t n
{Ix1" • "Xn Ax1'''Xnlt1" .. tn Here in
V2I
pn
Atl...t n
must not be proper parameter of another preceding in-
ference, and must not occur free in assumptions on which In
kE, If
tl,...,t n X
must be free for
Xl,...,x n
is absent, we use the notation
second order term as ~ 9
in
kxl...x n A
metamathematical, and in
A(P n)
depends•
A. and the concept of V2E , A(T n)
should be
understood as an abbreviation for the formula obtained from
A(X n)
replacing each occurrence of
Tn
xntl..•tn
by
B*tl...t n,
if
by
is the
317
metamathematical ed from
B
expression
kxl...x n. Bxl...x n , and where
by renaming bound individual
tl, ..., tn
for substitution
The corresponding
in
additional
variables
B*
is obtain-
so as to make free
B*Xl...x n. reduction rules are given by
tions and (for the first version)
V 2 - contrac-
X- contractions:
n(P) A(pn) VX n A(X n)
contr.
E(Tn) A(T n)
A(T n) k- contraction H At I ...t
E n
contr.
tXXl..-x n A X l . . . X n } t l . . . t n
Atl.°°t n
At1...t n 4.5.3.
Formalizing
the proof of the normalization
The proofs of normalizability both be formalized
in
HAS.
of Prawitz
We verify this here for Prawitz's
this is somewhat easier to describe, followed Prawitz's
theorem.
1971 and Martin-Lof
especially
1971A proof,
can since
since we already have
treatment for the first order case ; but for Martin-LSf's
proof the details are similar. The crucial point in the whole formalization HAS
of the predicate
is to show definability
"strong validity relative
~"
in
for all deductions
with a conclusion with a bound on the complexity relative
~.
Hence we
restrict our attention to this point. An assignment assignment
~
in the sense of Prawitz
of regular sets to occurrences
sarily to all occurrences, occurrences is an
then
the premisses of
~
If
is given for
is automatically
Below we are going to define
~
~
the same regular set to
extended
Con (N) ,
and
Con (N)
SVd(~ , E) ,
of complexity
strong validity relative ~ d
is counted as logical complexity,
rences in the domain of
~
relative
~.
Param (H) , obvious way.
of
Subst
~,
(Complexity
but where the term occur-
are counted as atomic,
i.e. of complexity
Some conventions: (a)
Rule (9)
to the conclusions
Rapp (~) .
for derivations with relative
of second order terms (not neces-
and not necessarily
of the same term).
I - rule,
1971 may be conceived as an
are extended to the second order case in the
0 .)
318
(b)
Regular (N) :
N
is a regular set.
Dom (~) : domain of
degree (T) : number of arguments of (c)
In clause (v) below,
~ U (~)
extended by assigning stituting
T
N
7.
T .
is shorthand for the assignment
to all occurrences of
for the occurrences of
Param (H)
T
~,
obtained by sub-
in
Con Prd1(U).
Now we consider the following closure conditions: (i)
Rule (~) = --I & VH'(Con H' = Prem (Con H) & SVd_1(~,~') --
(ii) (iii) (iv) (v)
Rule (9) Rule ([) Rule (n)
(vi)
jl Rule (H) =
= = = ~ule (~) =
--SVd_ 1 (~, Sub (Prem (Con ~), Z ' , Prdl~, Ass ( [ ) ) ) ) . V~I ~ Vt (SVd_~(% Subst (Param n, t , Prdln)). XI & Con H 6 Dom (~) & U E ~ (Con H) . ~ a Con H / Dom (~) & SVd_I(~, ~ r d l n ) V~I & ~ [degree (T) = degree (H) & ~ (Regular (N) --
"SVd_ I (~ U (~), Subst (T, Param (n), PrdIH)))] . Now
SVd(~,9 )
0 & [Norm (U) V N]]'(Red1(H',~) " X(~,~'))] .
should satisfy:
A(SV d, ~, ~) -- SVd(~ , ~)
(1)
VII I~(A(R, 9, ~) -- R(~, [)) -- VII for
R
in the language of
~(SVd(% ~) -- R(~, n)) A(X, ~)
HAS , where
is
(i) v (ii) v ... v (~i). Now we put
SVd(~ , H) ~def VX [VII'V~'(A(X,
~', ~') --
X(~', H')]
SVd(~, H) satisfies (I).
and
4.5.4.
Construction of a satisfaction relation.
Actually, what we shall define below combines characteristics of a satisfaction relation and a truth definition as used for the first order case in § 4.4.
(See remark 4.5.5 below.)
We first define, in of
~2(S)
HAS , a satisfaction relation
of logical complexity bounded by
system with constants
O, =, s
n,
Sat n
for all
for formulae
n.
S
is the
(zero, equality, successor).
Sato(X,x ) , the satisfaction relation for prime formulae,
x
the godel-
number of a prime formula,is defined by cases such that :
Sato(X, ~ where
t'l
Val ( r ~ )
= q ) ~Val
denotes the term = x,
ti[~ ]
(~) if
:
val ( ~ ) ,
t i ~ ti[x] .
by definition, and
Sato(x' rvPi ~I"" "~'~p) ~ x(p,i)(val ( ~ ) ' Here
x(p,i)(t ~.... , t~)
abbreviates
..., Val .C~-'p )).
X(j(j(p,i),
519
Finally we put
Sago(X, /~ Now Satn+ 1
) ~
A.
i s d e f i n e d by c a s e s from
Sat n
$atn+1(X , r A o B ~) *-* Satn(X , rA~) for
o ~-%
&, V,
such t h a t o
Satn(X , rB~)
and
Satn+1(X ' r (Qvi)A(vi)-) (_~ (Qvi)Satn(X, rA(~i)~ ) , for
Q -= V I, 31 , and Satn+1(X, /
r(QvP)A(viP) I) +-> (QY)VZ1((Vyly2(j(yl,Y2)
j(p,i)
/
-~ Z 1(y~,y~) : X(y,,yz) ) & Z(p,i) = y -4 Satn(Z , rA(ViP)n) ,
for Q = V2' ~2 ; T'o = ~% abbreviates VX(~oX ~ T~x) In the presence of the X - o p e r a t o r we should also add Satn+1(X, ~{Axl...x p- Axl...Xp}tl...tp ~) ~--> Satn(X , rAtl...tn)p " Below we shall leave out the
~-operator,
and also disregard a possible
distinction between (bound) variables and parameters. To complete our specifications,
we add
-I Satn(X, m) if
m
is not a godelnumber of a formula of complexity
< n without free
numerical vari able s. 4.5.5.
Remark.
In defining a satisfaction relation
seemed more natural, tension of that
if we would have defined
Sat
n
Sat n , it would have in a conservative ex-
HA~S with function variables as a relation
Satn(~ , X, n)
such
if we put (x
if
rt ~ = rsX0 ~ = ~x ~,
Val (~, ~t ~ ) x + a~y if
rt~
rs v
Y
,
Sato(~ , x, rt I = t2~) ~-> Val(s, "t1~ ) = Val(~, rt2~ )
S a t o ( ~ , X, r Vit P I. •. tp) +-' X(p,i ) (Val (&,%~) , . . V. a l.( ~ ./ t p ) ) and Satn+ I (~, x,
for
Q ~ VI' ~I' Actually,
~(Qvi)A(vi )~ )
and otherwise as before.
for any given
coincide, as can be shown in
n , the two definitions of satisfaction relation HAS
itself ; and for our applications
it is
52o
more convenient
to deal with numerical
manner as we did in the first-order 4.5.6.
Construction
quantifications
of a satisfaction
relation,
Let us denote the class of all formulae of mutual substitution
in exactly the same
case. continued.
M2(S)
obtained by repeated
from the set of formulae of complexity
We wish to extend the definition
of our satisfaction
we achieve this by first defining from formulae of complexity
Sat
relation to
, a satisfaction
n,p by iterating
< n
~ n ,
by
Fm (n) .
Fm (n) ;
relation obtained
substitution
at most
p
times.
We put Satn,o(X' r A~) ~def Satn(X' rAY)' and we define I
Sat
from
n,p+1
Satn'p+l(X'
Sat
n,~)'
rA~) ~-* Satn'p(X'
such that rA~) i~
V SB~T ° ... Tm~IY[Satn(Y , rB1) & & Sio' "''' Im' " Po .... ' pm(rA~ = r [V
(I)
& Nk~m Vw(Satn,p(X , rTk((W~)o . . . . .
vPm/ / To, "'''
im
Tm]~ ) &
~ m )~)
~-~ Y(ik,Pk)((w) o . . . . , ( ' ) m ) ) ) ] • Po vPm Vio, ..., im is supposed to be a complete list of the second-order
Here variables free in
B.
As it stands,
(I) is not formally correct,
because
it is not an expression in the language of ~HAS" " B " , " To, ..., Tin" in " ~B ", " ~T ° ... T m " should be understood as variables for gSdelnumbers of formulae and finite sequences of second-order The right hand side of (I) may be abbreviated
terms respectively. as
C(~x. Satn,p(X , x), X, rAt) . From this we see that we can define
Sat(n)(x, m) = O Sat n (X, m) p ,P Sat(n)(x, m) Erief ~Z ~y (Z ° = kx. Satn(X , x) & & Vi
Here
4.5.7.
Zi
abbreviates
Lemma.
by
(Zi+ I = kx. C(Zi, X, x)) & Zym) .
kx. Z(j(i, x)) .
Sat(n)(x,
r
A(vil .....
pl Vik' Vjl ....
¢-~ A(Vil, • . -, Vik, X(jl,p ~ ) ..... X(jm,Pm)) )
'
vPm)I)
Om
is provable in
HAS
for
A E Fm (n) . Proof.
We first note that ~Z (Z O = Xx. Satn(X,x)
is in fact equivalent
to
& Vi
Satn,y(X , m)
as defined before.
521 Now it is sufficient of second-order
to show that if
terms
p /
Sat
replacing
A
is obtained by iterating
times, then the assertion
of the lemma holds with
~
Sat ~n) .
This is proved by induction on
n,p is provided by the corresponding
property for
lar to § 4.4) proved by a straightforward of
substitution
p.
The basis
Satn(X , rA~) , which is (simi-
induction on the logical complexity
A. Consider as an example
to be obtained by Satn,p(X' A ~
i
V~BV
substitution.
B(vi)1) ; or there is a formula
i ~ [V~/T
Vvi[V~°/T 0,...] To, ...,
Satn,p+1(X , rVvi B(vi)~ ) , and assume
p+1 - fold iterated
0 ,...] Vv i ~ ( V ~ ,
C(V~°, . . . .
vi) ,
obtained by substitution
Vv i B(vi)
Then either already
C(V~:, .
, vi)
such that
.... vi) ~
(C
a formula of l o g i c a l
iterated at most
p
complexity
~ n ,
times, and not con-
taining
v. free). i In this case, we use the induction hypothesis
for
together with
Vv i Satn(Y , rc(
Satn(Y , rvv i C(
.... vi)~ ) ~
Satn, p
relative
To, ...
.... vi)-~) ,
etc. etc. 4.5.8. Lemma. In HAS (i) Sat-~-T(X, r A C ~ )B for (ii)
we can prove ~ Sat(n)(x, r A t ) o Sat(n)(rB ~)
o = "~ , &, V .
Sat(n)(x, r (Qvi)A(vi)~)
e-~ (Qvi) Sat(n)(x, rA(~i)~ )
for (~ = VI, ~I' (iii) Sat(n)(x, r (~Vi)A(Vi)) p P l
e-~
(•Y) ~Z1(VYlY2(j(yl,Y2)
~ j(p,i)-
ZI(YI'Y/) = XyI'Yz) &
& Z'(p,i) = Y "~ Sat(n)(z' ~A(vP)I))' for Proof.
Q =- V2' ~2"
Immediate,
(Abbreviations
as a corollary to the previous lemma.
4.5.9. Lc,~m~ Le~r=IC I .... , Cml of
as before.)
M2(S) , restricted
and let
to deductions
Proof (n)
denote the proof predicate
involving only formulae of
Fm (n) .
Then
HAs b P r o o f ( n ) ( x ,
rCl(• 1 . . . . ), . . . .
Cm(; I . . . . ) = A(; I . . . . ) u ) - *
., [Sat(n)(x, rc1(~ I .... )i) & ... & Sat(n)(x, ,Cm(~ I .... )l) .4 ., Sat(n)(x, r A(~I .... )~)] . Proof.
By induction on the length of
4.5.10.
Corollar[.
(Partial reflection
HAS ~Proofn(x, Proof.
r = A(~I
, .
x,
principle for
"'' ~n )-~)
Combine lemma 4.5.7 and 4.5.9.
of. § 1.5.
"
A(Vl,
Fm (n)). •
". '
v n)
•
322
4.5.11. Applications
of the preceding results.
We note that now we can obtain various closure conditions on the set of theorems,
also for cases with free parameters,
exactly as in § 4.4 for first-
order arithmetic. To see this, we note that for any deduction of complexity
! n , the formulae occurring in any derivation
tree of a derivation improper parameters For example, IPR
H , in which all formulae are
[al, a2, ... /tl, of
~)
t2, .;~I H
all belong to
in the reduction
( a I, a 2 . . . .
Fm k J
we can now obtain closure under Church's rule
(with parameters)
for
being
CR,
and under
HAS.
As an example of an application
which does not only use the partial re-
flection principle,
but also the satisfaction
relation itself in its rSle as
a truth definition,
we discuss closure under the rule of n o i c e
in the next
theorem. 4.5.12. ~Vx Proof.
Theorem.
HAS
~xPA(x, X p)
=
is closed under the following rule of choice w~P+IVxA(x,
We sketch the argument:
~yl...yp.y
p+l. (x,y I .... ,yp)) o
full details are long and tedious,
and
hardly instructive. Assume
~ V x ~ X hA(x, X n) .
closed.
complexity
ization theorem in (I)
p
we assume
Vx~X hA(x, X n)
Suppose all formulae in the given derivation
have logical
Tm
~ n .
HAS
H~8 f f V x ~ y ~ z
where
For simplicity,
and
by substitution
Vx~X hA(x, X n)
Then, using the formalization
to
of the normal-
(similar to § 4.4)~ ~ Tmn, p
Proofn(Y,
is class of gSdelnumbers
n,p arguments,
of
to be
~(CB(xP)~,
X p, z)
of the second-order
~(~A(x, ~P)~,
of second-order indicates
X p, z))
terms in
the g~delnumber
term with g~delnumber
z
Fm (n)
with
resulting
for
Xp
in
B(xP). Using Church's rule and combining this with lemma 4.5.9, we find a provably - recursive (2)
f
HAS-
such that
HAS ~ V x ( T m n , p ( f X ) & Sat(n)(v,
~(rA(~, xP)~ X p, fx)) .
Now we make use of the following facts. (a) For all
A E Fm (n) , and fixed
itself be shown to be (a formula
B
is said to be
holds for all
X, Vl,
(b) The replacement equal one of
in
HAS,
m - satisfiable
of lemma 4.5.7 can
for a suitable
if
m > n
Sat(m)(x, rB(~1 .... )i)
... ).
of a certain
Fm (n)
n , the assertion
m - satisfiable
second-order
term of
Fm (n)
(where equality of second-order
by a provably
terms is defined as
323
being
coextensive),
shown to be fx all
yields
a provably
m ' - satisfiable
in (2) may be supposed x,
since
(3)
to represent
V x ~ X n A ( x , X n)
was assumed
~v I ... v p . Sat(n)(z,
for any arbitrary
(a), the equality
provably
m - satisfiable
we find that, Sat(m')(V,
formula ; this fact can be
m' . a closed
Intuitively
X p, fx)
the required
~v I ~.. vpYp+l(x,
for some
m.
m' , in
v I ... Vp) .
Combining
this with
fx
is
(2) and fact
using 4.5.7
~X~
I ... ~ p ~'
X~, fx)))) .
repeatedly :
Vx A(x, kv I . . . Vp S a t ( n ) ( x z . 0 = O, ~(rX~v I . . . ~pn, X~, fx))). For
A
containing
species
variables
only slightly
more complicated.
Remark.
that,
Note
available, argument
induction
besides
if for the provably
it is irrelevant N
defined
whether
P~S
and induction
Xp
recursive
we must work in a definitional
~2(S) , with
the argument
function
extension
of
is formulated provable,
becomes
no symbol
HAS .
is
For the
as an extension
show closure under
a rule of dependent
= ~ v x n ~ z n+1 (XVl...Vn.Z(0 , v I ... Vn) = & Vy A(%v I ... Vn.Z(y,
of
or as a system with an
axiom.
One can similarly
(b),
HAS :
~(rA(~, xP) ~, X p, ~Vl...v p . Sat(n)(z, obtain,
term for
of (3) and the term with g~delnumber
for suitable
Thus we finally
second-order
to be closed.
~(rxPv I ... v p~'
Z , represents
By fact
equivalent
for suitable
choices:
X £
v I ... Vn )' %Vl "'" V n " Z(sy,
v I ... Vn))) .
Chapter V APPLICATIONS
OF KR!PKE M O D E L S
C. A. Smorynski
§
~.
~=~2~"
5.1.1.
Discussion.
semantics
In Kripke
1965 , S. ~ripke
for the intuitionistic
study this Heyting's
set-theoretic Arithmetic.
predicate
machinery
Kripke's
proofs
model
despite various
calculus.
systems,
istic reasoning.
interpretation
will certainly
calculus.
This fact,
combined
them an extremely useful
of Heyting's Before
Formally,
however,
model.
with the ease in handling
in order
the same logical
to motivate
logic of "positivistic
we consider
research"°
there is a collection
of objects
relation,
of a given
on the basis
e.g.
state of knowledge°
of a given
then,
order°
Obviously,
state of knowledge, of the given
A larger
new objects.
Also,
if it is seen to be true
state of knowledge.
properties
Further,
than atomic relations. of the logical
this property.
existential
Conjunction,
quantification
are straightforward
true on the basis of some state of knowledge Aa
constructed.
it must be seen to be true on
which preserve
such that
logic as a
may or may not be seen to hold on the
is to find an interpretation
a
of a Kripke
"states of knowledge",
construct
quantifiers
object
one of these inter-
definition
At each state of knowledge
we have mentally
an equation,
should hold for more complicated
structed
modeis,
investigation
is that of intuitionistic
may require us to mentally
the basis of any extension
problem,
the formal
We have various
into a partial
state of knowledge
let us consider
somewhat
which form themselves
basis
predicate
the Kripke
tool in the metamathematical
the Kripke models,
The interpretation
an atomic
reasoning of intuition-
Arithmetic.
defining
pretations
on
change his mind
laws are valid in the Kripke models and in the intuitionistic
makes
5.1.26
can be Tecovered.
to intuitionistic
to make it a plausible
this chapter.)
we
of
approach may seem out of
of some of the results
(The reader who disagrees
by the time he finishes
In this Chapter,
we remark in Section
theory bears no resemblance
attempts
a set-theoretic
and apply it to the investigation
Since the set-theoretic
place in a study of intuitionistic how intuitionistic
introduced
The
connectives
disjunction,
- e.g. we see
iff we have
this
~xAx
and
and to be
some mentally
con-
is seen to be true on the basis of this
state of knowledge. The other connectives that the interpretation cation
A ~B.
If
A~B
and quantifier
are problematical
loses its plausibility. is adjudged
Consider,
and it is here e.go,
the impli-
true on the basis of a state of
325
knowledge,
then
ledge and, if
A~ B A
is also true in any extension of this state of know-
is true in such an extension,
that, if, for every extension of our knowledge, we also know
B
to be true, then we know
A~B
so is
B.
A
to be true
to be true, is not at all
obvious~ but a usable condition to define the connective we accept it.
The converse,
once we know
~
is needed and
Negation and the universal quantifier are treated similarly.
The interpretations of
A, V,
and
~
seem natural enough, but those
of the more negative connectives and quantifiers are a little forced.
The
net result is that, to show that we cannot assert the truth of a statement on the basis of a given state of knowledge, we appeal not to the lack of positive knowledge - but to the fact that some extension of our knowledge contains false assertions.
Modifying the treatment of the negative connec-
tives might make the interpretation more palatable.
Such a task, however,
lies beyond the scope of this Chapter and we turn now to the formal definition of Kripke's models. 5.1.2. ruple set,
Definition.
By a Kripke model
~ = (K,~, D, I~) ' D
(K,~)
is a non-empty partially ordered
is a non-descreasing function associating elements of
empty sets, and
I~
no free variables D~'s)
where
(Kripke 1965) we shall mean a quad-
is a relation between elements of
i)
with non-
(where small greek letters denote
K) :
for if
ii)
K
and formulae with
(but which may possess constants denoting elements of the
which satisfies the following
elements of
K
A(Xl,...,Xn) ~Ib
~I~A~B
atomic,
~ ,
A(a I .... ,an) ~ then iff
iii) iv)
ikAvB iff
v)
tl- A iff
~I~A
tkA
iff
~ l ~ A ( a I ..... an) ;
~I~B;
or
~Z~(~IkA - ~lkB);
Z I tFA1,
vii) alkVxAx i f f The relation " a l k A
and
al,...,a n 6 D~,
!SZ~ " may be read " A
familiar with set theory,
" ~
forces
is true at A "
~
" or, for those
The elements of
K
will be
denoted by small greek letters and will be called nodes in order to avoid confusion with the elements of the domains of the nodes - i.e. elements of the sets
D~.
The triple
model structure (or
qms ).
(K,~, D)
is often called a qRantificational
If we restrict our attention to the proposition-
al calculus, a propositional model structure ordered set
(K,~)
( pms ) is just a partially
and a propositional model is a triple
Ik satisfies (i)- (v).
(K,~, I~) , where
326
As in classical model theory, one may define a notion of validity : A
will be called valid in the model
be called valid
(universally valid)
More generally, if A
r
iff A
~I~A
for all
FI= A ,
iff
We shall prove later on :
~6 K.
F
entails
iff
will
~.
A , written
is valid in every model in which every formula of
rt-A
A
is valid in every model
is a set of formulae, we say
valid.
5.1.3.
K if
~
is
rt-A.
Some basic properties of Kripke models.
Before giving some examples of Kripke models, let us remark on some of their basic properties. forcing relation
The first is that conditions
(it) - (vii) on the
I~ constitute the recursion clauses for an inductive
definition of a forcing relation on a qms.
In particular,
if we specify
which atomic formulae are forced at which nodes of the qms (in such a manner that (i) holds), then the relation extends uniquely
(by using clauses
(it)-
(vii)) to a forcing relation on that qms. A second remark is that the first condition on atomic formulae specifies a property that holds for all formulae. B I~A.
I.e. if
~ I~A
and
~ ,
then
The proof of this is by induction on the length of a formula.
atomic formulae, A = B & C.
the result is immediate.
Let
A
be a conjunction,
say
Then
8 I~ B ~ B
I~C,
by induction hypothesis,
Disjunction and existential quantification are handled similarly. cation, negation,
A = B~C
For impli-
and universal quantification, we use the fact that we have
required our condition defining let
For
and
~I~A
to hold for all
8~.
For example,
8 ~ ~.
since Negation and universal quantification are treated similarly. A final remark is that the truth of which are
only to those for
<
~6 K
and
~ I~ A
depends only on those
> ~ - each clause in the definition of the forcing relation refers 6!~.
Let
K
be a model and define
by :
D~
are the restrictions of
<
and
D
to
K ,
and
H-~
is
defined
327
by letting
8 I~ A
iff
expect that, for any
~ I~A , for
A,
~I~A
in
simple induction on the length of in
K
for all
those
~>~
5.1.4.
BE K~.
Thus,
A
A
atomic and
_K
iff
~I~
shows that
to verify that
(i.e. we may restrict ourselves
Examples.
in
8 I~A ~I~A,
•
We should
_K . Indeed, in
K
iff
a
~ I~A
we need only look at
to the model
K
) .
Let us first consider examples of models for the proposi-
tional calculus.
We indicate
the model by drawing a graph,
which determine nodes of the model.
A node
ordering if the vertex corresponding
to
~
~
the vertices of
precedes a node
~
in the
is connected by a series of
ascending lines to the vertex corresponding
ff
~E K A
to
~.
E.g.
~< 8
in the pms :
•
We indicate the forcing relation by writing atomic formulae next to the nodes forcing them. letting
~I~A,
E.g. using the pms just given, we obtain a model by
~ I~A,B
:
13I A'B ~
A
Observe that, in the model just given, but
C¢I~B,
(iv)
whence
~I~
~ I ~ (C'~D) V (D-~C)
-I-~B--B ;
(i)
(iii)
for any formulae
~ I~B V ~ B ; ~
(ii)
~ I~ -~ ~ B ,
forces any tautology;
and
C, D.
One can get more complicated models by allowing the graphs to branch :
For the quantificational to draw models for classical tionistic
theories.
theory, we must add domains. theories,
For simple cases, however,
we may indicate the domains
by listing their elements at each vertex of the graph. {a,b}
1
tal
~e may use this qms to construct a model :
{a,b} A,~a
l
Just as it is hard
it will be hard to do this for intui-
E.g. :
328
Here
A
is a propositional
sentence•
Observe that
~I~Vx(AVBx)
, but
A v VxBx As one may easily verify,
the formula
all models with constant domains function),
V x ( A V B x ) -~ A V V x B x
(i.e. models in which
where we again assume that
x
D
is valid in
is a constant
does not occur free in
A.
(It
is known that this class of models is complete for intuitionistic logic with this scheme added.
Cf• Gabbay 1969 A or G3rnemann 1971. )
Another interesting classically valid sentence which is not intuitionistically valid is
m mVx(Ax V mAx) .
Consider the model :
I IO, I, 21
A0, Aq
l
Io, ll I Iol
Ao
loe. we have a sequence ~ml~An
iff
m>n.
~o<~I<
Suppose
...
~
of nodes with
I~ m m V x ( A x V
D~ n - IO,...,n}
mAx) .
q-heo ~
and
~ c~o
O
IN mVx(Ax v max) . ~>~o
In particular,
~ I~Vx(AxV mAx) .
~n l~An
or
~n+1 I~ A n " that
~n I~ m a n .
Let But
~ s ~n" ~n I N An
It not only follows that
do I N m V x ( A x v m a x ) . Letting
xs n,
But then
~n I~ A n V m A n
by definition and
,
~n I N m A n
Re I N m m V x ( A x v m A x ) ,
i.e. since
but, in fact,
~o I~ m Vx(Ax V m A x ) .
~Vhen we have a classical model,
e.g. the standard model,
•,
of arith-
metic, instead of listing the domain and the atomic formulae to be forced, if we wish to force those atomic formulae true in the model, we simply place an
w
at the vertex.
E.g. if
~+
and
w'
are non-standard models of
arithmetic, we will write + (D t
W
for the intended Kripke model. We could continue to give several further examples of Kripke models, but feel it would be more instructive for the reader to construct some of his own.
E.g. he may wish to construct countermodels to
((A-~B)-~A)-~A,
(A-~B) V ( B - ~ A ) .
mA V m mA ,
We should like to stress that he should
pay close attention to the geometry of his countermodels. the Kripke models is the basic tool used in this Chapter.
The geometry of
529
5.1.5-5.1.11. 5.1.5.
The completeness theorem.
So far we have constructed a model theory for the intuitionistic
predicate calculus and used this model theory to demonstrate the failure of certain basic laws of classical logic which are not intuitionistically valid. It is now our job to demonstrate how closely the model theory fits intuitionistic reasoning.
Formally,
5.1.6.
(The completeness theorem.)
Theorem.
The proof of soundness ,
the fit is exact :
F ~A
implies
F ~A
F I=A ,
iff
~ I= A .
is long but easy.
One
merely has to show that each axiom is valid and that th~ rules of inference preserve truth.
E.g. consider the rule
~ = (K,~, D, I~)
is given and
by the definition of
~I~A~B,
~EK
PL2:
A,A~B
is such that
it follows that
~B .
If
~I~A,
~I~A~B,
~I~B.
Hence,
then,
this rule
is sound. The more ambit~us reader may prove the soundness theorem for any of the formulations of the intuitionistic predicate calculus given in Chapter I. We now turn to proving the completeness theorem. A
iff
~ A , is due to Kripke 1965.
The weak form,
The form we shall prove, often called
a strong completeness theorem, is due independently to Aczel 1969, and Thomason 1968. Thomason's treatment.
1968 , Fitting
For the sake of subsection 5.1.26, we shall follow
These proofs are modelled on Henkin's proof for
classical logic. Let
M
be a first-order language containing
i)
a denumerable
set
VM
of individual variables;
ii)
a denumerable
set
CM
of individual
iii) for each
j~O
, a denumerable
set
constants,, F~
of
and
j - ary predicate letters.
Formulae are to be built up from atomic formulae by using and
V.
Fm N
will denote the set of such formulae.
&, V, ~, ~, S,
Note that
F~ i
is
denumerable. Sn~ will denote the set of sentences - ice. the formulae with no free variables° 5.1.7.
Definition.
i)
F
ii)
A ~ S,~
and
iii) A , B E S n ~ iv)
if
A set
F~Sn M
is called
M - saturated if
is consistent ; F ~A
& AVB6
Ax 6Ym~i,
then, for some
x
= A E F ; F =A6
F
or
B 6 F;
and
is the only free variable in c 6 CM ,
A
and
~xAx 6 F,
Ac 6 F.
Those familiar with the algebraic representation theorems may consider a saturated set utive lattice.
r
to be a sort of counterpart to a prime filter in a distribBasically~
these prime filters will yield nodes of a model
and their inclusion relations will yield an ordering.
}~tters are slightly
complicated by the necessity of introducing new constants to successively enlarge the domains.
53o
5.1.8.
Lemma.
Let
F U {A} ~ n
be a denumerably infinite obtained by adding is an
{01, c2, ... }
M' - saturated
Proof.
Set
Case 1.
Fo = F
k
not already treated
Case 2.
superset
Let
= rk U { B I
Finally,
Let
F
CM
such that
inductively
Fk ~ Z x B F k.
B VB,
and let
{ci, c2, ...}
and let
of
M.
MI
be
Then there
A~ F.
as follows :
c
Then set
sentence of
M'
be the first constant in
~k+1 " Fk U {Bcl.
be the first disjunctive
~k b B VB' .
Otherwise
set
Let CM
be the first existential
not already treated such that
rk+ I
of
~B
such that
is odd.
F~A.
to the constants
Fk+l
net occurring in
k
and suppose
F
and define
is even.
{c I, c 2, .-.I
M
set of symbols disjoint from
If
formula of
F k U {BI ~ A
,
M'
pul
rk+ ~ = F k U IB'}.
F~ , k ~ o F k .
We must show that
F~
satisfies conditions
(i) - (iv) of 5.1.7 above. (i).We show by induction F2n+1 = F 2 n U {Bcl F2n. But
Thus
F2n,BC ~ A
,
PL5
k
k>i,
Fk ~ A .
that
(iii) , (iv).If some edd
~
BVC B V C
Fk+ 1 = F k U {B}
whence
Let
where
c
F2n ~ B o ~ A
or
(ii)' .
5.1. 9 .
and, by Q4,
F2n ~ S x B ~ A
.
But
F
~A
iff
F2n+2 ~ A ,
Fk ~ A
then
for some
k,
from
~A. E F,~
then
~i ~ B V C
for some
i.
is the first disjunction not treated. F k U {C}
i.e.
B6 F
or
CE F
(D
Be E Fw for some c F w ~ A , then Fw ~ A VA a n d ,
.
Hence, for Thus
Similarly, if
0D
then
If
Theorem.
K = (K, <, D, Ib) ,
Z : IA In fact,
Then
a contradiction.
'
ZxB 6 Fw
F2n+1 ~ A .
does net occur in any formula of
allows us to conclude that, iT
F2n+l ~ A . Hence, for all which it follows
Fk~A.
B, c
F2n ~ A ,
F2n ~ Z x B , whence Similarly,
that
for some
~
If
_F
is
)~-saturated,
and for some
by ( i i i ) ,
A6 F w.
Q . E . D,
then for some Kripke model
~6 K,
a l~A}
may be assumed to be a minimum element of
K °
Proof. Let Mo = N and let ~li+I be obtained from ~.l by adding the set S ~ i+I c i+n 1 = i = ICl ' "" " ' n ' "°° to CN[i , where S i O C~i ~ . Set K = IA:FcA and A is )~.-saturated for some il. We define A < A' iff 1
A_~ A' , DA = C]~i , where A(c1,..o,Cn)
a
with
A
is
}~i - saturated.
for atomic formulae
c 1,°°°,cn~ CMi , let
IbA(ct,...,on)
iff
A(c I . . . . .
Cn) ~ A .
We wish to show that this last equivalence formulae
Finally,
(i.e. formulae with no free variables
the proper language).
holds for all applicable and whose parameters
For this, we need the following
are from
33~
5. I. 10.
Lemma.
(a)
B~CE
(~) (c)
A6 K o
V~'_~A
~B~
if~
W'_~
VxBxE a
iff
VA'_~A
tion,
A
Let
iff
Proof.
A
(a) If
(BE A' = C E A') B¢~'~ V c e DA'
B - ~ C E A,
C E A' .
B c 6 A' .
B E A' , and
Conversely,
suppose
A' .~A, B~C{
A.
M.l- saturated, there is, by lemma 5.1.8, an
such that
C { A' .
A' ~ C
then Then
and, by satura-
U {B} ~ C
~i+I - saturated
This contradicts the assumption
and, for A' D A U {B}
B E AI = C E A' .
(b) Similar to (a). (c) Again, one direction is trivial. VA' D A
Vo6 DA' BC E A'
A~VxBx,
and
VxBx ~ A.
we conclude, by
Q I, that
there is an
~i+I - saturated
A'~A
Suppose, conversely, Let
A
A ~Bc
be for
such that
that
~i - saturated.
Since
c E CNi+I - CNi °
A' ~ B c
, i.e.
Hence,
B c { A' .
Q. E. D° We may now complete the proof of theorem 5.1° 9 by proving by induction on the length of
A
that
a I~A(c 1 .... for
,
Cn)
c I, .°., cn 6 C~i , where
follows by definition.
A(e I, ..°, Cn) ~ A ,
A
is
The case
IbB
iff
Let
iff
iff
B E ~
iff
BVCE
M i - saturated.
A = B&C
The case A is atomic
is trivial.
Let
A = BVC
:
or or
CE A ,
A,
by induction hypothesis by saturation.
A = B-~Ciff
The cases Let
A = ~B
iff
~A'~A(B
iff
B~CE
and
VxB
E A' ~ C 6 ~') , by induction hypothesis
A,
by lemma 5oi.10.
are similar°
A = ~xBx :
lW
Bx
iff
Zo D
iff
~c6 DA B c 6 A ,
IWBe by induction hypothesis
iff
~xBx E A,
by saturation°
This completes the proof.
Q.E.D.
We may now complete the proof of the completeness theorem.
5olollo F ~A .
P.roof o.f theorem 5oio6. Let
F ~A
theorem 5oio9, there is a model all
B ,
We have yet to prove
and find a saturated
f_DF
F I=A
such that
K = (K, <, D, I~ ) and
implies
f ~A
~E K
o
By
such that for
332
In p a r t i c u l a r ,
~I~B
5.1.~2-5.1.~8. 5oI.12o model
The Aczel
and
and a node
The converse, ~
~
~l~A.
Hence
E-saturated
set
Q.E.D.
r, there is a Kripke
such that
~t~-A}.
that every such set is
by extending
easy v e r i f i c a t i o n
rl4A.
slash.
By theorem 5.'1.9, for any K
r o {A: from
B~r
for
CM
I~'-saturated,
where
~'
is obtained
to include names for all elements of
w h i c h we leave to the reader.
we can obtain more i n f o r m a t i o n
on
~-saturation
As observed
D~,
is an
in Aczel
1968 ,
from the proof of t h e o r e m
5.1.9 than just this. Observe element
that
in the
structed
r = IA: ~ I~AI pms
for some
constructed.
and add a n e w node
Thus,
implies
that
~
is a m i n i m u m
let us start with the model
K
~ < ~ for all ~ 6 K , let o o-D ~ o = CM , and extend the forcing relation by defining, for A atomic,
%I~A The Aczel
slash is defined by
define
iff % I ~ A l(r) = IA : F ] A1 .
5.1.13.
Lemma.
Proof.
CI ear.
5. I. 14.
Theorem°
Proof.
Let
implies (i).
If
AE
I(F)
I (~)
a)
E-saturated
and
is a maximal
I(F) ~
~ - saturated
,A
{A: F ~ A I
.
subtheory of
N-saturated.
W e show that
I(F) o
A
is atomic, ~ F~A
, (iii). A = B & C ,
(iv). Let
is
I(F) c_ A_c {A: F ~ A I
A~A (ii)
such that
iff r ~ A
r IA Also,
~
~
~A6
I(F) .
BVC.
These cases are trivial.
A = B-*C:
l(r)~-Bo Then ~ B
and so
But % A' D F 8~d ~o B 6 A~ = C E A' (v) ~ -,15 . S ~ l a v ~.o ~iv). (vi) . Let A = ZxSx : A ~xBx
= ~ a E CM A ~ B a
,
~bc.
Th~s l(r) l - c .
a contradiction.
by
M - saturation
F. A~A
con-
353
b ~Bx ~ Ba ~ I(r) A = VxBx.
(vii) 5.1.15.
Corollary. F
is
Q.E.D.
Similar to (iv). Let
F
be closed under deducibility.
M - saturated
iff
F= I(F)
if~
r~
Then
I(O.
5.1.16.
CorollarM.
The intuitionistic predicate calculus is saturated.
5.1.17.
Corollar[.
AIA
in the sense of Aczel
iff
AIA
in the sense of
Kleene (cf. § 3.1). 5.1.18.
Theorem
definition). (i)
(Characterization of the Aczel slash by an inductive
The relation
For atomic
FIA
is inductively defined by the following :
A
rlA if~ r b A ; (ii) (iii) (iv) (v)
r pB~c r lBvc r lB--C rl~B
(vi)
F I SxAx
iff rib and rlc~ if~ rib or rlc~ iff r b B - C and (rIB~rlc) l iff r~ and r ~ S , iff P I Aa for some a6 CM ;
(vii) F l VxAx Proof.
iff
(i)
(iv) Let
P b VxAx and
by definition ~
riB--C,
i.e.
F}Aa
for all
(ii), (iii), and (vi) are obvious.
~o I ~ B ~ C "
Then
rIB=r}c. Since I ( r ) ~ r , r b B - c . Conversely, i f r X B - c , i . e . % I~ B - c , ~O l ~ C or
A~B, A~C only be true i f r~B-- C;
a6 CM .
~o l~B = C~ol~C'
the~ either
f o r some s a t u r a t e d A ~ F . the former if FIB , F X C°
i.e.
% I~B
(v) and (vii) are similar. 5.1.19- 5.1.21. 5.1.19 .
The operation
and
The l a t t e r can Q.E.D.
( ) ~ (Z)' o
The Aozel slash, like the Kleene slash, may be used to prove satu-
ration results (often called explicit
definability results).
In Aczel 1968,
Aczel used the inductive characterization (theorem 5.1.18) to give a version of Kleene's slash- theoretic proof of the
E D - property for
HA.
However,
our interest in this chapter is primarily in the model theory and in model theoretic proofs.
Thus, let us ignore theorem 5.1.18 and reconsider what we
did in proving theorem 5.1.14.
334
The proof of the completeness theorem involved our constructing a model of a theory
F.
We observed
(i) that
F
is saturated iff it is the set of
formulae forced by a minimum node of that model, and (ii) that, if we added a minimum node, we got a maximal saturated subtheory of
F.
We shall gener-
alize the model-theoretic construction of (ii). Let
~
be a Kripke model and let a language
of constants be given.
We will let
obtained by adding a new node
~
K' to
M
with a non-empty set
denote any model K
CM
(K',~', D', ID)
such that
O
(i) (ii)
% ! '~ for a l l D'4o=c~;
(iii)
i~
Then, f o r
~K;
A is atomic, ~eK
4 0 ID'A =~IPA
and any formula
A,
~ID'A
foralZ ~ K . iff ~lffA.
of special
in-
terest is the case in which the implication in (iii) is replaced by an equivalence°
This is the case we most often encounter.
By theorem 5oio14, if the class of models of a theory the operation
K~K'
, then
proof of this shortly. Let
zZ,
F = IK
U
M - saturated.
F
is closed under
We shall give another
be a family of Kripke models,
The disjoint sum,
! - (K,!, D, ID) defined by
(i)
K~
(ii)
( ~ , ~ N-( ~,~)
(iii) (iv)
D(~,~):DJ;
K ~I~j~
for atomic
E.g. suppose
is
First we must introduce another operation on models.
: ~ E N}
of the model
F
iff
A,
~= v
and
(~,~)iDA
4<
~ ;
~Ib~.
iff
is the family consisting of the following models (where
cM= la}), fa,b} --K1 : ] 4 ta} Then
ZF
Pa
~ --K2" 4
Pb
is the model :
(~,1)
t~,bl
(4,1)
I fa}
~I + ~2 =
Pa
(~,2)
I~,bt
(4,2)
I ta,b~
The relation (iv) in the definition of hold for all
ZF
Pb
may be shown by induction to
A :
(4,~I LD* Remark.
{a,b} I {a,bt
iff
Alternatively,
~ID~ 4. we may use the final remark of subsection 5.1.3 to
prove this without another induction.
335
If
~
is a family of Kripke models, we can apply the two operations
successively:
F ~ Z~ ~
ta,bl I faf
(Z[)' .
E.g. for the family
Pa
{a,b} I ta,bl
[
given,
(Z[)'
is:
Pb
o
5.1.20.
Theorem.
the operation Proof.
Let
Let the class of models of the theory
~ ~ Ax
(Z[)' . Then
contain only
Then, for each
a 6 CM ,
node, say
and
Let
~a'
F
be closed under
M - saturated.
free and let, for each
we can find a model
and let
Then, for some
~a } ~ A a ,
x
is
~&
a 6 CM ,
such that
~a
F~Aa. has a least
~a I ~ A a "
~ = I~a : a 6 CNI
~o I ~ x A x "
F
~o
be the least node of
a 6 CM = D ~ o ,
~o I~ A a "
But
(ZF)' .
Suppose
~ > ~ a-- o
and so
a contradiction.
(Recall that forcing at
~ a in ~ & is the same as that in (Z~)'.) Disjunction being handled similarly, we have the required result. Q. Eo D. Observing that the class of models of
~ Z~, 5.1.21.
Corollary.
under the operation Remark.
r
is closed under the operation
we have the immediate Let the class of models of the theory K ~ K'.
Then
F
is
~
be closed
N - saturated.
The difference between using theorem 5.1.14 and theorem 5.1.20 to
prove that
F
is saturated is that, to apply theorem 5.1.14, one has to
show that a particular model
~
of
~
yields a model
~'
of
F,
~
of
F,
theorem 5.1.20 requires one to show that, for any model a model of
F.
M o d e l - theoretically,
while ~'
is
both tasks should be equally difficult.
Theorem 5.1.18 makes the first task easier - but, it is the second approach that we will find more useful. 5.1.22 - 5.1.23. 5.1.22.
Models with equality°
In working with Kripke models, one may treat equality as a binary
relation satisfying certain axioms.
One doesn't always have the option one
had in classical model theory to assume that equality is interpreted by actual identity - if equality is interpreted by identi~y, Vxy(x=y
V ~x=y)
then
is forced - but there are intuitionistic equality relations
which are not decidable
(e.g. the equality of the reals).
When equality is decidable, however, it suffices to consider the class of
336
normal models - i.e. models in which the equality predicate is interpreted as actual identity. 5.1.23.
Theorem.
Let
r
have a decidable equality.
Then
F
is strongly
complete for the class of models in which the equality of two constants is forced iff they denote the same object. Proof.
Let
~ = (K,!, D, I~ ) be a model of
spending normal model
Let
~m
[x] : {y : x m y }
F.
~e shall define a corres
by using the following equalence relation
be the equivalence class of
x
under ~ .
~KD~
Define
:
f
by
(ii) i n = i (iii) Dn~ = {[x] = x~ D~}, (iv)
for atomic
A,
and
~l~A([al]
.... , [an])
if
[~A(a~ .... , a') , where a~ ~ a . and a~ 6 D ~ . 1 1 l nBy the standard induction on the length of A , the equivalence seen to hold for all
(iv) is
A .
Q.E.D.
Since Heyting's arithmetic has a decidable equality, we shall, in the sequel, only consider normal models. 5.1.24.
Function symbols.
Another device we could use is function symbols.
While we can show proof - theoretically that function symbols are eliminable, we cannot conclude from this that the theories determined by the classes of models with and without functions coincide.
(To do this, we would have to
prove completeness of the theories possessing function symbols with respect to their models possessing functions.)
We shall, therefore,
indicate the
m o d e l - theoretic proof of the eliminability of function symbols for the special case of a theory with decidable equality. Let
F
symbols°
be a theory with the language
choosing a family ef functions f
M
An interpretation of the symbol
: (D~) n ~ D ~
and, if
~!B,
and let f
{f~ : ~6 K1 f~r D~ = f
M
in a model
possess function ~
such that (if
.
is given by f
is
n - ary)
The interpretation of atomic
formulae involving terms constructed by the use of such function symbols is handled as in classical model theory. Suppose we now replace
M
by a language
function symbol is replaced by an § 1.2o
If
F,
M'
in which every
is obtained by translating the axioms of
adding the function axioms,
n - ary
n + 1 - ary relation symbol, as discussed in r
into
M'
and
then, just as in classical model theory, there is
a natural correspondence between models of
F
and models ef
F' °
33?
This is proven by mimicking the classical proof.
Thus we may restrict
our attention to models with functions replacing certain relations their graphs)° 5.1o25o
(namely
The details are left to the reader.
Conventions.
concerning models of
Let us finally make, in addition to our convention HA
that they be normal, a convention that they do not
possess functions and the simplifying convention that they all possess minimum (or least) nodes, which we shall call origins. usually be denoted by ~E K o
~
An origin of
and has the defining property that
O
(Observe that such models are not closed under
F~ZF
~
~ < ~
O--
will for all
and, hence,
we shall have to apply theorem 5.1.20 rather than its corollary.) 5.1.26. What,
Intuitionism ? one may ask, does all of this set-theoretic machinery have to do
with i n t u i t i o n i s m ?
We shall not attempt to answer this question - instead
we merely outline how certain proofs obtained by the use of this machinery can be transformed into intuitionistically meaningful proofs°
(See e.g.
Nints 1969.) The key to this transformation lies in the H i l b e r t - Bernays completeness theorem (cf. e.g. Kleene
1952), by which certain outwardly set-theoretic
constructions may be replaced by arithmetical ones.
Specifically, by arith-
metizing the completeness theorem for classical logic, one can show that, for any r.e. theory
T,
if
Con(T)
is added to classical arithmetic,
then
a provably arithmetical model exists - i.e. there is a model with an arithmetically definable domain and arithmetically definable relations such that the translations of the axioms of
T
are all provable.
The same is true of the completeness theorem given above (especially in the treatment by Thomason).
Thus, if we use the completeness theorem to
prove (say) an independence result, we can prove the result in classical arithmetic augmented by some consistency statements. results of this chapter.
If, in addition,
This is true of all the o the result is H 2 (e.g. as in the
case of an independence result), we know from a previous chapter that the proof in the classical system can be transformed into a proof in the corresponding intuitionistic system. We shall not prove this result here, however,
since most of the results
we give can be obtained constructively by less devious means and since the only results which we need for our classical proofs are (i) the existence of arithmetically definable models for any r.e. theory (intuitionistic or classical) and (ii) the fact that the models are provably arithmetical if we add the statement of consistency of the theory to classical arithmetic. classical theories,
this is the H i l b e r t - Bernays completeness theorem.
For
338
For intuitionistic theories this almost reduces to the Hilbert- Bernays completeness theorem as follows : Observe that a Kripke model is a classical model when viewed as a structure in its own right. and
~,
let
M'
be obtained by replacing each atomic formula
by a new formula K(~) , and symbols°)
~
That is, given
A(~, x1,...,Xn)
~o
~, F,
A(x~,.°.,Xn)
and adding new atomic formulae
D(~,x),
(Let us assume for simplicity that there are no function
The relation
~I~A(x1,o..,Xn) .
A(~,x1,o..,Xn)
is to be interpreted by
We then translate all statements about
~
into
M'
as
follows : (i)
for
A
atomic,
(it)
( ~ ] ~ A & B ) T = (~ I~A) T & ( ~ I ~ B ) T ;
(iii) ( ~ I b A VB) T
(iv) (v) (vi) (~ii) We define
=
(~IbA-B) ~ (°I~A) T (~ I~ ~ ) T ~ (~ IP V~x) T = F~
(~ l~A(x I ..... x~) T = A(~,x I ..... Xn) ;
(~l~i) TV(~ItB)
by taking, in addition to axioms asserting that we have a
Kripke model (e.g.
(~I~A)T&~!8
axioms
Then
A
of
T
v~!~((~ I~A) (~i~) ~), V ~ > ~ ( B IbA) T ~(~(~,x) a (~ l~Ax) ~) , V~Z~ Wb(~,x)-- (~ IbA~) ~] °
F.
metical model of
F
~
-- (~ I~A)T), the axioms
is r.e. iff
from one of
F'
~'
(~I~A) T
for
is r°e. and we obtain an arith-
The only problem at this stage is
that the provable arithmeticity of the models depends here on the consistency statement for
F'
rather than for
cause us no trouble.
F°
However, this loss of precision will
339
5.2.1-
5.2.4.
5.2.1.
The o p e r a t i o n
( ) ~ (Z)'
So far, aside from specializations of the form of the models used (to
being normal,
to not having functions,
and to having origins),
the only
results which we have proven concern saturation or explicit definability. The result we wish to apply first to Heyting's arithmetic is theorem 5.1.20 which implies that, if we show the class of models ef the operation 5.2.2.
Theorem (Explicit definability).
F-~Ax, 5.2.3.
th~n
th~n
~An
for some
to be closed under
or
If
Ax
has only
x
free and
n.
Theorem (Disjunction property).
~ A
HA
( ) ~ ( Z ) ' , then we may conclude the following
Let
A,B
be closed.
If
HA ~ A V B ,
~B.
To prove this, we shall have to choose a formulation of one fcr our purposes is the one with constants number, relations
S(x,y), A(x,y,z) , and
O, I, .°.
M(x,y,z)
~.
The simplest
for each natural
defining the functions
of successor, addition, and multiplication° TypOgraphically, letters
n, m
we find it convenient to reserve in this chapter the
(possibly indexed)
chapters, where
n, m
to denote numerals
usually stood for numerical variables,
were written with a bar : ~, ~, ~, y, °o° The axioms of
HA
are, in addition to the axioms of the predicate calcu-
~S(x,O) , ~x
= 0 ~
~y S(y,x)
,
s(x,y) ~ s(x,z)
- y = z,
s(y,x) ~s(z,x)
- y= z,
s(x,y) (ii)
A(~,y,z) ~A(x,y,~)
-
z=w,
ZZ A ( X , F , Z ) , ,
i(x,O,x) (±ii)
A(x,y,z) ~ s(y,w) ~S(z,v) -. A(x.w,v) ~(x,y,~)aM(x,y,w) - z:w, Zz ~i(x,y,z) , M(x,O,O)
,
M(~,~-,z) ~ s ( y , w ) (iv)
S(n, n+1) ,
and numerals
etCo).
lus with equality : (i)
(in contrast to the other
ti(z,x,v)
- M(x,w,v)
for each constant
n ;
340
and the scheme, for any formula not include
(v)
A
whose free variables include
x
and do
y :
A0 ~Vxy(Ax & S ( x , y ) ~ A y ) -- V ~ x .
Aesthetically,
it is more pleasing to use a formulation with function
symbols and, as shown in 5.1.24, we may do so.
However,
that would require a
little more care in defining various structures and a little more work in proving results about them°
We shall, occasionally, however, freely use the
fact that there is a natural correspondence between models of our official system above and the system with function symbols (or, if one prefers, we shall abuse notation by using function symbols). Our first step is to prove the following 5.2°4.
Theorem.
The class of models of
HA
is closed under the operation
( )-(~ ), Recall that, in the definition of
K--K' , we left open the problem of
deciding which atomic formulae to force at ~e I ~ A
iff
~I~A
for all
~ 6 K°
~o ' stating that we usually have
(Recall also that the proof of theorem
5.1.20 merely required us to have some model of the form
For ~ ,
K' .)
there is no ambiguity - closed atomic formulae a~e decided by the theory and, if
~'
is to be a model of
HA,
we must have
~o I~A
iff
A
is true in
the standard model. Thus our operation all nodes of
ZF,
~
(Z~)'
setting
D~
is given by tacking on a new node = {0, I .... I
A
then (using the graphic representation of subsection
HA
of each model in
The assertion that
~
is a model of
is valid in every member of ~).
For
(ZF)'
since then
~E K
for some
F
K 6 F
~o
(Z~)'
is a model of
each axiom
A
HA.
of
The only non-trivial ity, we assume that
Ax
H~,
means that
~,
some node
Obviously, we cannot have
(making the obvious identification -
i.e. ignoring the operation used to make members of prove that
~
(ioe~ forced at each node
not to be a model of
(Z_F)' must fail to force some axiom of
> d° ,
are non-standard models
is
Proof of theorem 5.2.4.
of
~*
below
A .
every axiom of
and
iff
is true, for any atomic
(~++ ®*)'
~+
~o
~o !~' A
of classical arithmetic, 5.1.4)
E.g. if
and letting
F
disjoint).
it suffices to show that
case to consider is the induction axiom. has only the variable
x
free°
Thus, to
~o I~ A
for
For simplic-
The general case is
left to the reader (i.e. we let the reader verify the validity of the universal closure of the scheme with free variables)°
341
Let
~o I ~ A O & V x y ( A x & S ( x ' y ) ' A Y )
B I ~ A O & Vxy(Ax ~S(x,y)--Ay) , but since then
~E K
for some
" VxAx.
Then, for some
~ I~VxAx .
~E~
and
~
~>_%,
Now we cannot have
forces all axioms of
~> =o'
HA.
Hence
~O I~AO ~Vxy(Ax ~S(x,y)~Ay) , but ~O I ~ V x A x ° Since =o I~VxAx' there is some ~ o and some b E D~ such that ~ I ~ Ab . Again ~= ~o and h is some natural number. m
Let
is a successor, say
that
~o I ~ A b '
m
be the smallest such number.
n+1 , and, since
~o I~An"
But
~o I~ T M ~ S ( n ' n+1)--A(n+1) . diction°
m
Since
~o I~ AO '
is the smallest number
b
such
~o I~Vxy(Ax ~S(x,y)--Ay) , whence
Thus
@o lffA(n+1) ' i.e.
=o I~ A m '
a contraQ.E.D.
Theorems 5.2.4 and 5.1.20 immediately yield theorems 5.2.2 and 5.2.5 as corollaries. 5.2.5-5.2.7. 5.2.5.
Appliqations of the operation
( )~ (Z)' o
The closure of the class of models of
HA
under
( )-- ( Z ) '
of the basic tools of the Kripke model approach to studying have already used this to prove
l~A.
is one
E.g. we
ED , the explicit definability property.
Its use here is simply that it allows us to take countermodels to and put them together to construct a countermodel to
AO, AI, ...
ZxAx o It is in this
construction of models that this operation is so useful.
Consider, e.g.,
the old result of Kreisells (Kreisel 1958) : 5.2.6°
Theorem.
Let
Ax
have only
denoting derivability in ~ VxAx V ~ - ~ A x
~.
iff
(Cf. also
~ @ 5
~y[~VxAx-~-~Ay]
and
and the
~
~O
~Ax) ,
Ay]
We shall show that
~ -~ VxAx-~ Xx-~Ax
whence
AyJ ~
Suppose
Then, ~Ano
implies
for each
By the decidability of
DP , ~ A n .
On the other hand, Let
[ WAx
y[ VxAx--
~-TAn-~ (-IVxAx---~An) ,
~Vx(AxV
~ -l VxAx-~ Zx-~ Ax
and leave the rest to the reader.
VxAx But
)o
free and suppose
Then
iff Proof.
x
K
~VxAx-~[~VxAx-~-~AO] ,
be a model of
E D~
~ I~Ab o
Now consider especially
K8
~8 ), : K
l %D
HA
with
~E K
By decidability
and so
such that '
~VxAx.
~ I ~ VxAx. ~ I~ -~Ab
Then
and hence
(recall the definition from subsection 5.1.3) and
A
n
.
342
Observe
that
~d
Y I~ m V x A x .
implies
°
Also,
y= d
~o I ~ VxAx
or
since, if
one has a contradiction.
Thus
We n o w use the fact that
~ mVxAx~xmAx
D ~ o = IO, I .... I 5.2.7.
y> ~
n,
for all
Let
of models.
~ = (T,~)
~
~o I~ ~ A n ,
implies
then ~e
to conclude
~ I~ VxAx
and
d°
( )~(Z
let
Ter
)'
in the direct
By a terminal node of the t r e e we shall
denote the set of terminal nodes of of
T o
For any node
w
if
dE Ter,
if
sCTer, ~(d)=(~S(d)~(~))'
5.2.8.
Theorem.
Preefo
We
~T =~(so) ~T
for terminal nodes. that
completes Note.
~(d)
Obvicusly,
A mA ,
If
we may replace
d
is the origin of
~(d)
T
is a model of
is not terminal, HA
for all
arithmetic.
respectively.
~.
The t h e o r e m
apply theorem 5.2.4°
dE K.
Letting
restriction
on
~
d= o Q. Eo D° by the
d O a °
I~AVmA
theorem that there is an i n d e p e n d e n t Thus there are models
Associating
o
that
the f i n i t e n e s s
we k n o w by G o d e l t s
of classical
we have the model :
e.g.
to each
We n o w associate
~.
that
is a model of
of the tree,
Observe
arithmetic
~E Ter.
restriction.
As an example,
and
to
the proof.
well-foundedness
sentence
d
is a model of
show by bar i n d u c t i o n
It follows
dE T - Ter ,
(viewed as a one-node Kripke model) ;
where
is trivial
shall
as follows :
(li)
define
~(d) = w d
~(s)
(i)
Finally,
A
is assigned
a Kripke model
We
do
Let us assume that we have assigned models of classical of the terminal n o d e s - say sET
But
W e turn our a t t e n t i o n n o w to this task.
be a finite tree.
will denote the set of successors
with each
o
~. E. D.
mean a maximal node of the tree - i.e. a node with no successors.
S(d)
and
I~VxAx.
s° l ~ x m A x
a contradiction.
We have not r e a l l y used the basic operation
construction
~) ~
s o I~ V x A x ,
~ l~VxAx
and, for some
and that
.
wI
and
~2
of
these models with the terminal nodes
343
A stronger version of G~del's theorem allows us, for any sentences tic. in
AI,...,A n
n= 3
iff
to find
O
ZI
which are mutually independent over classical a r i t h m e A0
is true
(We shall discuss this further in secticn 3.)
Letting
In particular, ~i
n,
we can find models
i = j.
and relabelling
At, A 2, A 3
as
w1,.o.,~ n
A, B, and
such that
C , let
w1' ~2'
and
~3
be
associated with the terminal nodes of the tree
T : O
Then
~T
is
~I A
w2 B
~3 C
O
% Iff~(~A~Bvc)~ ((~A--B) V(~A--C)).
Observe that
(Sse chapter m:,
section 2 ~ 6 for an application.) Let
w1' ~2'
and
w3
be as in the preceding example and let
T
be:
\o) f 0
Associating
ml, m2,
and
~4~
~3
with
~I' ~2'
and
~3 , we have
~2~,/~~3
O
Observe that, although
d°
and
~4
both have copies of
them, they de not behave alike, e.g.
w
associated with but
% 1}+ ~A, ~ ~ (~vc) 5.2. 9 - 5.2.12. 5.2.9.
If
£
Formulae preserved under
( ) ~ (Z)'.
is a set ef sentences, we may ask whether or not various meta-
mathematical properties of
HA
also hold for
I~A+ £.
For instance,
ask whether or not the explicit definability theorem holds for whether or not 5.2.6.
~+
£
to prove these results for HA+ £
or
is closed under the derived rules given by theorem
Since the only property used in deriving these properties of
the closure of the class of models of
models of
one may
HA+ 7
H A + £,
HA
under the operation
~
is
( )~ (Z)' ,
we need only show that the class of
is closed under this basic operation
Of course, to prove explicit definability,
one could use the Aczel slash
- its inductive definition makes it fairly usable.
The operation
( )~ (Z)'
544
has the advantage that, if
F
and
A
preserved - i.e. if the validity of ~
(Z~)'
H~A+ F
and if the same holds of
by this operation.
are preserved by it, then
~+
r+ A
is preserved by the operation
A,
then
~+
F+ A
is also preserved
Thus, the class of sets of formulae preserved by this
operation exhibit better closure properties than the class of sets, formulae which yield saturated extensions, 5.2.10. (i.e.
Lemma. A
Let the sentence
A
is a Harrop sentence, see
under the operation Proof°
is
H ~ + ~,
of
have no strictly positive I 10 5
).
F ,
of
HA.
Then
V
or
A
is preserved
A .
To carry out
( )~ (Z)' .
We shall prove this by induction on the length of
the induction step corresponding to (v), we must make a convention involving free variables. that
Let
A(ml,...,mn)
A
have
xl,...,x n
as free variables - we shall prove
is ;reserved for all numbers
ml,...,m n.
The result
then follows trivially for sentences. (i)
The preservation of atomic formulae follows by the decidability of
atomic formulae in (ii) and
Let
HA.
A(m I .... ,mn) & B ( m I ..... mn)
B(ml,...,mn)
are valid in
Z ~ (ZZ)' , whence (iii) Let
A&B
~.
is valid in
be valid in
A(m I ..... mn)--B(m I .... ,mn)
he valid in
(ZZ)' , we must have
~C l ~ B ( m l .... 'ran) °
A(m I ..... ran)
~O 1 ~ B ( m l ' ' ' ' ' m n )
(iv)
Similar
(V)
Let
For
But then
is valid in
'
5.2.11. H~A+ F
Again
Z"
is valid in
B(m I .... ,mn)
For this impli-
~o I~A(ml ..... mn) ' Z,
whence
is preserved, whence
(iii).
A(x,m I ..... mn)
s O l ~ A ( m , m I .... ,mn) ~
A(m I ..... mn)
a contradiction.
to
VxA(x,ml,...,mn)
valid in
~.
Then
(ZZ)' .
cation to fail to be valid in
B(ml,...,mn)
~.
But each of these is preserved under
be given,
VxA(x,m I ..... mn)
to fail to be valid in for some
valid in
m6 D~ ° = {0, 1 .... } .
But
A(m,m I ..... mn)
and is preserved, leading to a contradiction.
Theorem°
The class
~
of sets,
is preserved by the operation
Z-
(Z~)' , we must have
F,
is
Q.E.D.
such that the validity of
( )~ (Z)'
has the following closure
properties : (i)
D
(ii)
if
(iii) if each numeral Proof.
is closed under arbitrary union ; F6 ~ F6 9, n,
and A then
A
is a Harrop-sentence,
has only the variable
x
then
F U IA}6 ~;
free, and
~+
F ban
for
F U {VxAx}6 ~.
The only case we haven't proven already is (iii).
The proof of this
is basically the same as that of case (v) in the preceding proof.
345
5.2.12.
Corollary.
(Friedman
A)
Let
F6 ~ .
Then
ED
and
DP
hold for
H A + f. 5.2.13 - 5.2.25. 5.2.13.
Examples.
Reflection principles and tr%nsfinite induction.
Condition (iii) in the definition of
~
was introduced in Friedman
A for the purpose of proving results like corollary 5.2.12.
By it, if we
have an axiom scheme for which we wish to prove a preservation theorem, we need only prove the theorem for the scheme without free variables.
For in-
duction, AO & V x y ( A x & S(x,y) --Ay) --Ax , we need only prove the preservation result for each instance,
If we examine the proof we gave, we notice that we reduced the problem to proving the preservation of this last sentence. further schemata and apply condition
We shall now consider some
(iii) to prove preservation theorems
for them. Let l~x
<
be a primitive recursive
(or even provably d e c i d a b l e -
< y V ~ X < y ) well-ordering of the natural numbers.
Tl(<) , of transfinite induction on
~
i.e.
By the scheme,
is meant the following :
A0 ~ Vx[ Vy < x A y - - A x ] ~ VxAx , where, for convenience, 5.2.14.
Lemma.
striction that for
Let Ax
F
0
is taken to be the first element of the ordering.
be the subscheme of
have only
x
free.
TI(~)
determined by the re-
Then the preservation theorem holds
H A + F.
Proof.
Let
~
be a family of models of
genuine well-ordering on
~.
Thus, if
~+ F
F
and observe that
is net valid in
4
is a
(Z~)' , we have
4 o [~AO ~ V x [ ~ < ~ y - - A x ] ,
Letting
no
for all
~> % ,
be the least such
whence
n,
~o l ~ A m
~o I~VY~"o Ay'
for all
whence
m < n O and
8 I~ VxAx
% LbAn o. a contradiction. Q. E. D.
5.2.15. Proof.
Theorem° The scheme Let
F
TI(~)
is preserved.
be as in lemma 5.2.14 and let
B(Xl,...,Xn)
denote the
instance,
A(o,xl, °.. ,Xn) ~ Vx[ Vy < ~A (y,x I .... ,Xn) -- A(x,x I ..... ~n ) ]- V~(x,x I .... ,~n ) , if
TI(~<) . For each choice
ml,...,m n
of numerals to replace
346
Xl, . . . . Xn,
B ( m l , . . . , m n ) e F, whence ~ + F bB(ml, . . . . mn) . condition ( i i i ) that HA+ F+Be ~ . Thus H ~ + T I ( ~ ) =
It follows by
u {m+ r+sls~ Tz(~)}~ ~. 5.2.16.
Corollary.
Let
transfinite
induction
isfies
and
DP
5.2.17. of
y,
~
function
be the canonical
is decidable,
by the addition of some schemata of Then
~
sat-
let, for an r.e. extension
proof predicate.
The properties
of
and
I~ ~ P r o o f T ( n , rA")
rA ~
we let
H~
recursive well-orderings.
which we use are that
Proof T
(ii) where
extend
the next set of schemata,
H~, P~OO{T(x,y )
(i)
~
on primitive
ED o
To discuss
Proo~(x,y)
Q. E. Do
iff
~ffA,
is the godel number of r a y ~ denote
A.
s(y, rA l) ,
If
A
contains
where
s
is a primitive recursive
the free variable
such that s(n,rA I) = r [ ~ / n ] A ~
obtained by replacing
the variable
We may use this notation Local reflection for
RF(T)
is the godel number of the sentence
RFN(T)
for
~, ~ ( ~ )
reflection for
RFN'(T)
co~(~)
~x
by the numeral
Ao
, for
A
containing
only
y
free.
A
containing only
y
free.
~, RFN'(~) : for
~, CON(T) :
PrOOfT(X,~O=l~).
W- C o n s i s t e n c y
~,
of
w- C
(~) :
m-CON(T) ~LxProofT(x, r m V y A y T ) , m VyZxProofT(X,~A ~ ) Feferman 1962, theorem 2o19 gives an i n t u i t i o n i s t i c 5.2°18.
The schemata
Lemma.
RFN(~)
Thus, we need not consider these reflection 5.2.19.
n.
schemata :
:
Vy~xProofT(X,rA~l ) - V F A y ,
Consistency of
A
f o r sentences
Vy[~x ProofT(X,rA~1 ) ~ A y ]
Uniform'
in
~, RF(~) :
~x ProofT(X,rA1 ) - A ,
Uniform reflection
y
to list the following
principles,
Theorem.
CON(~)
and
RFN'(~) . see Feferman
and
~ - CON(~)
RFN'(Z )
, for A containing only y f r e e . proof of the following: are equivalent.
For the relative
strengths
of
1962 and Kreisel- Lev;/ 1968o are preserved by the operation
( )- ( z ) , . Proof.
CON(~)
and
~ - CoN(~)
have no strictly positive
V
or
Zo O E.D .
347
5.2.20.
Proof.
Lemma.
Let
Observe
~ ~A
A
be a sentence° implies
~
If
~ ~A ,
then
H~+RF(~) ~ A .
ProofT(X, rA1) ° RF(~)
~x
yields
v
5.2.21. so i s
Theorem.
If
T
i s p r e s e r v e d by the o p e r a t i o n
Proof. Let ~ be a f a m i l y of models o f be a model o f ~ + ~ F ( ~ ) . Then
Pr°°fT(X, rA1),
% I~ ~ for some sentence But
( )~(Z
)',
then
~+~F(~).
Proof T
5.2.22.
so is 5.2.23. so are
Thus
~
Thus, for some
is valid in
~o I~ A '
Corollary.
and l e t
(~Z)'
fail
to
% L~ A ,
is decidable, whence
By lemma 2.4.6,
(Z~)'.
A o
~+RF(~)
~ ~,
n,
dO
1~ProofT(n,~A~)o
~ ProofT(n,rA~ )
and
whence, by hypothesis,
a contradiction.
~ ~A . T
is valid in
Q.E.D.
If
~
is preserved by the operation
( )~ (Z)' , then
If
T
is preserved by the operation
( )-~(Z )' , then
~+~FN(~). Corollary.
T+RF(T) , T+RFN(T) , T + R F ( T + R F ( T ) )
, etc.
348
5.3.1.
Statement of de Jongh's theorem.
In addition to its use in proving explicit definability results and the validity of an occasional derived rule, we observed in 5.2.4 that we could use the operation
( )~ (Z)'
to construct Kripke models of
models of classical arithmetic.
a simple proof of the propositional Jongh. Let
In the sequel, A(pl,...,pn)
positional variables
Pp
~
out of
This last application has, as a corollary, case of an interesting theorem of de
denotes intuitionistic propositional logic.
be a propositional formula constructed from the propl,...,p n .
In an as yet unpublished paper (de Jongh A,
see de Jonah 1970), D.H.J. de Jongh proved the following 5.3.2.
Theorem°
sentences
If P~7~A(Pl ..... pn) ,
BI,...,B n
According to this theorem, tautology,
then
~pLA(BI,
if
there are arithmetical
sentence underivable in
H~.
A(pl,...,pn)
is not an intuitionistic
substitution instances resulting in a
Alternatively,
we can view this as a complete-
ness result if we define the validity of a formula be the validity of the scheme Actually, instances
A(BI,...,Bn)
of
pl,...,p n
A(pl,...,pn) in
determined by
de Jongh proved a stronger result:
BI,...,B n
.... Bn) , for some
of arithmetic.
~
to
A(pl,...pn ) .
The choice of substitution
can be made uniformly in all
A(Pl,...,pn) . A proof of this by means of Kripke models is more difficult and will be given in section 6. Another result of de Jongh's is a completeness theorem for the predicate calculus.
To date, the only proof of this result is de Jongh's original
proof, which combines the use of Kripke models and realizability. 5 . 3 . 3 - 5.3.~. 5.3.5.
Preliminaries on the propositional calculus.
The proof of the completeness theorem given in 5 . 1 . 6 - 5.1.11 special-
izes easily to the propositional calculus.
Kripke's original proof (Kripke
1965) also yields the completeness (but not strong completeness) tuitionistic propositional calculus, underlying
pms
is a finite tree.
of the in-
p~ , for the class of models whose Our first task is to retrieve this result.
Ne do this by starting with a countermodel to a formula finitely many nodes needed to falsify
A
and pluck out
A , splitting and ordering them into
a tree in the process. We will let quence. denote
~, ~
denote finite sequences.
< >
denotes the empty se-
denote the sequence whose only element is
the concatenation of
~,T - i.e. if
a.
~* T
will
a = <s1,...,Sm> , ~ = ,
349
then
= <Sl,...,Sm, tl,...,tn> .
~T
5.}.4.
Theorem.
origin
%
(Finite tree theorem.)
such that
%INA.
(K ,~ , I~*)
such that
Proof.
S
s(s)
Let
: {Be s
Set
In particular~ Let
~ * = <Sl,...,Sm, a>-
(K,!, I~)
be a model with
Then there is a f~nite tree model
< > IN*A.
be the set of subformulae of
A , and, for
8 ~ K , let
: B Iffst.
~< > = %.
Given
~q , let
~.<1>,...,~.
he a maximal set of
71,...,y k
such
that
(i)
8q~i
(ii)
S(~)
(iii)
if
(iv)
S(Ti) / S(Tj)
Now let
~iTi?
for all
for
i,
S(~) = S(8~)
i , then
K* = {~ : 8~
ordering.
i ,
for all
/ S(Ti)
has been defined},
For atomic
B , define
and let
Let
(iv)
be
Let
IN D
B
B
Conversely
~a~e_~.
~ I~"~C .
B,
~
be the usual tree
for
B E S,
that
a I~*B
iff
Let
B
, ~
INC~D
be
there is a Thus ~C.
?~ N A
iff
The proofs are trivial°
v Ib* C,
T I N * D.
But then
~T Ib C'
I~C--D°
.
~tN*D.
such t h a t
T~*~
~l~*c,
~IN*D
Y I~ C ,
7 I~D •
S(~T) = S(7)
and so
8~!7'
such t h a i
Then ~ JN*o~D.
and ~ t N * c ~ D .
The proof is similar to (iv).
Corollary (Kripke).
models, i.e.
C VD .
G IN* C'D.
Then there is a 7
D/s(~).
(v)
Let
or
Then ~ IND and , t ~ * C ,
But, by c o n s t r u c t i o n ,
c~s(~),
C&D
such that ~!~T
9 let
41~c"
be
C~D.
~Za
and, since
~2_~-
5.3.5.
<*
The atomic case follows by definition.
(ii) - (iii).
Then there is a ~
and
s I~*B
We prove by induction on the length of
(i)
S(y) : S ( ~ i ) ,
or
i/ j .
P~ A
Q.E.D.
is complete for the class of finite tree
has a countermodel in a finite tree.
We shall find it convenient to work with a special class of trees.
To
prove completeness for them, we prove the following result (which generalizes a result of Gabbay 196 9 B). 5.3.6.
Theorem (Extension theorem).
the finite tree
(K1,!l) .
Let
Let
(Ko,!o) .
Then there is a forcing relation
all
and all formulae
~
K
I~oA
iff
A ,
~I~IA"
(Ko,So)
be a finite subtree of
I~o be a forcing relation defined on I~I on
(K1,!1)
such that, for
55o
Not eo
By "subtree"
we do not merely mean "tree which is a subordering of"
- the result is false in this case. be successors
in the tree
The successors of a node
(KI,~I) .
For convenience,
~6 K
must o we also require the
origins of the two trees to coincide. Proof.
For
~
K°
and atomic
86 K o,
choose a terminal node
Then there is a maximum
(i) (ii)
~%,
if if
where
t~
(i)
(ii) -(iii)
in the tree
~I
~IPoA
(Ko,~o) .
~"
Define,
.
For each
Let
~6 KI-K O •
for atomic
A ,
A ,
iff iff
~l~oA' t~l~o~,
A , the result follows by definition.
The cases
Let
(a) Let have, for
A=B&C,
821 ~
~IPoB'C'
some
predecessors
and so
~ IPI c
81~o C.
~
y
Then
thesis,
~'~ [PI
~
8-->I ~
I~o C , whence the in-
implie~ that, if
~ tb~ B,
It-I B- c.
~l~Ic
Let
t
and so
Hence
Then there is a
~81P1B ,
~E K1-K o.
Y~I ~"
iff
8 I~I C .
and we ha~e
Then
~ Ibl B° I f @6 K O , ~ I~O B by Thus ~ I~I C. If ~6 KI-K 0 , we t I P o B . Now t 1 7 1 ~ (since the
C.
are linearly ordered)
~ I ~ oB-~C .
(c) Let
are trivial
such that ~ IPo
8 Ip1B
?!i 8,
of
duction hypothesis yields
(b) Let
BVC
A = B ~C.
induction hypothesis
then
iff
is defined as above. For atomic
(iv)
~I~IA
such that
, ~I~IA , ~Ib~A
~1-~0
define
ts~ 8
Ko
86
We now show, for all
A,
~I
8Zo=
~6 Ko,
such that ~Ibo~'
and ~ I ~ I B - C . ~
is maximal in
and let K
~6 K o
such that
o
be maximal
Y--
such that
By induction hypo-
B iff N IPoB an~ ='~hC iff tyltC
IP1 ~" c
since, for terminal
iff
C~Ip I B = c~Ip1 c
iff
t l~oB=t
iff
t? IPOB-~C'
t,
the forcing
IPoC semantics is the same as in classical
logic.
(v) E.g.
The proof for negation is similar to that for case (iv).
Consider the trees
(Ko,
If we embed
(KO,<~)
(Ko,~o)
(K1,~I) :
(KI,-
o/ in
and
(KI,<1)
Q.E.D.
~%/~/
in the obvious manner,
and if we have a
351 forcing relation
I~0
we must decide how like
~,
because
behavior. -
on
¥ ~
However,
(Ko,!o) , in order to extend to a relation
is to behave.
we can make 8.
y
Now,
the same in
~
cannot distinguish
hence
~
In this proof, we need only assume (Ko,~o)
(Ko,~o , I~o)
and
(KI,~I)
terminal nodes are replaced by complete Theorem 5.3.6 has the immediate 5.3.7.
C orollar2o
Let
5.3.8.
Then
P~
Ikl ' behave ~'s
and
y
from
8
and
(KI,~I , I~I) .
(Ko,~o)
is finite.
are infinite. sequences
The theorem
In this case, the
(in the sense of Cohen 1966 ) .
corollary :
{(Kn,<_~)I n--
property that every finite tree (Kn,
y
behave like any terminal node beyond
Note.
holds when both
let
has an extra node beyond it which may affect
which in this case is behaves
We cannot necessarily
be a sequence of finite trees with the
(K,~)
can be embedded as a subtree of some
is complete for the sequence
(Kn,
Examples.
A) ~ _ ~ _ ~ 2 ~ " (n~1)
is
This is the sequence whose
n - ar N and of height
B) Th~_~a~}ow~ki_sequ~nce. (Jaskowski Gabbay
This economical
sequence,
1936 ) (see also Rose 1953 , Gal, Rosser~
1969
, and Mostowski
1966.
to ours for this sequence.), result of taking
n
n - th element
n :
Gabbazi 1969
and Scott
1958, Scott 1957,
gives a treatment
is obtained by letting the
copies of the
due to Jaskowski
similar
n+1 - st tree be the
n - th tree and dropping a node below them:
I ,
C) The modified Jaskowski .
.
.
.
.
.I
.
.
.
.
.
.
.
.
.
.
.
.
.
J2
.
.
.
.
.
.
.
.
etc.
sequence. .
.
.
.
.
J3
J4
, etc.
35~
Let us finish this subsection by remarking modified Jaskowski
trees (a property
on a useful property of the
shared, incidentally
by the diagonal
trees) : Every node of
J* is determined by the terminal nodes lying beyond n From this, we can prove the following lemma :
it.
5.3.9.
Lemma.
Let
~1,...,~n,
be the terminal nodes of
J*.
•
have a forcing relation, is a formula
A.
I~'
Suppose we
n
defined on
Jn*
such that, for each
i , there
such that
1
~j l~Ai Then, if
S
iff
j=i.
is a set of nodes of
J*
such that
~6 S
and
~< ~
imply
n
6 S , there is a formula
A
constructed
from the
A.'s , for which 1
S = I~ : ~ I ~ A I .
Froof.
In particular,
~/~o "
If
~
terminal nodes of nodes of
~.
~ I~A
~ I~A
those not beyond
(otherwise $
Let
~
and
Thus
~ .
8
~i
A
let
For
~ I~ ~A i 8)
~o
such that
it is also determined and let
ASo = A I ~ A I .
this is trivial.
either
for some
i).
B
includes
Hence the set of
is included in the set of terminal
be one of these. ~
or
Since the set of predecessors ~< ~°
But
~<~
of
would imply
have the same terminal nodes beyond them, a contradiction.
The converse is easy:
Finally, W
8~.
(i.e. beyond
(since no extension of
=
iff
A
, the set of terminal nodes not beyond
is linearly ordered,
that
A
there is an
~6 J* is determined by those ~i > ~, n ~i~o Let A = ~im~ ~A'I for *) ~/%
We must show that
~i
~,
Since
by those
Let
for any
S
B
forces
~i~ A i)°
have the property
implies Hence
~i~
~ I~A
and
~ I~ ~ A i
.
stated in the lemma and let
.
Q° E° D.
~S Let us comment briefly on the content of this lemma. is complete with respect to the modified Jaskewski ~ ~-~ A(Pl,...,pn) , there is a such that Si
~o I ~ A(Pl,...,pn) .
of nodes
~
such that
We know that
sequence.
P~
Thus, if
J* and a forcing relation, I~-' on J* n n With each Pi' we can associate the set
~ l~-pi .
The lemma gives a simple sufficient
condition on a forcing relation on behaving like
Pi "
Recall
J* that there exists a formula n that, if we have associated non-standard
~l,...,~n!
of arithmetic with the nodes
model
of
*)
~,
~j~
~
~.
models
~1,...,~n~ , we can define a
As long as the sentences
are used for repeated conjunctions
Ai
have no constants
and disjunctions
respectively.
353 denoting non-standard ~j~°
numbers,
the proof of the lemma carries through for
This is the key to our simple proof of de Jongh's theorem.
5.3.10-12 ~h~Godel - Rosser- Mostowski - K r i p k e - ~ y h i l l A straightforward independent
iteration
sentences
AI,...,A m
the simplest possible independent Mostowski
of the G o d e l - R o s s e r for any
substitution
sentences
AI,...,A m
1961 , Kripke
m.
instances
For the sake of obtaining in theorem 5.3.2, we want the
O
to be
1963, and N~hill
theorem. theorem will give us
Z I . This result has been proven by 1972o
We shall present Nyhill's proof of the following 5.3.11. Theorem. extensions sentence Proof.
Let
A
such that
Let
X,Y
ed by formulae
and Let
T , ~I' "'"
be an r.e. sequence of consistent r.e.
~O
of classical
(Peano) arithmetic, A
is independent
be recursively ~yR(x,y),
~c
sets and let
~yS(x,y) , respectively,
iff
~yR(n,y)
iff
~
nEY
iff
~yS(n,y)
iff
H~ ~ I y S ( n , y )
H~
~(~fR(n,y) & ~yS(n,y)) .
X I• =
t~i X*
i,
= X*UX'
~nv
=
and
X_cU.
t
Now,
U
(see e.g. Rogers
1967, p. 72~
such that
,
Vc_X'
For, if
and
nExnv,
Ti ~ ~yR(n,y)
since
nEY
But
n EU
T i ~ ~SyR(n,y) . contradicting
,
~,
U_cX*,
Y~V,
be represent-
in such a way that:
o
U, V
UUV
Also,
T. X,Y
~ZyR(n,y) ,
By the reduction theorem in recursion theory there are roeo sets
T i ~ ~yR(n,y)
ZI
b ~R(x,y) t
X~ = Ix: ~i ~ m ~ R ( x , y ) 1
Clearly
O
Then we can find a
over each theory
inseparable
n~ X
a n d conside
•
n ~ X' ,
implies implies
and, for some
contradicting
~i'
the consistency
of
T.
~i ~ ~yS(n,y) , whence T. ~ ~yR(n,y)
for some
i , again
consistency.
and
Then, if we let
V
separate
X
and
Y , whence there is an
A = ~vR(no,Y ) , we see that
A
n o~Uuv
is independent
o
over each Q.E.D.
5.3.12. over
Corollary.
HA c .
For any
m,
we can find
m
Z~
sentences independent
T..
354
Proof.
Let
A1
be independent
HA C + ~ A 1 ;
A3
independent
~c+
HA ~ c ; A2
over
over
H~C+AI+A
independent
over
2 , H ~ C + A I +-~A 2 ,
I ,
~c+~I~
2,
Q.E.D.
~ A 1 + _~A 2 ; etc.
5.3.13-5.3.15.
~C+A
de Jongh's
theorem.
Let us now combine the results of
5.3.3- 5.3.12 to prove theorem 5.3.2, which we restate here:
5.3.13.
Theorem.
sentence Proof.
Let
%
AI,.oo,An! of
Ai +
lock at
If
BI,...,B n
pp ~ A ( p t . . . . , p n ) ,
l ~ A ( P l .... ,pn)
be independent A~ m A .
then
~
~ A ( B 1 . . . . . Bn) ,
for
some
of arithmetic.
Associate
Jk*
for some forcing relation on
over
~c
and find,
for each
and let
i , a model
these models with the terminal nodes of
~i J~
and
j#i --j~ °
rag. ~J7 is
Let, for each
Pi '
Si
we can find a sentence
be the set of nodes forcing B. l
of arithmetic
Pi "
By lemma 5.3.9,
built up from the
A's 9
such that
Now, a simple induction can be used to show that, for any formula C(Pl,.o.,Pn)
and any node
B,
~tffC(PI,'-,Pn)
iff
~ibc(B 1
under the two forcing relations.
I~A(~I . . . . .
%
. . . .
,~),
In particular,
Q. E. D.
~n)
The sentence corresponding
to
S
is (except in the trivial case
doE S)
of the form, ~ ~Aii, J whence we have the following corollary 5.3.14.
Corollary.
The substitution
may be taken to be disjunctions Observe
that one cannot use
Starting with
AI,.o.,A m
o 91
of o ~I
due to ~ h i l l
:
instance~ BI, ..., B n o ~I sentences. sentences,
because,
and independent,
if
in theorem 5.3.13
B
is
o 91 ,
we have the following
555
5.3.15o
Corollary.
The substitution instances
BI,...,B n
may be taken to be disjunctions of double negations of Since the only properties of
H~
used in proving
were the closure of the class of models of ( ) ~ ( Z ) ' , the consistency of
HA
under the operation
HA+ F
for
HA+~.TI(<),
for any r.e.
F6 ~
(as in section
In particular,
H~A+RF(HA),~ HA+~..R F N ( ~ )
, etc.
If
not consistent with classical logic 9 the independence of
A
the independence of
exist.
models
wl,...~w m
~A ,
so that models of
wl, .... wm
HAc , we can conclude that de
5.2.11) which is consistent with classical logic. theorem holds
sentences.
de Jongh's theorem
with classical logic (so that
could be chosen), and the incompleteness of Jongh's theorem also holds for
HA
in theorem 5.3.13
O ZI
A
and
~A
de Jongh's HA+~_ F
is
is replaced by Then the
are replaced by Kripke models.
5o3.16. de Jongh's theorem for one propositional variableo In Nishimura 1960, Iwao Nishimura characterized the lattice of formulae in one propositional variable in the intuitionistic propositional calculus. It happens that there are close relations between these lattices and
pms's.
From Nishimura~s work, it is not hard to prove the following 5.3.17. Theorem°
Let
(K,!, I~) be the model s h o ~
formula in the variable
p
such that
P~ ~ A ( p )
below ~ d
let
. Then for some
The proof of this lies beyond the scope of these notes°
B
is true.
Consider the model
be
dE K ,
A proof avoiding
the use of lattices may be found in de Jonah Bo Let B be g OI , independent over H~O . Then there is a model which
A(p)
w+
in
35~
+
K:
B
~
~B
This allows us to prove the following result, due independently to de Jongh and ourselves. 5.3.18. Theorem° Then
HA ~ A ( B )
Proof.
Let
Z o1 , independent over
B be
in the first model iff
~I~C(B)
C(p)
that, for
Then apply Q . E . Do
Recall theorem 5.2.6, by which e.g. we showed
l~ ~ ~VXAX--Zx ~ A x ,
when
~
We may restate this equivalence as
5.3.18 readily applies.
In
~
~VxAx V~x ~Ax
iff
~Vx(Ax V ~Ax) .
HA ~ ~ ~ ~xAx~Zc~Ax , for decidable
~ ~xAx,
.
~E K ,
in the second model.
theorem 5.3.17.
~
~A(p)
.
Prove by induction on the length of
~1~C(p)
HAc ~ , and let
~ZxAx A .
~ A x V
whence
~c
~xAx
is true in
~ ~uxAx,
~c
~xAx, ~
V ~ZxAx
iff
But, for this formulation, then
~ XxAx.
~
~xAx
theorem
and
The decidability of
A
and there is a non-standard model
w
+
implies that of
~xAx °
Now, applying the proof of theorem 5.3.18 to the sentence
and the propositional formula The case
~
~.~
~xAx V ~ Z x A x
~ ~ p~ p, is trivial.
dependence proof for Markov's schema cursive.
- Markov's
~xAx
we have the result. (In particular,
~ZxAx--~uxAx
,
we have an in-
Ax
primitive re-
schema is studied in section 4, below.)
Another point worth stressing is that, for one propositional variable, we o have a best possible result : uniform E I substitution instances. Finally,
observe that we do not have the result for all
since we must have considering models :
F
valid in
~o
H A + F,
F6 ~ ,
One can get around this slightly by
357
+
++ B
w
-~B
W
I"/" / I We leave
the i n v e s t i g a t i o n
to m e n t i o n 5.3.19.
Then,
the f o l l o w i n g
Theorem.
if
Remark.
Let
~p ~/
following: in w h i c h a simple 5.3.20.
If p
we h a v e
theorem pms
In the last p a r a g r a p h one wants include
formula
as well
of complexity.
there
formula
some
~ ~B,
theorem)
(if at all).
using
~B
~B
This will
Kreisel
of b o u n d e d
and n e g a t i o n s
of n e s t e d
alternating
Kreisel
formulae
asked
B ,
one
occurring
tells
in one v a r i a b l e that
instances
there
- i.e.
not all of the f o l l o w i n g
in a
in one's
lattice
for a proof
for the s u b s t i t u t i o n
that,
complexity,
quantifiers
of the N i s h i m u r a
many p r o p o s i t i o n a l
of Kreiselo
and Levy m e n t i o n
implications
sentence
give
pms.
to solve a p r o b l e m
for f o r m u l a e
Pp o
the
tree model
the N i s h l m u r a
of n e s t e d
HA
~_. ~
to prove
definition
over within
time only
(digression).
The i n f i n i t u d e
truth d e f i n i t i o n
HA ~
for some finite
nodes
without
of de J o n a h
as the n u m b e r
are n o n - e q u i v a l e n t
and
~o I ~ A ( P )
5.3.18
are i n f i n i t e l y
Tx
(the e x t e n s i o n
then
of K r e i s e l - LevTf 1968~
a truth
taking
~.~ ~7~A(B) .
5.3.6
was u s e d by de J o n g h
the n u m b e r
measure
such that
only at terminal
of t h e o r e m
Another
must
,
to the reader,
of de Jongh's :
be a sentence
theorem
is forced
The N i s h i m u r a
that
B
Pp ~ A ( p )
proof
when
of such results result
us which is no
for any
equivalences
are derivable: T(~A(B) ~) ~ A ( B ) where
A(p)
rA(B) i
ranges
denotes
W e present 5o3.21. only
x
over all p r o p o s i t i o n a l
the gSdel
de J o n g h ' s
Theorem. free.
,
Let Then,
number proof
B for
~ T(~A(B) ~ ) ~ In other words~
be
of
formulae
of this r e s u l t o ZI ,
in one v a r i a b l e
and
A(B) . here :
independent
some p r o p o s i t i o n a l
of
~.HA c ,
formula
and let
Tx
have
A(p) ,
A(B).
for any i n d e p e n d e n t
0
ZI
sentence
B ,
there
is no truth
o
35a
d e f i n i t i o n for the set of p r o p o s i t i o n a l formulae in B o o Since B is Z I and independent, B is false in the standard model.
Proof° Let
~+
be a n o n - s t a n d a r d
of the N i s h ~ m u r a
model of
HA c + B ,
and assign levels
to the nodes
model as follows :
+
B
w m B
• /
Let
,,,~
level 2
C(Xl,...Xn)
be a f o r m u l a with free variables
prove by induction that, for any n C,
then (i)
ml,...,m n,
C
( i i ) - (iii) (iv)
on the length of
C(ml,...,mn) Let
Let
level 0
if
C
C
be
as indicated.
that there is a level
C(ml,...,mn)
W e shall
n C~I
such
is forced by a node of level
is forced by all nodes.
be atomic. If
C
Then
is D~E
D&E o
n C = I. or
DVE,
Then take
no : max(nD,nE)
will do the trick.
n C = max(nD,r~) + I.
To see this,
label the n o d e s as follows :
Let
n = max(nD,nE)
and, if
~n I ~ D '
D-~E.
If
force
D-~E .
~,~ I ~ D ,
If, for some
I~.
and let
~n I ~ E
~n+1 I~ D ' ~ E "
all nodes force
%',
%'I~ D - • E ,
If the level of or
First,
and all nodes force
8
whence
forces
then D
%
this is the case,
~n
On the other hand,
there is no node of level
8n+I I~ D-*E
E,
~n I k D '
or
8>%'
~n
that
~n Ik D-~E
w h e n c e they force whence
all nodes
such that
is
>~°
6 I~D,
But, if
and all n o d e s force the implication.
Hence all n o d e s force D - ~ E . The case in which
observe
and
then there is a
is > ~ , 8n
D,
D
is similar.
< n
which is not
> ~n+1 °
359
(v)
C
(vi) Since
is
Let
~D .
Similar to (iv).
C(Xl, .... Xn)
be
~n D I~ ~xD(x,ml .... ,mn) • nodes force EnD
~xD(x,x I .... ,Xn) .
nD ~ I , the domain at level
nD
is
Then, for some
Then
n C = n D.
IO, I, ...I. m,
Let e.g.
and l~D(m,m I ..... mn)
D(m,ml,...,mn) , whence they force
and all
~ x D ( x , m I .... ,mn) .
is handled similarly. (vii)
C(x I .... ,Xn)
is
VxD(X,Xl, .... Xn) °
Then
n C=n D
and the proof
is similar to that in case (vi). Thus, any formula
Tx
will have a level
a way that, for all
FA(B)I , if
then
for all nodes
~ I ~ T ( r A ( B ) ~)
nT
associated with it in such
~nT I~T(~A(B) ~) ~
or
in the model.
~nT I~T(FA(B)I) ' The proof is completed
by the following lemmao 5°3.22° Lemma.
For each level
forced at a node of level
n,
n,
there is a sentence
but at no nodes of level
A(B)
which is
n+1
or higher.
The proof of this is not difficult, but is rather long and we omit it. The reader is referred to de Jonah B, for the proof.
Alternatively, if he
is willing to accept theorem 5.3o17 and the infinitude of the set of inequivalent formulae in one propositional variable, lemma 5°3.22 for arbitrarily large
n
follows by a simple cardinality argument - with only
finitely many nodes of level ~ n
to distinguish these formulae, we can
only find finitely many inequivalent formulae.
By either approach, the
proof of theorem 5.3.21 is completed.
Q.E.D.
5.3°23. Further results on de Jongh's theorem~ In theorem 5.3.13 (theorem 5.3.2), we proved that, for any underivahle A(Pl,...,pn) , arithmetical is underivable in BI,...,B n
2"
BI,°..,B n
can be found such that
uniformly in all
A(Pl,..o,Pn) .
Friedman A improved this by
showing that, where uniformity is desired, any collection sentences independent over work.
A(BI,...,Bn)
de Jongh's original proof (de Jonah A) gave
~c
augmented by all true
H oI
B1,...,B n
of
o ~2
sentences will
Friedman's proof made use of his generalization of the Kleene slash.
We shall present a (Kripke) model-theoretic proof of his result in section 6, below. In terms of the simplicity of the substitution instances, corollary 5.3.15 shows that
BI,...,B n (in the non-uniform version) may be taken to be disjunctions of double negations of Z oI sentences, which, classically, would o o bc E 1 . We obtain Z I substitution instances in section 6. When restricting one's attention to a particular number of variables, we o outlined a proof of the existence of uniform Z 1 counterexamples in 5.3.165o3o19 above° Using an alternate proof involving the arithmetization of the Kleene slash, de Jqn~h 1971 and B reproved his theorem 5.3.19 (cf. also 3.1.16).
3~0
§ 4.
?{]arkov's schema
5.4.1 - 5.4.3 . 5.4.1.
Markov's
schema.
~e have already encountered
Kripke models.
~rkov's
schema in our discussion of the
In section 2, we presented a model-theoretic
Kreisel's version of the independence
of };~rkov's
proof of
schema (theorem 5.2.6) and
in section 3, we observed that this result also came as a corollary
to a
special version of de Jongh's theorem for the special case of one propositional variable.
Both proofs are off-shoots
schema is not preserved by the operation Before discussing Narkov's
of the simple fact that Markov's
( )~ (Z)'
this last fact, let us consider
(i)
Vx(A~ v ~A~) ~ ~ ~ ~ - -
(ii)
Vxy(Axy V ~ A x y ) Q Vx ~ 7 ZyAxy-- Vx ~ A x y
where
of
X~A~,
(iii) Vxy(Axy V ~ A x y ) ' V x [ ~ Z y A x y - - Z y A x y ]
(iv)
several formulations
schema :
W [ V y ( A x y v ~Axy) ~ ~ A
, ,
~ A~y] ,
~Axy-
contains only the free variables
shown.
Observe that the schemata
obtained by replacing the present
x~ y, or z by finite sequence of variables reduce to ~o~. schemata via a pairing~V~-Thus, there is no loss of generality in
considering
only (i) - (v).
methods,
see also Io11°5
5°4°2°
Lemmao
Proof.
(v)-
(v) ~
(iv).
(iv) -~ (v). ~I~Vz[ and
For the treatment
of N~rkov's
and § 3.8°
(iv) -~ (iii) -- (ii) -- (i)°
Trivial°
Let
K
be a model of (iv) and let
Vxy(Axyz V-TAxyz) & V x - 7 - 7 ~ v A x y z - ~ V x Z y A x y z ]
bED~,
schema by other
we have
~ [~Vxy(AxybV-TAxyb)
,
46 K .
be such that
Then, for some
8 l~Vx-7-7~Axyb,
~>~
and
B [~ vx ~ Axyb But then there are have
y> ~
standard primitive recursive Then
and
c E Dy
~{l~Vy(AcybV ~ A c y b ) & - 7 - ~ Z y A c y b
o
such that Let
we have
J2 d) , whence,
~ IbZyA(Jld' Y' J2 d) '
( i v ) "~ ( i i i ) .
Let
[b -~-~.~yAby ,
applying
o
~ l~XyAbyo
is the
A'xy:
A(JlX, Y , J2x) ,
a contradiction°
Then there are Now
j
J1' J2 "
and (iv) to
_K be a model o f ( i v ) ,
~[~Vx[-77[yAxy-~Z~fAxy]
where
pairing function with inverses
~ Ib Vy(A(jld, Y, J2 d) V ~ A ( j l d , Y, J2d))
YI~-7-~ZyA(Jld'Y,
~ I ~ Zy Acyb o We also
d = j(c,b),
~ ]~ Vxy(Axy V -7Axy) 81(~
and
~ t b V y ( A b y V -TAby)
[b Vy(Aby V -TAby) ~ -7 -7 ZyAby-- ~y Aby,
whence
bED~
and
such t h a t
and, by ( i v ) ,
[ ] b ZyAby ,
a contradiction.
391 (iii)
** ( i i ) .
Let
K
be a model of ( i i i ) ,
~I~Vxy(AxyV
mAxy) ,
--
and
/3 I~-
~7~69
J~V~Vx m m ~yAxy-~ Vx ~y Axy . Then there are ~ > c~ and b E DB such thaC~V~ and S I~-JL~yAbyo But, by (iii), S I b m m ~ y A b y - ~ y A b y , a contradiction. ( i i ) -* ( i ) o Trivial. Q.E.D. Unfortunately,
we c a n n o t
model-theoretic operation that
all
independence
( )-* ( Z ) ' five
settle
any of the converse
proofs
are ruled
by one w h i c h p r e s e r v e s
schemata are preserved.). (iv) i s
Proof.
The proof is based on a remark of Kreisel's
derivable
in
allows one to add free variables.
~ Vx ~[z ProofHA+(i) (z, rvy(A~y V 7 A ~ y ) & Be(X), o°.
be a primitive
Vy(Axy V 7 A x y ) & m m
the
s c h e m a , we w i l l
see
H o w e v e r , we c a n p r o v e t h e f o l l o w i n g :
Theorem.
Let
(simple
o u t - when we r e p l a c e Narkov,s
5.4.3.
tion principle
The scheme
implications
recursive
~+RFN((i)). that the uniform reflec-
Ne show
7"7 ~" A~y-~ ZyA~y I) . enumeration
of all instances
of
~yAxy-~ ~yAxy .
a)
H A ~ f f V w P r ° ° f H A + ( i ) ( r B w ( 0 ) 7 ' rBw(O)~) , i ° e .
b)
Te t
VwXz P r o o f s m + ( i ) ( ~ , % w ( ~ ) ~ ) .
every axiom is
Also, let
f
its
own p r o o f .
be p r i m i t i v e
recur-
sive such that
h%(x+l) By well-known
~Bfw(X)
properties
•
of ~roof,
B~ bZz ProofHA+(i)(z,rBw(X+~)l) ~xProofHA+(i)(z,rBfw([) ")
But c)
and so
~-e ~z ProofHA+(i)(z,rBfw(x)~)
.
~zProofttA+(i)(z,rBw(~7~+1)').
Thus
HA _ bVw~z ProofHA+(i)(z,rBw([)
~
)-~Vw[zProofHA+(i) (~,r Sw(X+1)-
)
.
This and (a) yields HA ~ Vw Vx ~z ProofHA+(i) ( z , r B w ( X ) ~ ) . RFN'[~+
(i))
implication
(which is equivalent RFN-~RFN' , however,
to
RFN(~+
(i))
by lemma 5o2.18 - the
is trivial) yields,
for
w
the index of
Vy(Axy V mAxy) & 7 m ~ y A x y - ~ X y A x y ,
~+
RFN(~+
[i))
b V x [ Vy(Axy V m A x y ) &ram ~ A x y - ~ ~tyAxy]
Thus, if schemata equivalent°
(i)
these schemata,
we shall allow ourselves
of the schemata
(i)- (v) (for the present chapter). MP
Q.E.D.
- (iv) are not formally equivalent, they are almost
Combining this with our model-theoretic
We note that
.
may be formulated
inability
to be sloppy and let
as a rule of inference
to distinguish MP
denote any
(see3.~.~).
3d2
5.4.4-5.4.6.
The i n d e p e n d e n c e
As remarked above, we have This time, however, 5.4.4. Then,
Theorem.
let
Let
Proof.
with
~xAx
r ~
Let
~
NI°
be r.eo,
F~ ~
of
NP
twice.
(as defined in section 5.2.11).
is not derivable
be i n d e p e n d e n t
with
in
Ax
~+
F.
primitive r e c u r s i v e
(so
Then
m-~ ~x A x - ~ x ~ x
~xAx
a node
.~+ F
of
~ Vx(Ax V m a x ) ) .
~+
MPo
we shall be more direct.
some instance
In fact,
of
already proven the independence
.
be i n d e p e n d e n t
such t h a t
of
HA+ F
and let
~ I~ ~L~_&x and c o n s i d e r
K
be a model of
HA+ F
(K~)' :
_KB
I 0
We will show IP~xAx some
So I ~ m m X x A x - - X X A X o
implies
n.
dO l f f m m Z x A x .
As usual,
Since But, if
this means
HA ~ A n
~Z~o
implies
s o lff ~ x A x , and so
~= ~o
then
HA ~ x A x
or
yZ~ ,
d O lffAn
,
for
contradicting
independence. But
F6~
and so
For example,
~IP
H~+F
is preserved by the step from
is independent
of
In addition to outright i n d e p e n d e n c e form of the a x i o m a t i z a t i o n m,
K 8
, ~+TI(<)
to
(K~)'.Q.E~).
,
~ etco
+ co~J(~ + ~ )
ure,
~+RFN(~)
of the c o m p l e x i t y
of
~P
results,
as follows.
of a formula
one can obtain results on the First,
let us define a meas-
of number theory.
W e do this in-
d u c t i v e l y as follows, (i)
if
A
is atomic,
(ii)-
(iv)
(v)
m(mA)
m(A~B)
(vi)
m(XxAx)
= :m(Ax) ,
(vii)
m(VxAx)
=
m(A) = I ,
= m(* v s )
= m(A-~S) = m a x ( m ( A ) , m ( S ) ) ,
= m(A) , A x = ~-Bxy
for some
B
for some
B
m(Ax) + I , otherwise,
Observe
]m(Ax) , [m(Ax) + 1
Ax
:
VyBxy
, otherwise.
that, for classical
arithmetic,
HA c ,
in the sense that a truth d e f i n i t i o n for formulae be given. ~.
m(~A)
this is a r e a s o n a b l e of bounded
We have observed in 5.3°20 that no such d e f i n i t i o n
(To o b t a i n one, redefine
= m(A)+lo)
m(A~B)
= max(m(A),m(B)) + I
measure
complexity
can
can be given in and
3~3
5.4°5.
Theorem.
to formulae of
~
H~+MP
A
is not axiomatized
for which
m(A)~n o
of bounded complexity
Proof°
by any restriction
for any
no .
can axiomatize
of the schema
I.e. no set of instances
MP
(ever
5)°
We know from recursion theory that all formulae whose complexity is
of measure
lie in a certain level of the arithmetical
o
also know from the hierarchy classical
logic)
theorem and the completeness
that there is some non-standard
model
hierarchy°
theorem
w+
of
Ne
(for
~c
in which
the truth of a sentence at or below the given level of the hierarchy agrees with truth in the standard model, but truth at higher levels does not. Thus consider
(w+)' :
t w
o
We first show by induction on m(A)~no,
~, ~o I~A
iff
use the facts that
Atomic V
A
are decidable
I~B'C
whence .
iff
w + I=A
iff
w
But
@O I ~ B ' C
Also
,
B
and on the length of ~
A , that, if
w ~ A )o
For this, we
I= A .
Consider Since
B~C.
~o I ~ B ~ C '
~o I ~ C o
whence
Again
The connectives
If
m(B--C)!no,
is less than that of
whence
(written
and there is no problem.
w+I=A.
SO suppose
and the length of
m(A)
is true in
also offer no difficulty.
~I~B'C, %
~I~A
A
~o I ~ B ~ C
wl=A. ~
or
B~C
If
wl~ C ,
whence
and
, then ~l~B--C,
~o I ~ B ,
, whence
&
l~C"
~I=B ,
then ~I~B
~o I~B "
~ [~B~C.
~I~B~C" B
is handled similarly.
Now consider the quantified
formulae.
For convenience,
we only exhibit
one quantifier,
although there may actually be a block of like quantifiers.
Thus, consider
~xBx o
% = w+l=~xBx w ~ ~xBx , by choice of
w+ o
Conversely,
by induction hypothesis° For
VxBx ,
% I> vxBx =
vx3x w+~ VxBx w ~ VxBx
by choice of
w
+
°
~64
If
I= VxBx,
•
suffices
then
to show
m+ 1= VxBx
~o I~ Bn
~I=VxBx
we see that,
%lffA
for
tence
w+
IFBn,
To conclude
~o lh VxBx,
it
n . But
by induction
if~
m+ I=A,
hypothesis.
d tffA I= A .
•
is not an elementary
A,
Ik VxBx.
m(A) < n ° ,
iff But
~
= Vn w l = B n
= Vn % Thus,
and
for all n u m e r a l s
w~A
.
Let
extension A
of
m
and, for some prenex
be such a sentence
for which
re(A)
senis
minimal. Then observe
A
is of the form
that,
for which
by minimality
m(C) < m(A) .
%tbc and
C
A
of
re(A) ,
m(B) ( m ( A ) .
the above
argument
To see this,
holds for all pre,~e~C
In particular,
iff
~lffc
iff
• I= C ,
is decidable
Suppose
~Xl...XnB , where
(since doI~C~-~Iffc
is o f t h e f o r m
arid dol ~ m C * - ~ I ~
VXl...XnB(X 1 . . . . .
Xn) ,
mC
)o
m(B)< m(A) °
Then
®+I = V x 1 " " X n ~ X 1 " ' ° x n = ~ I F V x 1 " " x n B
V a l . . . a nE D ~ { ~ l f f B ( a l . . . . . an)) = Vm1"''m n ~ ( ~ I F B ( m l . . . . 'mn)) Vml...m n Eo3[~ o IhB(m I , o o ~ ,ran)), since ~ ( B ) <
m(A) .
lffVx1"'°XnB
This last fact,
implies
together
dO l~Vx1"''XnB
°
with the fact that
But, more importantly,
since
m(B) < m(A) , ~o l~-B(ml .... 'mn) ~ ~I = B(ml' .... ran) ' whence
w 1= VXl.°oXnB
Hence we have
and
A
cannot be of the form suggested.
~+ I= [x1°..XnB , w I= m ~x1°..XnB , and
O~o I~- Vx1"'°Xn(B(Xl . . . . . Xn) V mB(x 1 . . . . . Xn) ) (by the fact that re(B) ( r e ( A ) ) o But ~ l F ~ l " " X n B ' and eo % lff m~ ~ 1 " ' ' X n ~ ° But we cannot have ~o I F ~ X l ° ' ° X n B ° Contracting q u a n t i f i e r s , we have an i n s t a n c e , VX(BIX
V
mBIx)
Atom
of (i) which is not forced Finally,
for
at
~XBIX-~ ~XBTX , %.
m(C) ( he-1 , we have
m(~x C) (
no
i s decidable i n the model ( m(~xC V-n~xC) = m(~xC)< % Ie ~ c o= C~° l b m 2xC, we h a v e
, whence
no
)o
~xC
But, whether
365
Hence,
~
is true in the model for instances of low complexity, but not
for high complexity. enough to include
The fact that we can make the "low complexity" large
n
yields the theorem.
Qo E. D.
O
5.4.6.
Corollary.
MP
is not derivable from the subscheme,
Vx ~ Z y A x y ~ where
A
Vx~yAxy,
is primitive recursive.
The above proof was rather long, but the idea is simple. a model
~+
of
~c
which agrees with
w
If we start with
in the truth of formulae of low
complexity, but not for formulae of high complexity,
then formulae of low
complexity are decidable in the model :
L ~O whence
~D?
~
'
holds for sentences of low complexity.
complexity are not decidable and, in particular, which is not decidable at ty.
Hence
~P
~o ' but for w h i c h
B
But,
sentences of high
there is some sentence
~xBx
yields a decidable proper-
fails in some instance of high complexity.
We might also comment on the measure of complexity used.
One might object
that we should consider an intuitionistically meaningful measure - i.e. one for which the bounded truth definition can be given in HA c • As observed above,
such a measure
m'(A~B)
= max(m'(A),m'(B)) + I
formula
A,
and
m'
m'(~A)
m'(A) A m ( A ) , whence a bound on
the result follows from theorem 5.4°5. choice of a measure
m,
~
as well as in
is obtained by defining = m'(A) + I . m'(A)
Further,
But then, for any
yields one on
re(A) and
concerning the specific
theorem 5.4.5 can be shown to hold for any measure
for which truth definitions for formulae of bounded complexity can be given in
~c
(and hence for those measures whose bounded truth definitions can
be given in
~)o
The above theorem 5.4.5 and corollary 5.4.6 easily generalize to any r.e. ~+F,
where
F6 ~
5.4.7 - 5.4.9. 5.4.7.
and
F
is true in the standard model,
MID
results for
to be preserved by the operation
to prove its independence and to prove that it cannot be replaced
by a bounded set of instances of itself. operation
~+RFN(.~).
A comment on proof-theoretic closure properties.
We have used the failure of
( )~ (Z)'
e.g.
( )~ (Z)' HA
to
In ~4.|O-I@,
we will replace the
by one which will allow us to extend many standard HA+~P.
We have also given a derived rule (whose proof was based on this failure to be preserved) :
366
5.4.8.
Theorem.
Let
A
contain only
x
free and let
x
free and let
b v~(A~ v max)
Then the following are ~quivalent :
(i)
m b ~:':V m~Ax,
( i i i ) ~ b ~y[-~m~.x--Ay]. (This is theorem 5.2.6.) An almost trivial derived rule is 5.4.9. and
Theorem.
Let
HA b m m ~ x A x .
Proof.
A
contain only
Then
Observe that
~
~
HA b V x ( A x V m A x )
b ~LxAx o
b mm~ocAx
implies that
~xAx
is true in the
$O~e
standard model and hence disjunction
An
property yields
is true ~orVn. ~
bAn
or
~
But
HA ~ A n Y m A n
b man.
Hence
and the
HA b A n ,
b ~Ax.
Q.E.D.
From an earlier chapter,
we know that this last result holds when
allowed to have other free variables. tension to this generalization. prove this directly. ~l~Vx ~
~yAxy.
Suppose
That is, let
Then, for some
~ ,
~
bE D~,
Since
~ Ib m A b c
~ l~ZyAby.
mAxy) , ,
But
and we obtain no contradiction.
~
is
then that we decide to attempt to
be a model with node
bVx(AxyV
cE D ~ - D ~
we see
~
we can find some
A
The present proof admits no easy ex-
HJ~AAb V x m m ~ y A x y
the model
i.e.
for all
~> ~
and define a nonstandard
and
cE DB. cE D~
Also,
such that
We cannot pull
model of arithmetic
such that
since ylbAbe. D~
out of
on it, because it
does not follow from the fact that an axiom is forced at a given node that it will be true in the classical model determined by the domain and atomic formulae forced at that node° would not do to show
~
(Note : We have not here done anything that we
b Vxy(AxyV ~Axy) &Vymm~xAxy~Vy~xAxy.
we should not expect to get anywhere.
Also, unfortunately,
Thus
there seems to
be no place at which to apply the trick used in proving theorem 5.2.6.)
5.4oiO-5.4.w. 5.4.1o.
( )~ (E)t
by
~
to be preserved under such as the
~ ~
is preserved.
,/
has its
we must give a similar such
Fortunately,
is a family of models of
( Z ~ + ~ ) ~ . Eogo let
( )--(~ )'
E D - theorem required preservati~
and, to obtain such results,
problem is simple : If ~
~
But applications
operation under which
on
+®),
The failure of
applications. under
( )-(z
~ = I~1,~21.
the solution to this
MP , define an operation Then
(~+~)'
is:
3~7 5.4.11.
Theorem.
If
HA+MP
- i.e. the validity of
is valid in
~+~P
Proof.
We consider the schema (i)°
look at
GO •
Let ~.
Then
and so, for some
n,
~l~Ano
%IbAn, i.e. %Ib
5.4.12. Corollary.
G I~m
xo
H~A+~
then it is valid in
It being valid in .
~o I~ V X ( A X V ~ A x ) ~ A x
corresponding to
Proof.
£,
is preserved by the operation
~xAx,
Let
whence
~
E ~ + w, G> ~o
I~xAx"
A is decidable and so
(ZZ+ ~)'
( )-- (E + ~)' . we need only
be the node DG = {0, I ....
HA ~ An,
l
whence
hus
possesses
ED, DP.
We can't quite quote theorem 5.1.20, but we can observe that the
proof carries over easily, the additional summand being used only to guarantee the validity of
MP.
Q.E.D.
Regarding closure properties of the class validity of
~+
F
~w
of sets
is preserved by the operation
almost exactly the properties corresponding to
( )~ (Z)'
The difference is that we must also insist that
F
such that the
( ) ~ (Z +w)' , we get
F
(theorem 5.2.11).
be true in the standard
model. 5.4.13. Theorem.
~+
r
The class
~w
of sets,
F,
such that the validity of
( )~(~ +w)'
is preserved by the operation
has the followin~ c l e s ~
properties: (i)
~w
is closed under arbitrary union ;
(it)
if
~E ~
(iii) if
and
FE ~w,
A
A
is a Harrop- sentence
has only the variable
for each numeral
n,
then
(Note that, in (iii), the fact
+r~An Proof.
for all
n
and
and
x
wI=A , then
free, and
~+
F~An
F U {~kAxlE ~ . ~I= VxAx
follows from the facts that
~I= r . )
The proof of theorem 5.4.15 is identical to that of theorem 5.2.11
and we omit it. The results of 5.2.13- 5.2.23 carry over easily.
We leave the verification
to the reader. 5.4.14. Consider
Remark.
The proof of de Jongh's
theorem
J3: G~
G2
G~
G~
~c
~6
does not carry over :
36e
If we wish associate
~I
to
force
R~
when we turn this into a model of
the standard model with either
associated with one of
@3
and
~
~I
or
~2 o
and with one of
~,
Similarly, ~5
and
~
we must it must be
.
Thus, we
end up with something of the form : ~I ~ I
%
~
~3
2
But now the proof of de Jongh's does not apply since
~2' ~
/4
~
3
6
theorem does not go through : Lemma 5.3.9
and
~6
all behave identically.
tion of our technique in section 6 will allow us to get a r o ~ d Also,
it will yield a method of generalizing
need not be true in the standard model.
A sophisticathis problem.
theorem 5o4.13 to cases where
F
369
§ 5.
The schema
IP c O
~==::==========
5.5.1.
In addition to
MP,
the schema
IP~ : Vx(Ax V ~ A x ) ~ where
(VxAx~ ZyBy)--[y(VxAx--By) ,
A, B have only the free variables indicated, is valid under G~del's
interpretation (see 5.5o~O). of
As in section 4, where we considered variants
MI° , we may consider variants of this schema.
IP
, however, is simply O
not as susceptible to study by means of the Kripke models as
~4P, and, thus,
we shall only consider the schema as presented (i.e. with no free variables). The reader may consider variants as he pleases (in particular, 5.5.2- 5.5.3. Let
~
5.5°2° able
Proof theoretic closure results°
and
~w
Theorem° y
IP ).
free.
be as defined in sections 5o2o11 and 5.4.13, respectively° Let
Let
A
have only the variable
F6 ~.
If
H~+ F~
x
Vx(AxV ~Ax)
free, and
B
only the vari-
~+
F~VxAx--ZyBy,
then
Proof.
Suppose
H~+
K
with origin
of
(EK_Kn)'. By the decidability of
-'11
for all
n,
~n
r b ~ ( V ~ x - By) . Then, for each n there is a model such that Bn IbWA~, ~ n l ~ Bn" Let ~O be the origin
~o t~VxAx"
8n> ~o
and so
5°5°3.
Theorem.
m+
Vx(AxV ~Ax)
rb
Hence
~n l ~ B n , Let
Proof° Replace
and
and the fact that
and, f o r some
n,
8n I~ V x A x
~o l ~ B n "
a contradiction°
A, B and
m+
r b V~x~rBy,
F6 ~ w
Q.E.D.
( ~ n + ~)'
Mutu%l independence o f
If
then
F = MP , TI(<) , RFN(~),
There is one useful property that
But
Q.E.D.
be as in theorem 5.5.2 and let
(Z~n)' by
For instance, we may let 5.5.4- 5.5.7.
A
~o I ~ y B Y
lV~
IP c
and
RFN(HA+MP) ,
etc.
IP~.
has : It is not preserved under
O
( )-- ( Z ) ' 5.5.4. Ax
or
( )-- (Z + m)' .
Theorem.
and a formula
~+
Let By
F6 ~
Because of this, we may prove the following
be roeo
Then there is a primitive recursive
A
(each with only the free variables indicated) such that
r ~ (wmx- ~By) - ~ ( V ~ x - By).
370
Proof. Cz
Let
~zCz
and
VxAx
primitive recursive.
corollary 5°3.12).
be formally undecidable in
Also, let
VxAx& ~zCz
H~ c + F,
Ax
and
be consistent (e.g. use
Define
~y - ( y = o ~ c ~ v I F = ~ C z ) . Let ly.
~+ , • ++
be classical models of
Also, observe that
(~+m++m++),
•
VxAx& ~zCz
is a model of
and
-~ Vx~x,
VxAx~-~zCz.
respective-
Consider
: ® V~7~zCz
G 2 w+ VxAx
XzCz
G3~++~VxAx
o
Now
~o I~ F
by theorem 5.4.13.and
recursiveness of
and
I B1,
GI BUt
Ax.
Also,
~o I~Vx(Ax V ~ A x )
~o I W V x A x ~ y B y '
~O I ~ ~ v ( V x A x ' B y ) ' ~2
since then
I~ vxAx~B1
Corqllar~.
However, which
Iv~
5.5.6.
IP c o
~o I~ V x A x ' B O
GI I~VxAx~BO
or
IP~
Qo E. D°
F=}{~ :
is not derivable from
N~
@
is preserved under a special case of
( )~ (E)'
,
under
is not preserved :
Theorem.
Let
w+
be a model of
H~ c o
Then
H A + IP c
Suppose that, in the model
is valid in
o
(®+), . Proof.
~o I~ V x A x ' B I °
and, in the second case,
' both implications leading to contradictions.
It is worth singling out the case 5°5.5-
GI,G 2 I~VxAx
l Bo.
In the first case, it follows that that
by the primitive
since only
(w+) ' ,
+ G
l so w ~o
does not force an instance of
IPoc :
s o [~# Vx(Ax V ~ A x ) ~ (VxAx-. ~ B y ) - . ~ ( V x A x ~ By) . IP c
being valid at
~,
%
Ib Vx(Ax V ~ A x ) & ( V ~ x ~
%
I N ~(VxAx-- ~y) o
By this last statement,
I ~ BO. all
n.
we must have
In particular, It follows that
~yBy) ,
~o I N VxAx--BO
~lbVxAx ~o I~ VxAx.
and, for some
and, But
A
[~-->~o'
being decidable, Go I b V x A x - ~ v B Y '
~ I~ V x A x '
G° I b A n whence
for
371
~o I~ ~ B y
and, for some
n,
~o I~-Bn"
Thus
~o I~ V x A x ' B n
and so
~o I~ Zy(VxAx'~By) ' a contradiction°
Q° E. D.
The immediate corollary is 5.5.7.
Corollary.
some instance of
Let NP
~+
F
be r.e.,
FE ~,
is not derivable in
fi~c + F
consistent.
Then
HA+ T+IP c . o
Proof.
f1~+ T + I p °
is valid in
(m+)'
for any model
as shown in the proof of theorem 5.4.5, is an elementary extension of 5.5.8.
N]~
m+
of
HAC¢~. But,
is not valid in I~+)'unless
We
~+
Q.E.D.
Final comments on
Ipc ° The non-preservation of IP c under o o ( )~ (Z)' allows us to prove for IP O an analogue to theorem 5.4.5. We o may also generalize theorem 5.4.5 by using the fact that IP c is preserved o under w + ~ (w+) ~ Also, aside from such results, and formulations of such corollaries as the independence of
IP~
from
I~+NP+RFN(~+NI
°) +
60N(I~+ IP:) +TI(<) , etc., we may observe that we can obtain subtler results such as the following : 5.5.9.
Theorem°
There is a formula
By
and a primitive resursive
Ax
(each with only the indicated free variables) such that (i)
m+~
~f (WAx-- ~ B y ) - - ~y(VxAx--By) ,
(ii)
~ + IP ~
ZxAx-- ZxAx°
(In other words, the same formula Proof.
Observe that
A
A
works in both independence proofs.)
may be taken in both independence proofs above to be
arbitrary up to the requirement that
ZxAx
be independent of
~c
o
Q.E.D.
Beyond this, there is little we can do model-theoretically since (i) the only models of
IP:
we have so far are (except for those given by the com-
pleteness theorem) models
w+
of
~c
and models of the form
(ii) the only models to which we know we can apply the operation and preserve
IP c o
are the models of
HA c.
(m+) ' , and ( )-- ( )'
372
D= =e=f: :i:n= =a=b~ =i:l~ =i:t: ~y: = ~ o= ~f= : =models of F~ o : a p p l i c a t i o n s ==---~::=~==:::=~:::::=
§ 6.
5.6.1.
The o p e r a t i o n
The o p e r a t i o n certain
( )~ (E)*o
( )~ (Z)'
purposes.
,
while extremely useful,
In p r o v i n g de J o n g h ' s
is too restrictive
theorem, for instance,
for
we had m o d e l s
of the form
o Z I s u b s t i t u t i o n instances since ~I' ~2 and o Z I sentences (and similar results for models on
This cannot possibly give us ~3
must all force the same
the other modified Jaskowski noticed that, H~+N~+
F,
Let
to apply
trees).
( ) ~ (Z
we had to assume that
~
w+
and (ii) atomic formulae whose any node (Z~)*
F
in proving
~
w+
E D - property for
be a n o n - s t a n d a r d
is contained constants
in the manner in which we defined
in
De
name elements
exactly when they are true in
(Z~)*
reader will also have
(say) the
was true in the standard model.
be a family of models and let
such that (i) the d o m a i n of
Then
The observant
+ w)'
~+
for all in
~+
model of ~E K ,
H~ c
~E~,
are forced at
Then, we can define a model
(Z~)' . E.g.
let
~ = {~I,~2}.
is
o
Unfortunately,
we don't know if
For, what have we got to guarantee at
~ ? o true in
Truth in w +,
a model of
~+
placed at
~
will always be a model of
that the induction
is not c o n v i n c i n g
but not forced at H~Ac
(Z~)*
~o"
conclude
In
( )-- ( Z ) '
AO&
~y(Ax&
x
in
(Z~)'
(iii),
to
we had not to look at the schema without A x , but we only had to look at all instances
S(x,y)--Ay)--An.
This is obviously v a l i d in
(El)*
- but,
we cannot conclude
%I~AO~Vxy(Ax&S(x,y)--Ay)
~o I~ AO' AI . . . . .
themselves°
axiom to be valid in
consisted in o b s e r v i n g that, by theorem 5.3.11
other than
standard integers,
, we did not merely have
- we had the natural numbers
that induction was valid,
free v a r i a b l e s
schema will be forced
- the law of the excluded middle is
Let us consider how we proved the induction Our "second proof"
HA.
that
since the domain at
S°
has non-
from the fact that
~o l ~ A a
for all
aED~o,
b u t only that
373
The actual proof we gave in p r o v i n g t h e o r e m 5.2.4 was based on the following r e a s o n i n g : a least
n
If
~o I~ A O & V x y ( A x ~ S ( x ' y ) ~ A y )
such that
antee the existence (i)
w+= w,
of
w+
-in
~o I ~ A n
°
of a least element
and (ii) the c o n d i t i o n other words,
and
sO I ~ VxAx,
there is
There are two cases in which we can guarin
w+
" ~o I ~ A
A
is not forced :
is expressible
in the language
d e f i n i t i o n for a formula + in the Kripke model can be g i v e n w i t h i n the classical model w . 5.6.2 - 5.6.7. 5.6.2.
if the truth
of which "
Definabilit,y.
In this subsection,
we formally define what we mean by the definabil-
ity of a Kripke model in a model of which
HA
~E K,
a
(or, forcing)
is valid, E D~,
~
w+
~c.
a nonstandard
denote a n u m b e r in
have formulae as follows,
Suppose
_K
is a Kripke model,
model of arithmetic. w+
indexing
with the free variables
a~
in
Let, for each
We assume
that we
as indicated :
K(~) , D(~,x) ,
~<~, S(~, x, y ) , A ( ~ , x , y , z) , M ( ~ , x , y , ~) o Also,
~, 7, ..o
will denote indices of
O, I, . . . .
Let us suppose that there is a o n e - t o - o n e ~E K
and elements
a, b
are associated ~< ~
a
of
with
iff
w+
for w h i c h
~, 8 ,
W+ I= a ! b
correspondence
w + l=Ka,
respectively,
b e t w e e n elements
in such a way that, if
then
,
and
w+ I: Vxy(xiy'Kx~Y) Then,
obviously,
the domain of
we may identify
w+
and
~
elements
of
K
with the definable
with a definable
partial
ordering
subset of
on this subset.
We also assume that
and w + I= D ( ~ , a )
= a
is an index
~
for some
We may assume either that the set of constants I$8: b s E D S ~
or that there is a definable
f(~, 8, ~ ) In what follows~
is an index of
however,
a
{~
function
a :a
E D~° E D~}
f(x,y,z)
is contained such that
E D8 [c~ % ~ .
we shall ignore this m i n o r distinction.
in
374
Finally,
if, in a d d i t i o n
I= for
all
to all of this,
....
atomic
B,
iff
nodes
~,
and
definable
in
we have
dlff (a
.... ,and),
elements
a1~ ,...,an
E D~,
then
we
say
+
that
the The
model
~
is
definability
definability
of
of the
w
a classical
one
node
w ++ ,
model,
Kripke
model,
or
in
can
w+
be
can
taken
be
in
taken
the
as
the
obvious
manner. Three 5.6.3°
rather Lemma.
A(Xl,.o.,Xn)
obvious
lemmas
Let
be d e f i n a b l e
~
w i t h free v a r i a b l e s
with free v a r i a b l e s
5.6.4. Lemma. + w , then ZF
in
+
.
as shown,
iff
_F = --nI~l"'''Kl + is d e f i n a b l e in w . Let
K
for any f o r m u l a
is a f o r m u l a
from
w+
A*(x,xl,...,Xn)
such that,
be such that each
be d e f i n a b l e
+
Then,
there
for all
w + 1= A * ( d , ~ I d ..... ~ n d ) o
Let
Lemma.
w
as shown and parameters
l b A ( a 1 ~ ..... a n d )
5.6.5.
are
in
w
++
--IK
W++
and let
is d e f i n a b l e
be d e f i n a b l e
in
in
+
w
Then These
this,
K
is d e f i n a b l e
lemmas
we will
is d e f i n a b l e
will
need in
in
w
be applied
to prove
w+
(~
. shortly
that
in the c o n s t r u c t i o n
(Z~)*
finite).
is a model
But, b e f o r e
of
of models.
~
when
we can do this,
For
each
~E
we n e e d
the
following : 5.6.6.
Lena.
Let
a least node O
-i.e.
preserves Proof. I
of
~
the atomic
that,
cation
of the
in
w+
of
sXo
in
in
of
D~o ,
definable embedding
into
etc.
D~
(in ~ )
under
w+
under
there to
0
in
of
w+ w+
which
O
of
But, "y
of
~
and let into
~
have
the domain
of
is o n e - t o - one and
with
to
O "
(i.e.
the successor
The
the
0
for n o n - s t a n d a r d
is an element 0 °
truth
equations
of atomic x
~
we must
is expressible. and by induction
which
is the
formulae with
the
x - fold appli-
y=sXo)
D~ o
Dd ° ,
elements,
of the
of other atomic in
of
function,
in
associates
of the truth
of the r e c u r s i o n
~+
is the result
the map which
The p r e s e r v a t i o n
f r o m the v a l i d i t y
w+
function
of successor
preserved
D~ ° °
of
the r e l a t i o n
Dd ° x
fold a p p l i c a t i o n is o b v i o u s l y
I
H~,
for all
~
formulae.
successor
By the closure
of
is a canonical
one can match up the
w i t h the
observe
in
be a model
There
o a map of the d o m a i n
Obviously w+
~
.
x-
S(a,b)
the object formulae
and i n d u c t i o n
in
follows + w
Qo E. D.
375 (Note :
It is in steps
are i n t r o d u c e d these
like
in the model.
functions
in favor
We
5.6.7.
Theorem.
K1
that
shall,
of a more
the same w a y that a l g e b r a i s t s We m a y n o w prove
this
the e m b e d d i n g however,
informal
suppress
approach
avoid m e n t i o n i n g
functions
f(e~ ~, x)
further
mention
to the proofs,
such m i n o r
of
in much
difficulties.)
the f o l l o w i n g
_F = tK 1 . . . . ,K_n}
Let
.o.
be d e f i n a b l e
in
w+.
Then
(ZF)*,
K
\o+/ is a model
definable Proof.
of
HA
Z~ = ~
w+
5.6.4,
Z~
To define
o
w+
is d e f i n a b l e
in
w ++
(ZF)*
is
is d e f i n a b l e
(ZF)*
in
~
w+
in
w+ °
first
,
Obviously,
recall
that,
w+
is defin-
formally,
where
K = K1X I I } Thus
~o = ( 0 , 0 ) ~ K
Thus
K ,~
explain
domains
U K2~ t2} U ..o U Kn~ In} , and we may l e t
are definable.
~(~,x) ~o
in addition,
aJ
By lemma
able in
If,
++
in
and
"xE
~ w +",
is we have
will
denote
be the node f o r
recall
that
choose
a formula
its index.
w+
and d e f i n e
Let
(~= % ~ x~ ~+)vC~> %
so we must
that
~o
eteo
the elements
special
singling
(Recall
D(~,x)).
indices
of
w+
out the indices
that
w+
index
to d e n o t e for
is d e f i n a b l e
elements
elements w+ in
If w+
of
in all ~+
-
a E w+,
and c o n s i d e r
what we mean by this.).) To complete h o w to define as a Kripke definition
the proof the atomic
model in
B+
B*(~,Xl,..o,X
) ~(~=
NOW,
5.6.3,
by lemma
(~)*
is definable Let us assume
d ° )°
its d e f i n i t i o n
%~B+(%,x
Let in
f o r any node
iff
is not a model of
w+
B
in that
w + , we need w+
only
is d e f i n e d
be atomic :
B2
show
in
denotes
w+ its
Then
1 .... , X n ) ) V ( ~ > % & B Z ( ~ , X l . . . . . Xn) ).
for any f o r m u l a
such that,
lffA(a 1, . . . . an) S~ppose
_ (ZF)*
(say with node
Z~;
A*(X,Xl,...,Xn)
that
formulae.
A , ~,
there
is a f o r m u l a
and elements
al,..o,a nE D~,
w+ I= A * ( ~ , a 1 . . . . ,~n) o
~.
Then, f o r some A(x,x I . . . . . x n)
only
X,Xl,...,x n
~o I ~
VXl"''Xn[A(O'Xl .... ,x n) & Vxy(A(X,Xl ..... x n) & S ( x ' y ) ~ A ( y ' x
with
free, I .... , X n ) ) ~
VxA(x,x I . . . . ,Xn) ] •
37~ + Then,
for some
al,...,a nG w
~o 1~TZA ( 0 ' a l . . . .
,
'an) & V x y ( A ( X ' a l . . . . . an) ~ S ( x , y ) - - A ( y , a
1..... an))"
-- VxA(x,a 1 .....
an)-
Then ~o I~ A ( 0 , a l ' ' ' ' ' a n ) '
Vxy(A(X'al ..... an)$ S(x'y)~A(y'al'
.... an))'
% 11"¢A(a,al, .... an) for some for
aE ~+.
Letting
I:
~+
A*(%,
~, al .... 'En ) '
(I)
~+
Now
®+ I~A*(%, E, ~I .... '~n ) " s*(%,~,~) ~--)s(~,y), whenoe (~)
I: v~y[A~(%,~,~, ....~n ) ~ S*(%,~,~)-A*(%,~,~ .....~n)],
(2) ®+ I: Wy[A*(%,~,~I But
the
~o I ~ A(a'al ..... an) ' we see,
be such that
% {bA(~,~1,...,%),
defini~
A*
a
map
a ~
hecomes
..... ~n) ~ S(~,y)-A*(%W,~,
is
definable
in
whence
w+ ,
.... ;n)].
there
is
a
a
least
o
such
that
~+ l d A * ( ~ o' a s ' a 1 . . . . . an ) ° ao~0 ,
whence
ao=b+l
®+ I= A*(%,
for
some
b
By minimality,
S, :I .... ':n ) "
By this last statement
and
(Z),
+
A*(a o, a o, a 1 . . . . . :n ) ,
I=
a contradiction.
Thus
(2~)*
is a model of
HA. ++
The final comment, in
w ++
that
(Z~)*
is definable
follows from the d e f i n a b i l i t y
of
(Z~)*
if
in in
~+
m
+
is definable
and lemma 5.6.5. Q.E.D.
5 . 6 o 8 - 5.6. 9 . 5.6.8.
The H i l b e r t - B e r n a F s
o q m p l e t e n e s s ' theorem.
In 5 . 6 . 2 - 5°6°7, we proved two important r e s u l t s :
are definable
(ii) if ~+ definable
w+ ,
, as above,
in
of definable
in
m++.
then
(~I + o.. + K--~)*
is definable
in
is a model
~ ++ , then
(i) If
of
(~I + ' ' " +
But, to be able to apply these results,
Kripke models and definable n o n - s t a n d a r d
is obtained by appeal to the H i l b e r t - B e r n a y s 5.6°9.
Theorem
(Hilber%-Bernays
sistent
r.e. extension of ~c.
completeness
theorem).
Then, for any model
and
~)~
~+
is also
we need a stock
models of
completeness
~I .....
~:
~c
. This
theorem. Let
of
T
be a con-
~%CO~(~),
377
++
there is a n o n - s t a n d a r d For a proof,
model
see Kleene
+
~
of
T
w h i c h is definable
in
1952, XIV, T h m 5 3 6 - 40, or F e f e r m a n
~
•
1960, t h e o r e m
6.2. The C o d e l - R o s s e r - M o s t o w s k i
5.6oI0 - 5.6.12.
-Kripke-Myhill
theorem .
revisited. 5o6.10.
Our first two applications
the existence instances
O
of
ZI
of the above results will be a proof of
substitution
in de Jongh's
theorem.
instances
O
and u n i f o r m
~2
substitution
For these results, we need two r e f i n e m e n t s o For Z I substitution, we need, for r.e.
of theorem 3.3.1 and its corollary° ~,
AI,..°,A m
for
such that
~+A i
is consistent
i / j . W e present K r i p k e ' s
5.6.11o
Theorem°
Let
an r°e. relation +Pn+
~xPx
~c
~ Ai~
~Aj
be a consistent
r.eo extension
of
HA c .
such that, for every natural number
There is
n ,
is consistent.
P r o o f . ( Kripke {el(x) = y .
T
P(y)
and such that
proof :
196~. ) Let
R(e,x,y)
Define a partial
numeralwise
recursive
represent
the r e l a t i o n
function as f o l l o w s :
(choosing the first theorem of this form if there are more than one). has an index,
e .
Let
Px
be
R(e,e,x) . We show that, for all
Then n,
Pn & ~! xPx is consistent First,
with
observe
~o that
~(e)
is undefined.
If not,
~(e) = n o
for some
no .
Then b But clearly,
~ ( R ( e ' e ' n o) ~ ~ z R(e,e,z)) . if
~(e) = n o ,
a contradiction°
Hence
~(e)
~ ~(R(e,e,n) Hence,
for all
n,
is u n d e f i n e d
and for no
n
do we have
~ ~! x R ( e , e , x ) ) o
~ + R(e,e,n) + ~! x R ( e , e , x )
is consistent.
Qo E. Do
Letting we have
A be P n , we have the desired result. One might m e n t i o n n HA ~ A i - ~ A j for i / j as well as HA c ~ A i ~ ~ A . ,
For the 5o6.12.
o
H2
substitution,
Theorem.
of arithmetic.
Let If
T~H
is an infinite family,
H
that
we need the f o l l o w i n g
denote
HA c
augmented by all true
is consistent {AI,...,An,... I
and has a of
O
H2
o Z2
O
HI
sentences
enumeration,
sentences
then there
independent
over
378
(in the sense of 5.3.10
that we may choose any subset of them to be true
and the rest to be false). If we observe that the proof predicate is
o Z2
are precisely those numeralwise representable in
and that the ~,
o Z2
relations
we can mimic the proofs
of theorem 5.3.11 and corollary 5.3.12 to obtain an infinite set of independent
Z~
sentences.
Replacing these sentences by their negations yields the
theorem. 5 . 6 . 1 3 - 5o6.16.
o
ZI
Substitution instances in de Jqngh's theoremo
5.6o13. Recall that the reason we used the modified Jaskcwski trees in proving de Jongh's theorem was that every node was determined by the set of terminal nodes not lying beyond it.
Thus, if each terminal node was the
unique node satisfying a particular sentence, it followed that every node was the least node satisfying a conjunction of negations of sentences corresponding to terminal nodes°
Then, any set which could be the set of nodes
forcing a propositional variable under a propositional forcing relation was now the set of nodes forcing a disjunction of such conjunctions of negations. 4o In proving the existence of Z I substitution instances, we will assign to o each node of a tree a Z I sentence which is forced only at and above that node.
The substitution instances will be disjunctions of these sentences o Z I )o
(and will thus be
Note that we no longer need to use the special property of the modified Jaskowski trees that every node is determined by a set of terminal nodes. Nonetheless,
it will still be convenient to work with them.
Consider,
e.g.,
J3 :
I\
3\ ;4
5\ }6
% v
o
Starting at the terminal nodes and working our way down the tree, we shall assign theories to the nodes. over
~c
Let
AI,.oo,A 6
be
Z oI
mutually independent
(or, let them be obtained by theorem5.6.11 ).
Assign to
~i
the
theory
~i = ~ c +Ai +
j~i ~Aj.
By the independence of the family choose
BI, B2, B 3
{AI'''''A61 ' ~iT
is consistent.
Now
individually independent over
H~ c + CON(~I ) + ... + CON(T6) + ~ A I + ... + o A 6
(which is true in
~
hence consistent)
i% j °
8i
such that
~c
~ Bi ~ ~ B j
for
Assign to
and the
379
theory 6 ~ CON(~ ) +
T! = HA c + ~. + ~I
Again,
T'
~
l
i=I
is consistent.
6 ~
i
Finally,
~o + CONCT~) + C O ~ ( ~ )
mA..
i=I
z
assign to
+ CO.(~)
Having assigned such theories,
~o
the theory
+ ~B 1 + ~ B 2 + ~ 3
we now assign models of
to the nodes.
~c
Place ~ at ~o" Now ~ ~- (]ON(T~) + CON(T~) + COf1(Tg) , whence there are models ml, m2, and m3 of T~, T~, and T~, respectively, such that each ~.
is definable in
m.
Now,
1
®i I= T,i : ~e ~ Thus, in
mi
models of T3, T 4
+B.
+
i
j :~1 CO~(Tj) ~
6 AA m A . .
+
j=1
there are definable models of
TI, T 2 , respectively,
definable
in
~2 ; and
definable
w31' ~32
J
TI, ..., T 6 . in
~I ;
models of
Let
m11, m12
~21' m22 T5, T 6
be
models of
definable
in
w3 •
Thus, we have
Wll% 1 //~12
c~21 \ m { 0U22
m31~ //m32 w3
w
Now,
successively
apply the lemmas 5 . 6 . 3 - 5 . 6 . 6
ing structure is a model of
HA.
Further,
forced only at the node corresponding BI
is forced only at
and above.
~I
by a
Z o1
that the result-
w11 , A 2
at
w12 , A 3
at
AI
w21 , ....
sentence and we may proceed from here.
For ease in assigning
theories and models to nodes in the general
case, and
for ease in giving the proof, let us use the notation for trees of finite sequences as described in 5°3.3. <1,1>
<1,2>
J~,
e.g., will be represented by
<2,1>
<2,2>
<3,1>
<3,2>
< >"
Let
J*
be given and let
~1,o..,~n,
be its terminal nodes°
Choose
n
A~I ,
, Ann ' °'"
~T
is
and above, B 2 at w 2 and above, and B 3 at w 3 o Z I sentence is forced at w. Hence each node is
Any provable
characterized
to
to conclude
as we shall prove below,
such that
•
= HA c~.~ + Aai + J~iA~"m A ° j
~c
+ A i +
~
~
7A
jli
Let
~1'''''a k
is consistent
and let
aj
be the non-terminal
nodes of
38o
length
m
and let
~ *, ..., ~ . . < i > 1
m+1.
Aa I .... ' i ~ k and such that each A
be chosen such that is consistent
k Tin+I = HA O + A~ "~ i= I (Observe that Let
w
( i = 1,...,k)
~
of
and, for each definable
~s
in
i
Ta
~,
then
Ta
assigned
(OM(Ta.<j>)
has a definable
and successor
~a"
-hA j
for
i/j
1 k 1 ~A C O N ( T ~ I . < j > ) + /~A A~ mAai.< j=1 i=I j=1 J> "
is a model of this theory and, thus,
is a successor of
every model
HA C ~ A
with
T + . "~i = Tm+1 A~i Thus, every node s gets a theory
a.<j>
be the n o d e s of length
1
Let
let
to it.
Further,
is provable in
model of
a.<j>,
H a v i n g defined
it is consistent.)
Tq.<j>.
w .<j>
these models,
Let
if
T~.
Thus
~< >
be
be a model of
T .<j>
assign Kripke models
K_~
to the nodes as follows : (i)
Kq = ~
(ii)
let
for terminal
q*<J>,
= (S~.)* K_a.
• . . W
If
Lemma°
s 6 Da .
Let
trivially
is p r e s e r v a t i o n
Of course,
trary extensions
Observe
is a model of
sentence.
~ I~A
recursive,
Then
~. iff
~
I=A
we know that
w T 1= B s .
Now,
~ l~Bs
it does not f o l l o w
the w e l l - k n o w n
characterization
under extension and r e s t r i c t i o n
in w h i c h all of the new elements ~tiyasevich
for some
for end ex-
are larger than
1970 now gives us the result for arbi-
There are three tricks we can use here :
the language
The lemma follows (ii)
and let
- but this is far from trivial.).
Proof of lsmma 5.6o14o Expand
Z oI
K_~
primitive
be terminal.
(i.eo extensions
the old ones.
(i)
be a B
each
wu ~ Bs - m o d e l - t h e o r e t i c a l l y ,
of r e c u r s i v e n e s s tensions
~,
K_~.
A
say
~
that
of
~
Let
a I~ZxBx,
be the successors
:
By the lemmas 5 . 6 . 3 - 5.6.6, 5.6.14.
q ,
o.., ~*
so that primitive
recursive relations
are atomic°
trivially.
that the definable
The lemma n o w follows,
because,
extension for
sEw +
m ++
of
Vx< s
m+
is an end extension.
means
the same in both
models. (iii) A p p l y the theorem of N a t i y a s e v i c h
HA ~ A *-~ ~xl..oXmD(X1°..Xm) , where follows
trivially.
D
1970 by which,
if
A
is q u a n t i f i e r free.
is
O
ZI ,
The lemma then Q.E.D.
38~
5.6.15.
Lemma.
Proof.
Clearly
~
~ AT
iff
~T 1=AT "
~T
. T I~AT ,
Then
whence,
CAT ,
if
¢ I~AT ,
whence
I=AT • Conversely, implies that ~,
I= A T
trees.
is
A T . First,
fact that
T . Hence
but ~,
~
T
T'~
is terminal. .
Then
wa I= A T
But, the only terminal
~ = • , by the property of the modified Jaskowski
I= A T ~ ~ I~AT
I=T~ ' = ~ c
let
~ < T . Then, for some
= a' I~AT = ~ ,
+ ~, ~ A T ,
+ ~'~ CON(AT'
c <~' ,
I= A T , contradicting
the
+ T"~T' ~ A T " ) , where
w',
range over terminal nodes.) Let
T
not be terminal
of length less than ~ for any
and let
T , because,
~
p
of length greater T
Now,
whence
wa, J=A~, ,
and
than that of
Assume
a~ T .
~ o Thus,
a~'
for some
o' I~A
, , whence
HAC ~ A ~ , ~ ~ A T , by definition, Note.
I= A T o
a
cannot be
by choice,
~Ap,
at least that of
But
assume
for all terminal
(To avoid using this property,
T = ~ , . ,
T"
consider wT, 1= A T
~'
the length of
~
of length the same as ~ I~Aa, , whence
w
a contradiction.
is T .
I=A , . Q.E.D.
The above proof could have been simplified by unifying the cases
- which could have been done by stipulating treating the terminal nodes. using theorem 5.6.11. antee that
AT
that theorem 5.6.11 be used in
The non-terminal
- Unless we know
nodes must be treated by
~.c ~ A
will be false in extensions
~
,~ ~ A T ,
we have no guar-
of
(This observation
~' .
is due to de Jongh, who found and corrected original
attempt at proving the existence
the corresponding error in our o of Z I substitution instances.)
We may now prove 5.6.16.
Theorem.
BI,°..,B n
Proof.
Let
such that
Let
J*n
(If no O
is
ZI ,
~
Lst, for each W
forces S I~Bi
C(pl,...,pn)
Z oI
sentences
~ A ( B 1 , .... Bn) -
be given with a forcing relation on it such that
<>l A(pl ....
B i ~-@
Pp ~ A ( p l , .. .,pn) . Then there are ~
Pi'
A~. Pi'
iff
let
Bi
: I~pi ,
be any refutable
Z oI
sentence. )
Then
and a simple induction on the length of
shows
i c(Pl .... 'Pn)
iff
t C(B I .... ,Bn)°
Q.E.D.
B. I
582
5 . 6 . 1 7 - 5.6.19o 5°6°17.
o H2
Uniform
substitutions o
The p r o b l e m with the
ZI
in de Jongh's
theorem.
substitutions is that we could not choose o Z I sentence to be forced. E.g.
the nodes at which we w a n t e d a p a r t i c u l a r consider the tree
Suppose we want have a model
A, B
~
A , B 6 Z oI .
forced as indicated,
Then,
at
~4
we must
of
~4
~ c + B + c o ~ ( ~ c + B + ~A) + C O ~ ( ~ ° + B + A ) Since
B
is false in the standard model, Im c + - ~
is also consistent H~ c + CO~4(H~ c + B +
and ~A)
nested consistency tences exist. sistency
must be i n d e p e n d e n t
statements
statements
B + CO~(HAC + ~ B )
is inconsistent°) BI,oo.,B n 9° I
as desired°
Theorem.
Let
Proof.
and mutually independent
sentences
of arithmetic,
over
~c
when
we can assign n o d e s to
This is the basis of the f o l l o w i n g m o d e l - t h e o r e t i c
o
~I
(K,~, IW)
< > I~ A(Pl,...,pn) .
A: Let
BI,Oo.,B n
sentences,
be
and let
be an a r b i t r a r y
o
9 2 , independent
P~ ~ A ( p l , . . . , p n ) .
Let
T
is terminal,
If
a
has successors ~
BI,...,B n
tree model of
P
over
~c
Then
B. +
T "~
=
HA c ~
and let
satisfy the h y p o t h e s i s
to nodes as follows :
If
+
E2
I .... ,Bn).
and assign theories
~
o
are
(Friedman).
augmented by all true
~A(B
Larger trees will require larger o and we just don't know if any such Z I sen-
that we cannot have as much i n d e p e n d e n c e from cono o H 2 - sentences, since, if B 6 Z I , then
proof of a result of F r i e d m a n 5.6.18.
over
.
as with
augmented by all true the f o r m u l a e
B
+ CON(HA c+B+A)
(Observe
If, however,
-~A) + CO~(~c+B+A)
+ CON(~ c+B+
+
Tl~p i
z
T
~ . , ..o, ~ . , k ~B i + ~£O~(T~.<~>)
i
.
z
of the theorem
383 Each
T:
is obviously consistent and we may define models
starting at
< >
with an arbitrary model of
and, finally, we have a model of
HA.
T< > °
Then
w~ -~K
as usual, is defined
E.g. with the tree featured above,
we have
m<2,1>
where
w<2,1>, w<2,2 >, w<2 > ~ B ,
5.6.19 . Lemma. Proof.
Let
A
~2"
w<2,2 > ~ A . ~ lffA iff
VT > a
wT ~ A .
Probably the simplest thing to do is to appeal to MatiyaAevich
or add new predicate Vx1°"Xn where
o
be
w<2,2>
C
symbols so that
A
SYI""YmC(XI"'''Xn'YI'''''
is quantifier-free
VT>~
1970
is of the form,
and decidable.
Ym ) ' Then
Vsl...Sn6 DT ~tl...tmE DT(TI~C(s I ..... sn, tl,...,tm) )
= VT>o-Vs1.o°Sn6
DT ~tl...t n w
I= C(sl, .... Sn, tl,...,tm)
w>_~%I= vx1...x~y~..ymC, i.e. ~ I=A. The converse is just the definition of forcing for an
VS
combination. Q.E.D.
To finish the proof of the theorem, observe that
The
Q.E.D.
rest is just the usual induction. De Jongh's theorem for
5.6.20 -5.6.22. 5.6.20. Just as
MP
was not preserved by
preserved by
( )~ (Z)*.
definable in
w +,
(Z~+w+) *.
MP
and if
If, however, MP
it is not in general
( )~ (Z)' each element of
is valid in
~,
then
MP
~ = I~I, .... K_.nl is is valid in
This is just a variation of the result we will need.
verification of this variant is left to the reader. will need is the following :
A direct
The general lemma we
384
5.6.21.
Lemma.
Let
~
placing non-standard every node
~ ,
is v a l i d in Proof•
be a tree model of
models of a r i t h m e t i c
HA
obtained by the process
at the nodes. T _> a
there is a terminal node
of
Suppose that, for
such that
~=
wT .
Then
MP
~.
Assume
MP
is not valid in
~ - take
MP
in the form (iv) of
section 4 :
< > Iff w[vy(A~y v ~A~y) ~ ~ A ~ y ~ Then,
for some
~<
> ,
s E D$ ,
I~'~ Vy(Asy V ~ A s y ) & ~
P~
Thus, for some
, whence
Let
T~p
DT = D p t
.
Also,
P
be terminal with whence
p I~Ast
~3"Asy-- ~:fAsy .
t E D
p I~ Vy(Asy v ~Asy),
But
~A~y] .
p I~
~Asy,
wT = ~ p "
Then
t ~ Dp° ,
pl~yAsy,
pl~Ast,
ice.
5.6.22.
Theorem.
Let
T l~yAsy,
say
v l~Ast,
tE DT.
Also,
V~Ast
whence
P If'/- ~ A s y
a contradiction°
PF ~ A ( P l , . . . , p n )
Q.E.D.
. Then there are sentences
BI,...,B n
such that H~A+MP ~ A ( B I , . . o , B n ) Proof• where
Let AI
J* n
be given and define theories Z oI , independent
is
~m+1 = p c where
Am+ 1
~+~ m m
of
~Tm+I , let
of
as f o l l o w s :
T I = ~C+A
+~Am
~1,...,~n, be Given a model
I ,
HA ~ c .
+ (ON(~m) + ~ A m + Am+I ,
is i n d e p e n d e n t
the terminal nodes of
w
.
of
~c
+ (OM(~m)
.
Let
J* Let w n_~ be a model of ~Tn~ n mW be a model of -mT definable in
to the terminal node
classical model a s s i g n e d
~
m
.
Wm+ I .
In g o i n g down the tree, assign to
to the r i g h t - m o s t
successor of
~ •
J5 ,
Assign a
the eig•~
looks like
~4
~
This gives us a Kripke model of ~P °
Finally,
w6
~
o
By the lemma,
it is also a model of
each terminal node is the u n i q u e node f o r c i n g a p a r t i c u l a r
385
sentence
.
The proof of de Jongh's theorem in section 3 now goes through
easily.
Q.E.D. o ZI
Note that H~+~
substitutions are impossible:
~ A ~ A
If
A
o ZI,
is
.
5.6.25- 5.6.25.
Other applications.
We first present a !emma. 5°6.25. Lemma.
Let
HA+~ TI(<) , each valid in Proof. TI
<
be primitive recursive,
--IK" definable in
~+ , and
~1,...,K_.n
w + I=TI(<) .
Use
TI(<)
in
~+
applied to
A(x,xl,...,Xn)
A*(~o, ~, x1,...,~n)
is forced at
By insisting that each theory TI(~) , every model
is
~
Q ZD
used in 5o6.13- 5.6.22 also contain
K_~
(where independence over HA+TI(<)
H~ c
(}~O+TI(<)
(~c
~+N[P+TI(<)
A similar proof does not work for was preserved if
was forced By
s°
in
for some natural number provable.
+ true ~ )
~+TI(~)
is replaced by independence
~
o
RF(T)
or R~N(T~ .
was, when we assumed
( ~ ) ' , it followed that n.
o ZI
Thus, the
+ true H~)) . Further, the unrefined version of
de Jongh's theorem holds for
RF(~)
to verify that
s°
encountered is a model of TI(<) . o substitution and uniform H 2 substitution results hold for
prove
TI(<)
(ZZ)*
applied to
over
models of Then
Recall that, to ~xProof T (x,~A ~)
ProofT(n,rA~ )
From this it followed that
We can no longer reason in this manner for
A
was forced
was indeed
(Z~)*.
We can,
however, appeal to the following result of Kreisel - Levy 1968 ; (Theorem 12,
p. 125): 5.6.24. Theorem. TI(~¢s )
For small ordinals
~,
~
together with the scheme
(transfinite induction on the canonical well-ordering of type
is equivalent to the system obtained from T -~ T + R F N ( T )
1+s
times ; e.g.
HA
by iterating the process
HA+RFN(H.~A) = ~
+ TI(~¢o ) .
5.6.25. It follows from theorem 5.6°24 and the above remark that the o and n 2 results hold for HA + RFN(H~A) , H~ + RFN(HA + RFN(HA)) , etc. results for
H~ + RF(H~)
e )
o ZI The
follow trivially from the results for the extension
It also follows that we get the unrefined version of de Jongh's theorem for
~+~U?+TI(<)
result for
, HA+~[P+RFN(~)
H~ + ~iP + RFN(HAA + MP)
+ ~ + RF~(~ + ~)
, KA+~KP+RF(HAHA) - we do not have the
because it is not known if
= ~+
M~ + T i ( < % ) .
386
In discussing
the operations
closure properties
( )~ (Z)',
of the classes
by these operations.
~, ~w,
In disussing
~*
+~)' , we gave some
respectively,
the operation
of models described in the statement the classes
( )-- (Z
of sets
( )-- ( Z ) *
F
preserved
and the class
of lemma 5.6.21, we should comment on
of sets of sentences valid in the Kripke model of the lemma
provided they are valid in all of the non-standard model is composed.
models of which the Kripke
We both lose and gain some closure conditions.
In both cases, we lose Friedman's
condition
(iii)
(theorems 5.2.11 and
5.4.13 above) which we restate here for convenience : Condition all
n,
(iii)
then
Recall that, if in
(Z_F)'
If
FE ~(~w) , A
has only
x
free, and
HA+ F~
An
for
F U IVxAxl E ~ (resp., ~w) .
or
F + VxAx
was valid in
(ZF+ w)'
~,
VxAx
when some instance
~o - which is ruled out by the hypothesis. valid reasoning in the present
An
could only fail to be valid was not forced at the node
Obviously,
this is no longer
situation where the new origins have non-
standard integers in their domains. For
( )-- ( Z ) * , the use of definability
to condition rebate.
(iii)o
For applications
to
does not give us an alternative
~P,
however,
Rather than to try to state an intelligible
we do have a slight
analogue to theorem
5.4.13, let us consider an example : 5.6.26.
Theorem.
(i)
~+~
(ii)
~+TI(~)
Let
~I
be the class obtained by the following :
~ ~I~ E ~I ;
(iii) the union of any r.e. (iv)
if
F E ~I ' A
HA c + all true Then, for any Proof.
sentences,
F E ~1'
~+
First, by (iii) all
sentences, Also, if
B,
then
F
has
F ~ ~I'
F
definable
fails to be forced°
A
~I
is in
~I ;
is consistent
with
DP.
F E ~I
are consistent with
is r.e. and, if
~I' ~ 2 '
and
T U IA} 6 ~I "
which includes the consistency
Kripke models respo
H °I
sequence of elements of
is a Harrop-sentence
~c
+ all true
O
HI
statements needed to define models.
~.~+F~A,
H~A+F~B,
in some model
~+
of
~+
we can find F,
in which
A ,
(To see that these are definable Kripke
models, use the reduction of the problem to the Hilbert - Bernays completeness theorem outlined in 5.1.26.) which
AVB
(~I + ~ 2 + ~+)*
is a model of
We cannot generalize
n o Second,
~+
F
in
Q.E.D.
would have non-standard for some
Then
is false° this to obtain integers and
ED.
First,
~o l ~ x A x
to have the definability
the domain at the origin
would not imply of forcing for
~o l~An
(Z~+ w+) *
in
+
w
, for an infinite family
~,
for all elements in the class°
we must have a uniform forcing definition But, the minute we have this, we have models
387
K --a
for non-standard
a
occurring in our description of
F - i.e. we are
not defining the model we want to define. Such esoteric results as theorem 5.6.26 are of little interest in themselves.
They do~ however, illustrate the differences in our ability to
treat
~
and
HA+MP
prove
ED
for
FE ~I )"
(as do such negative results as our inability to
388
5 . 7 . 1 - 5.7.2. 5.7.1.
Subsystems
of Heytin~'s
In van D a l e n - G o r d o n
to settle example, ', +,
1971 , van Dalen and G o r d o n apply Kripke models
some independence consider
questions
the system
, and axioms
arithmetic.
~
regarding
with
(in a d d i t i o n
subsystems
the constant
of
~.
0 , function
For symbols
to axioms of the i n t u i t i o n i s t i c
predicate
calculus w i t h equality) : x' = y '
-* x = y
-~X~
x+O
=
X
x+y'
x'O
=0
x - y'
=
x+y
= y+x
x.y
=y.x
(x+y)
+Z
=
x+
Then~
O
= (x+y)'
(xoy)
(y+z)
m x = O
=
x.y
• ~
x
+
= ~.(y.~)
ay(y'=x).
-
van Dalen and Gordon proved :
5.7.2.
Theorem.
Proof.
~ ~Vxy(x=y
(van D a l e n - G o r d o n
the field of real numbers. w h i c h are i n f i n i t e s i m a l l y *N = {x6 * ~ : Now,
consider
V mx=y)
1971). Let
o
Let ~
*~
be a n o n - s t a n d a r d
extension of ~ ,
denote the set of elements of
*~
close to some natural n u m b e r - i.e.
~ n 6 ~ ~infinitesimal
6 ( x : n + 6)}.
the model,
~I *N2
I ~O *NI
'
D ~ I = D ~ ° = *N
where herited
from
*~ '
the operations
~o I ~ a = b
a9 b
are close to the same natural number).
~o I ~ n = n + 8
Observe at
~I
GO I ~ n = n + 8 Vmn=n+8
h
and,
on
*NI, *N 2
are those in-
are i n f i n i t e s i m a l l y
close
The axioms of
~
of equality is not forced, since
~I I~ n : n + 8 '
are obciously since, if
6
~0 I ~ ~ n : n + 6 .
.
Q.E.D.
the standard model GI
of
(i.e. iff
that, in the model given in the proof of the theorem,
is, basically,
duction is forced at duction
and
but the d e c i d a b i l i t y
is infinitesimal, Thus
a
', +, •
actually denote the same element
and
G o,
iff
a, b
•~,
forced at
~I I ~ a = b
iff
w.
Thus,
and the double n e g a t i o n
the model
every instance
of in-
of every instance
of in-
is forced at
by induction,
~ . Hence, a l t h o u g h equality is provably decidable o it is not provably decidable by the double n e g a t i o n of i n d u c t i o n
- or by i n d u c t i o n on H a r r o p - formulae,
since, as the reader may easily verify,
589
if such a formula is forced at
~I ' it is forced at
~o
Thus, as one
might expect, one cannot use a negative form of induction to prove the positive result that equality is decidable. This raises the question : How much induction is needed to prove the decidability of equality? By induction on quantifier-free formulae, Vx(n=xV ~n=x) .
one can prove, for each
n,
(Thus, quantifier-free induction fails to hold in the
above model and quantifier-free induction is not derivable from the negative formulations of induction mentioned.)
Can one use induction on quantifier-
free formulae to derive the decidability of equality - Vxy(x=y V ~ x = y )
5.7.3.
E x t e n s i o n s of
H~:
?
Theory of s p e c i e s .
Aside from some comments on free choice sequences in Kripke's original paper
Kripke 1965 , the only discussion of Kripke models and higher systems
published to date is Prawitz 1970 in which Prawitz proves the completeness of the cut-free rules of the second-order intuitionistic predicate calculus with respect to second-order Kripke models subclass of these models, namely,
(and also with respect to a proper
the second-order Beth models).
Since the
induction scheme is given by a single axiom of this second-order language, this yields a completeness theorem for second-order arithmetic plus comprehension with respect to these second-order models. The simplest way to describe the second-order Kripke models is to say that the domain function
D
splits into two functions
DI
and
D 2 , each
satisfying the monotonicity condition, and such that there is a binary membership relation,
6 , between elements of
Further, when discussing comprehension, and every formula variable in
A(x)
with parameters from
A , there is an element
i~ W [ A x
X
and
D2~
(for given
~).
of
DI~ D2~
and
D2~,xthe
only free
such that
*--9 x 6 X] .
(In the presence of a little arithmetic, relations in
DI~
one assumes that for every node
we can restrict ourselves to unary
D2ffo )
In attempting to construct models of second-order arithmetic,
the obvious
approach is to mimic our procedure in constructing models of first-order arithmetic - but also insisting that the classical models being used be models of the second-order theory with comprehension.
We have not considered
this possibility thoroughly enough to say whether or not it will lead anywhere. Let us consider a simple example. (~+,A),
where
B
and
A
models.
We would define a model :
Suppose
(~++, B)
is definable in
are the classes of sets of numbers in the two
390
(~++, B)
I % (w +,
A*)
.
The choice of
B
is obvious ; but what do we choose for
simply choose
A
since,
extension of
for arithmetical %
C
~e cannot
(w+,A) J need not be
~++
is not an elementary
C(x)
for which these sets
The obvious approach is to start with species
XC
such that
I~ Vx[x~ X c + ~ O(x)] .
We cannot do this for A*
is true in
(E.g. as long as
w + , there will be arithmetical
will have to differ.)
what
Ix I C(x)
e.g.,
Ixl0(x) is fo~cedat % 1
A*?
is.
C
with species variables
since we don't know yet
Adding a species at a time, one can handle comprehension for
formulae
~EXI...Xn C(x,Xl,...,X n) , where
C
has no bound species variables ; but one cannot automatically
handle more complex formulae - each new species added changes species and hence the nature of universal Another possibility individuals
is the use of
problematic
~
comprehension,
is to guarantee
a E DI~,
x
free°
we need at ~l~aE
X
~o
Iff
behave like
automatically
S
Let
represents
~.
A
scheme, even
A simple way to guarantee species are in the domains.
(K, ~) , A
a set ~I~aEA
at node
The induction
that all possible
and let
= {aEDI~
X E D2~ ° s.
D2~ ~D2~ O be the set of all partially sets of natural numbers indexed by K S
- i.e. models in which the
will obviously be forced and the only
be a model, with partial order
with only the variable
any
formulae,
scheme is that of comprehension.
comprehension Let
quantification.
~-models
are precisely the natural numbers.
applied to second-order
the domain of
be a formula
: ~l~Aa}.
To guarantee
such that for any
To guarantee
satisfying
For any formula
~
and
this, we simply let
ordered systems,
and comprehension
Ax
~B
~ = IS ~a~ K ,
of
= S ~ S ~ , and let
A , the system
A = {A la~ K
is valid.
This latter type of model can be used to obtain certain formal results, e.g. it is easy to construct a model VxmVXI...VXnA(XE
Xl,...,xE X n)
is valid for any propositional in the intuitionistic example,
formula
propositional
A(x E X1,...,x E Xn)
an arithmetic
(the full binary tree) in which
counterexample,
A(pl,...,pn)
calculus.
which is not derivable
As a second-order
counter-
is as simple as they come - one might hope for or at least a version of de Jonghls theorem ;
but these models will not yield such results,
because all true arithmetic
391
formulae are valid in them. explicit definability 5.7.4.
Similarly,
or disjunction
Other set-theoretic
they cannot be used to prove the
theorems.
approaches.
The Kripke models only form one of several classical modellings tuitionistic lattices, sitional
systems.
Others include the Beth models,
and topological
interpretations.
calculus and the first-order
For higher systems, Beth bodels
Their applications
predicate
(and alse Kripke models)
arithmetic
to the propo-
Prawitz
1970 applies the
to the theory of species ; Scott 1968 interpretation
order of the continuum ; and Moschovakis pretation to second-order
in
calculus are well-known.
they have barely been applied.
and Scott 1970 apply the topological
of in-
interpretations
A
to the theory of the
applies the topological
- i.e. arithmetic
inter-
with quantification
over functions. In Scott 1970 , Scott proved the validity of Kripke's ~oschovakis,
in ~oschovakis
a system of second-order
~{~(~/
A,
showed the consistency
intuitionistic
arithmetic.
schema in his model. of this schema with
Kripke's
schema,
0 "--eA(x))] ,
is important in that it contradicts (see e.g. Hull 1967).
many theorems of classical analysis
Chapter VI ITERATED INDUCTIVE DEFINITIONS,
TREES AND ORDINALS
J. I. Zucker
§ I.
6.1.1.
Introduction°
This chapter contains
ordinals"
of first-order
inductive
definitions.
an investigation
intuitionistic
theories
A characterization
theory
(or rather tree) classes. 52
I D2(A )
of twice iterated
of the supremum of these ordinals
is obtained in terms of recursive functionals number
of the "provable recursive
on trees of the first three
Specifically,
we describe an intuitionistic
of trees of the first three classes and functionals
of finite
type over them, with a distinct ground type for each class, and a separate functional
for (finite or transfinite)
Although
the functienals
functionals
on ordinals,
mathematical
literature
st~active)
the word "ordinal" to denote,
cofinal
tive sense. theory
of previously obtained
is given,
is that certain results,
either in a classical
stated in
or in a construc-
respectively,
a classical
and a "constructive"
(i.e.,
interpretation.
In addition,
this work shows that proof-theoretical
formulated elegantly and intelligibly relations)
structure,
"ordinals".
An example of this is given in 6.6.3 (a) and (b), where the
~2
recursive)
can be interpreted
but in fact (con-
together with some additional
sequences
One advantage of the present procedure terms of trees,
is often used in constructive
not classical ordinals,
trees, or well-orderings
such as distinguished
recursion on each class.
on trees considered here are less familiar than
rather than ordinals
results can be
in terms of trees (or well-founded
(or well-orderings).
(For certain purposes,
on the other hand, it may suffice to consider only the ordinals
of trees, i.e.
to suppress the tree structure.) 6.1.2.
0utline of the contents of this chapter.
In § 2 we describe intuitionistio iterated inductively
defined sets
the pair of defining formulas
inductively
and
A ~ (AI,A2)
which is stricter than positivity, "well-known"
first-order
QI
Q2
theories
of (twice) where
satisfy a syntactic condition
but sufficiently
defined sets.
IDD2(A )
of natural numbers,
broad to include all the
395
With each
a6 QI
is associated an ordinal
IID2(A)I , the "ordinal of lID21 , the "ordinal of ordinal
IID~I
ID 2 " , as
IIADII and
We then define I.~D2(A) b QI ~}'
sup{ID2(A) : A 6 ~ }
corresponding
is defined for the
and the ordinals
laIA1.
sup{lalAI :
ID2(A ) " , as
IID~I
and
o Similarly, the
classical theories
IDa(A) ;
are defined analogously for the in-
tuitionistic and classical theories (respectively) of non-iterated inductive definitions° In § 3 thetheory
32
of trees
is described.
There are three ground
types : 0 , for the natural numbers (the "first number class"), trees of the "second class", and
2,
I , for
for trees of the "third class".
In § 4 the notions of conversion and reduction for terms of
~2
are
defined, uniqueness of normal form is proved, and the normalizability of closed terms of
22
is proved by a computability argument.
In § 5 we consider ~
computability, and hence prove the normaliza-
bility of all terms (not necessarily closed) of In § 6, well-founded (wf) models of wf
~2
22"
are discussed.
Four specigic
models are considered : (a) the full set-theoretical model
model
HR0 2
finite type over
01
and
02 , (variants of) the reoursive first
second number classes of Kleene and Addison, HEO 2 Let t6 CT T
of (b), and (d) the term model CT T
(b) the
its computability, and also the ordinal
say) for any ItlM ~ ItIc.
~2
of type
model wf
model
M
of M,
~2°
ItlN
Then for
but for
tE CT 2 ,
This failure of invariance of
T . With each
Itlc
T = I and 2) we can associate an ordinal
wf
and
(c) the extensional variant
CTNF 2.
be the set of closed terms of
(for
in a given
J2 '
(referred to above) of hereditarily recursive operations of
by
virtue of
of the tree denoted by
t
t6 CT I, ItIN = Itlc (= Itl ItIN ItIM
varies with for
t 6 CT 2
M,
and
points out
an interesting difference between non-iterated and iterated inductive definitions (discussed in 6.6.6). Now we define the "ordinal of
~2 " :
I~21 ~def sup{Itl : t6 CT1} . The main result of this chapter can now be stated :
(1)
t 2f = 1 21o The one inequality
Iz 21 ~
I~2]
is
proved
(6.6.8) by
two methods:
(a) formalizing the construction of the model
HR0 2
of the inductively defined sets
and (b) formalizing the proof
of computability of terms of
22
defined computability predicates respectively.
CI
and
@2'
in a theory
I D2(~ )
in a theory
I~2(C )
CI
for terms of type 1 and 2
and
C2
of the inductively
394
In order to prove the reverse inequality, and hence (1), we consider a functional interpretation (modified realizability) of This extends the
m r - interpretation of
~imrQit~ ~ P i ( ~ i , t F and
P2
(i: 1,2),
where
HA
ID2(A )
(chapter l l I , §
~i~is
a type
i
in § 7.
4) by defining:
variable, and
PI
are two new predicates (not interpretable by recursive relations
in
HRO 2 ).
is
~2
So
ID2(A )
is actually interpreted in a theory
augmented by the predicates
PI
and
2-~2['
which
P2 ' with appropriate axioms
(and also extensionali%y axioms).
By means of this interpretation, it follows that if some number proof in
a,
then from the proof we can find a t e r m
E-~2~
I t] Z [a[A1,
~2(A) b Q1 ~
of
P1(t,~) °
and so ( 6 ° 7 . 9 )
Further, for this
o b t a i n the i n e q u a l i t y
for
t 6 CT 1. and a
t , we can prove:
II.~D21 i
[221,
and
hence ( l ) ° It will also be clear, by a simplification of these arguments, that a corresponding result holds for non-iterated inductive definitions :
I~11 where
I~11
: I~11,
i s the " o r d i n a l
o f " a system
~I
of functionals of finite
type on trees of the second class only. In § 8 we turn to the problem of characterizing the ordinals IID~I
by means of functional interpretations of the classical theories
I~D~(0)
and
IDa(O)
turns out that for system
II~DII and
~I(~ )
of the inductively defined sets
01
and
02 .
It
IDa(0) , a Dialectica interpretation is possible, in a
of functionals, consisting of
constructive) number selection operator
~.
~I ' together with a (nonThen by means of a "majorizing"
technique (essentially due to WoA. Howard) we show that the ordinal of this system is no greater than
I~1(~)I
I~II ' giving the result:
However, attempts to prove I ~ 1 ~ 1221 by a f = e t i o n a l i n t e r p r e t a t i o n (modified r e a l i ~ a b i l i t y or m a l e c t i c ~ ) of ~ ( 0 ) , in which Q.t is trans1
!ated as
~ i P i ( ~ i , t ) , fail.
I~I
= I~21
or
So the problem of whether
I~I
> I~21
remains open. In § 9 we consider (briefly) extensions of the result (I) for intuitionistic systems
IDv(A )
of inductive definitions iterated
v > 2 , and corresponding systems However
(unlike the case for
T
v
of trees of the first
v = I ) it is not known for any
times, for 1+ v v> 1
classes. whether
the result analogous to (2) also holds for the ordinals of classical systems
395
IDa(A) , i.e. whether
~IDC
= I~Vl
(as stated above already for
NOW the proof-theoretical equivalence of these classical systems
V = 2). IDa(A)
with well-known subsystems of classical analysis has been established by Feferman 1970 , so a positive answer to the above problem would then also provide an interesting characterization of the "ordinals of" these subsystems of classical analysis (i.e. the suprema
of their provably recursive
well-orderings). By contrast (6.9.2), it i__ssknown, for (respectively classical and intuitionistic) systems nitions, that
ID~w(A ) IID~I
and
I D<w(A )
= IID<~I ;
of finitely iterated inductive defi-
and here we do have an ordinal character-
ization of a corresponding subsystem of classical analysis. Finally (6.9.3), we describe a (classical) system of iterated inductive definitions, which is proof-theoretically equivalent to classical analysis with the 6.1.3.
n I - comprehension axiom (modifying a result in Feferman 1970). Historical note ~ comparison with other treatments.
Systems (especially intuitionistic) of non-iterated "generalized inductive definitions" were formulated and studied in Kreisel 1963 C. An intuitionistic system of finitely iterated inductive definitions was formulated in Kreisel 1964 in order to prove the well-foundedness of Takeuti's ordinal diagrams of finite order. Feferman 1970 (as stated above) considered classical systems of inductive definitions iterated along primitive recursive well-orderings, and established their equivalence with subsystems of classical analysis. Other known characterizations of
IIDII (= II~D~I)
in terms of functionals
on ordinals (due to Howard and Feferman) and Bachmann's notations (due to Howard and Gerber) are described in 6.8.6.
There also we describe the
(known) (proof-theoretical) reducibility of
ID~(~ )
to
IDDI(O) , and state
a conjecture (also made independently by Martin-L~f) giving
lID21
in terms
of Bachmann- Isles notations. ~rtin-L~f
1971 analyzed an intuitionistic theory of finitely iterated
inductive definitions by an elegant extension of normalization results, rather than by an extension of functional interpretations, and gave an ordinal characterization of this theory (see 6.9.2).
However, the function-
al interpretation given here seems more amenable to a direct calculation of ]I~D21 .
In addition, the
wf
models of
~2
seem to be mathematical
structures of independent interest. As pointed out in Kreisel 1971 , § 2, syntactic transformations on derivations are suitable for obtaining derived rules (as can be seen in Martin-Lof
396
1971, § 9.5), whereas functional dence results, Vn(~ 6.1.4.
interpretations
can be useful for indepen-
cf. 6.8.8, where it is shown that the formula
OIn ~ @In)
is independent
of
I D2(0) .
Note to the reader.
Although many of the references intended to be self-contained, I, II and IIl.
can be read with profit,
or rather,
Some paragraphs,
this chapter is
depend only on (parts of) chapters
of a more or less technical nature,
are
enclosed in square brackets. 6.1.5.
Acknowledgments.
This chapter originally formed part of the author's doctoral (Stanford University, supervisor,
Prof.
1971 ) .
S. Feferman,
dissertation
~y sincere gratitude goes to my dissertation for his invaluable
and unstinting
guidance.
I also wish to thank Prof. G. Kreisel for his very helpful criticisms and suggestions,
and P. ~ r t i n - L ~ f
and H. Friedman for many fruitful discussions.
I also want to express my appreciation supporting my first year of graduate
to the Newhouse Foundation for
study at Stanford University.
397
§
2.
I~,D2(A) o
The systems
6.2.1.
Inductively defined sets of numbers.
Let
~
be the language
of
~
(1.3.2)°
languages formed by adjoining to and
X~Y
~
and
~X,Y]
are the
unary predicate (or set) variables
X
resp., but with quantification over number variables only.
Notation.
a 9 b, e, k, m, n, ...
denote number variables
s, t Let
~X]
denote number terms.
AI(X,a )
be a formula of
~X]
which is monotonic in
X,
i.e.
vx,x, c ~[Al(X,a ) ~XcX,-~AI (X, ,a)] (provably in A set
HAS
XcN
is
say), where
N
is the set of natural numbers.
A I- closed if :
V a ( A I ( X , a ) -~ a E X) . Let QI -=def r~ I X a N : x is A I - closedl . Then we can prove (in, say~ ~ ) : (i)
QI
(ii)
VXcN(X
is
At-closed
(using monotonicity of
is A 1 - c l o s e d - ~
(i) and (ii) characterize Now let when
X
A2(X,Y,a )
QI
be sets
Y
uniquely as the least
be a formula of
i__£sinterpreted as
A I ),
and
QIaX) o
QI "
~X,Y]
A I -closed subset of N.
which is monotonic in
?re define, similarly,
Y
A 2 -closed sets to
such that Va(A2(QI,Y,a) -* aE Y)
and
Q2 ~def A { Y c N : Y
Again,
Q2
is the least
is
We define, for ordinals
v< Q,
sets
QI,~ =def {a,: AI( ~ v QI,~' QI U QI , v " v<~
Then In fact, where
A 2 - closedl .
A 2 - c l o s e d subset of
~I
Q1,v
N o by induction on
v :
a)} .
QI
U Q v<~1 1,v is the first non-recursive ordinal (Specter 1961).
We can associate with each
a E QI
the ordinal
lalA1~defminl~'a~%,~l. In this way the elements of
QI
can be thought of as ordinal notations.
398
The theor~
6.2.2.
I~D2(A) ; definition of
II~D21 .
In the study of theories of iterated inductive definitions, visable
(at least to start with)
stricter than monotonicity, in a set
parameter
X
to consider conditions on
e.g. positivity.
(This
and
A2
A formula is said to be ~ositive
if it contains only positive occurrences of
i.e. there are no occurrences of plication.
AI
it seems ~d-
X
X,
contained in the antecedent of an im-
concept of "positivity"
is weaker than the one defined in
Kreisel and Troelstra 1970, § 4.4.) For classical theories the requirement of positivity is not really a restriction,
since it turns out that a monotonic inductive definition can be
reduced to a positive inductive definition unknown
(6.8.1) ; however, it is still
whether this holds in the intuitionistic case.
We will consider a syntactic condition stricter (even) than positivity,
C,
defined below, which is
although it is sufficiently broad to include
all the "well-known" inductively defined sets. Classes
Co, C I
and
C2
of formulas of
~,
~X]
and
respo are
~X,Y]
defined as follows. is the class of formulas of
~
built up from prime formulas
O
by
&, ~, CI
A
and
V
(i.e. the "negative formulas" of
(s=t) E C I
for any number terms
2.
Xt E C 1
for any number term
3.
If
A,B 6 C I
4.
If
A 6 ~I
Remark !o
in
and
B 6 C
O
t
A&B, then
s, t .
VnA, ~y~A. (B~A)
E CI°
were replaced by
"B
As it is,
CI
is a formula of ~", which are positive
would be precisely the class of formulas of
X. C2
then so are
If in clause 4, "B 6 Co"
~I
is a subclass of this class. *
is defined inductively by :
I.
(s=t) 6 C 2
for number terms
s, t .
2.
Xt 6 C 2
for number terms
t o
3.
Yt 6 ~
4.
If
for number terms t 2 A,B 6 C 2 , then so are A & B ,
5.
If
A 6 C2
Remark 2. ~X]",
1.10.6).
is defined inductively by :
I.
then
~,
(s = t)
and
B 6 CI
If in clause 5,
then
C2
then "B c ~I"
A VB,
(B ~ A )
VnA, ~nA.
E C2 .
were replaced by
"B
is a formula of
would be precisely the class of formulas of
~X,Y]
which
@
On the other hand,
~I
~t
.
.
.
neither includes, nor is included in, posltlvlty
as defined in Kreisel and Troelstra 1970 , § 4.4.
t!
399
are pesitive in
Y . As it is, O 2 is a subclass of this class. ×~ Finally, ~ Edef I 2° Let AI(X,a ) and A2(X,Y,a ) be formulas of ~I and ~2 resp. in which
is the only free number variable. predicate constants. ~QI] , A
resp.
~Q],
is
~
$~D2(A)
and
Q2
QI ' resp.
be two new unary QI
and
Q2"
(AI,A2) 6 ~.
consists of
cluding the equality axioms formulas of
QI
augmented by
denotes the pair of formulas T h@ the or2
Let
~
(1.3.3) in the language
a=b & Qi a ~ Q i b
(i = 1,2)
~Q]
(in-
and induction for all
~ Q ] ), and also the following axioms for
QI
and
Q2 :
%.I) A1(%,a)--%a QI.2)
Va[AI(F,a)'F(a)]
q2o 1) Q2.2)
A2(QI,Q2 a ) ' Q 2 a Va[A2(QI,F,a )-F(a)] -- Va(Q2a--F(a)) ,
where
F(a)
AI(X,a )
is any formula of
by replacing
Axioms
Qiol
(for
express minimality of are definable in ~ Q ] Alternatively, on
Qi
-- Va(Q1a--F(a)) °
Qi°2
Xt
by
~Q]
AI(F,a )
is obtained from
F(%) .
i= I or Q. 1 )o
, and
among
2)
state that
Qi
is
A i- closed, and
Qi.2
A.- closed sets (at least those sets which 1
can be thought of as expressing (transfinite) induction
(i= I, 2) .
The "ordinal of
I~D2(A) " is now defined as :
1~2(A) I hef supilalA~' ~2(A) b%~} and the "ordinal of
~ID2 " as :
II-~D2l ~def supll (In fact, as we will see,
2(A) l : A~to I~21 = I~2(A)I
For certain pairs of formulas special symbols, viz. for the predicates
A ~ C,
C m (01, 02 )
QI' Q2
I D2(C) ) f o r
(see below)
and
C ~ (CI, C2) ,
both
of the formal theory and for the corresponding
sets in the intended interpretation.
(or
for suitably chosen * ~ )
to be defined later, we will use
I D2(A) , 4 O ]
In such cases we will write
f o r the language
~[Q] , 4 @ 1 ]
I~D2(0 )
for
~[Q1] ,
etc. We now define a particular pair of predicates or sets which will be used in 6°6.4 and 6.8.
~ ~ (~I' G2)
They are (simplified versions of) the
second and third recursive number classes of Kleene 1955 and Addison and Kleene 1957 resp°
They are defined by the formulas:
Al(X,a ) ~ a=0 . v . a=3.5 (a)2 & Vn X l ( a ) 2 t ( n ) , A2(X,Y,a)
~ a:O . v. a:32.5 (a)2 ~ Vn(Xn--Y{ (a)a }(n)),
400
where
(a)x
tation of
is the exponent of the a,
and
XI(a)21(n )
x th prime in the prime factor represen-
means
~v(T(a)yny & X(Uy))
notation (1.3.9, A) (and similarly for It is easy to see that
in Kleene's
YI(a)21(n ) ). *
(AI,A2) E ~ o
Some further definitions. I~DI(A )
is the intuitionistic
defining formula A(X,a)
(i.e.
theory of one inductively defined set, with
~+QIo1+Q1.2 , in
~QI] )°
Again
1~I(A)1 ~def supflalA : ~1 (A) PQI ~t and
I~11 ID
is just
~def sup{l~1(A)I , A~¢1}" HA o
~O
Finally,
IDa(A),
IID~(A) I
and
analogously for the corresponding Remark° I~DI(A )
IIDCl
For an inductive definition is conservative over
plicitly definable in
~
(for
v = I or
2)
are defined
systems with classical logic°
2'
A
of the class
F
(chapter I, § 4),
since the inductively defined set is ex-
(1.4o5)o
Here and elsewhere dots are used to punctuate formulas in a well-known way~
so e.g.
A-~: B. V. C & D
means
A -~ [ B v ( C & D ) ]
°
401
§ 3. ~ L ~
~2 °
This is an intuitionistic
theory of trees of the first
three number
or
tree classes.
ductively
6.3.1.
structure. The set ........ by the two clauses :
(i)
~2
contains
(ii)
~'~2
0
Type
3 gr~d
T ~2
of type symbols
types : O, 1
and
class,
1
is the type of trees of the second class.
2
is the type of trees of the third is the type of functions
6.3.2.
Terms
Let Tm
~2
is defined
in-
2 ;
~(~)~ ~T~2"
is the type of trees of the first
(a)r
of
Tm T
of
numbers.
class° *
from type
~
to type
•
objects°
~2 o
~,
be the set of terms of type
is defined
or natural
inductively
(i)
Variables
of type
(it)
Constants
(see below)
(iii) s~m(=)~, The constants
T
belong
~2
: ~2},
~Ul~m
of type
(countably
belong
T
to
many for each
7)°
Tm
T
.
are :
0 °, 01~ 02 :
zeroes
SO
:
successor,
$I, S 2
:
sup
of type
O, I, 2 °
of type
(0)0 o
or join for trees of type I, 2
Their types are RO,T, R1,T, R2, T :
Tm T
to
t~mo =st~m of
and
by :
Constants
((0)I)I
for recursion
resp.
and
((I)2)2
on type
O,
respo
1, 2 resp., for
each
wc T (Their types can be seen from the axioms ~2 ° for them, below°) ~,T' 6.3.3.
Zp,~,T
: for all
Notational
~2 "
conventions.
T . x T , y T ~ xl, . (or. just . a, b, e, k, m, n, ... ,
p, ~, • 6
.x, y, Xl,
.)
denote variables
of (any) type
T
denote variables
of
type
0.
denote variables
of
type
Io
o
This notation conflicts with that of chapter I, where I and 2 denote the types (0)0 and ((0)0)0 resp. However, this notation will be used only with this meaning in this chapter, so there should be no confusion.
402
2
82
f(0)l
,
denote variables oae
9
f(1)2
of type 2.
denote variables of type
,e,o
(0)I,
(1)2, resp.
r, s, t, u, v, tl, t',.., denote terms. Type superscripts
R.1 , T
and subscripts may be dropped:
e.g. we write
Ri
for
(i : o , 1 , 2 ) .
Other conventions
stated in chapter I, § 6 are also followed here,
for representing a term as a function of many arguments
e.g.
(Io6.5)o
Formulas°
6.3.4. If
s
and
t
are terms of the same type, then
s=t is a prime formula. and quantification 6.3.5.
Formulas are built up from these using all connectives at all types.
Axioms and rules.
(a) Axioms and rules for many-sorted (b) Axioms of equality:
intuitionistic
predicate
the first 4 of 1.6.7 (b), for all
logic.
~,T6 T ~2 °
(C) Axioms for zero and successor or sup: S n / 0°
,
Som= Son -~
re=n,
s~f(°)I/o~,
slf=s1~-~fo~,
s~f(1)2 / o2
s2f = s2~
,
(d) Axioms f o r f i n i t e
--~
f
=
and t r a n s f i n i t e
°
induetion :
F~ - F(0 °) & Vn[F(n) ~ F(Sen)] ~ ~ F ( n ) ~Z 1 - F(01) a V f ( 0 ) I [ V n F ( f n ) ~ F ( S l f ) ] ~ V~F(~) ~ i 2 - F(02) ~ V f ( 1 ) ~ [ V ~ F ( f ~ ) ~ F ( S 2 f ) ] ~ V~2F(~2), F(n), F(~), F(~2) .
for arbitrary formulas
(e) Axioms for finite and transfinite FR
recursion :
: finite recursion. I Ro~TXy0°
=x ,
Re, Txy ( Son ) = y (Re, Txyn ) n , for all TR 1
:
I" E~2
, where
x6
I",
y6 (T)(O)T .
transfinite recursion on type { R I ,TxyO I
R1,jy(Slf) for all
~6T2,
I .
=x ,
=y(Z(n(R1,jy))f)f,
where
x 6 v,
Z - Z0,1, T . ~(See remark 6.3.6
yE((O)T)((O)I)T, (a).)
HeH(1)T,O
and
403
TR 2 : transfinite
recursion
on type
2 .
{ R2,Txy02 =x , R2,~xy(S2f) = Y ( Z ( H ( R 2 , ~ x Y ) ) f ) f , for all
T , where
(See remark
6.3.6
(f) Axioms 6.3.6.
(a) and
for
xE T,
yE ((1)T~(1)2)T,
and
Z E ZI,2,T.
(a).)
H
Zp,~,~:
and
as i n 1.6.7 (d)o
Remarks.
~he
- operator
can be defined
can then be written
TR 2
{ Rlxy01
{~
The axioms
TR I
as :
= x ,
,
R1xy = @ , and writing
01
~2 ' as in 1.6.8.
(more perspicuously)
R1xT(Slf ) = y ( ~ R l x y ( f n ) ) f or (putting
in
y
as a function
of two variables)
:
=x,
~(Slf ) = y ( ~ o f , f ) (where
o
denotes
composition)
R2xy0 2
= x ,
{ R2~(S2f) or (putting
; and similarly :
y( k~R2xY(f~))f ,
R2xY= ~ ) :
02
=x,
t ~(S2f) = y ( ~ o f , f) . (h) Let
~oT ~ respo
types are Then
0 , T
resp.
is just
~0
T 1 , be the restriction 0
and
N-HA w
of
~2 ' where
I . (chapoI,
§ 6).
~
(c) Note that
~2 ~ v~i[ ~ = °i v ~ , ± ( i : from the axioms for
S. i
and
TI. o I
sir) ]
(i ----1,2)~
the only grotuad
404
§ 4.
Computability
of closed terms of
~2 °
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Definition
6.4.1.
CT
Let
T
cT-=u{c%: (a)-
T , and
~c~2}. *
The relation:
2.2.2
of reduction ~ normal fqrm~ etc.
be the set of closed terms of type
% contr t'
(d) ( w i t h ,
t,t'6 Tm)~
(for
of course,
Ro,~
is defined by the clauses
in (d)), and also (for
RI, ~
and
R2, T ) : (e)
t = Ritlt201,
t' ~ t
(f)
t -= R i t l t 2 ( S i u ) ,
t'
t >'I t'
The relations of normal form
(NF)
(i = I or
2) ,
(i = I
2).
I
~ t2(g(n(Ritlt2))u)u and
t ~ tv
and reduction
(t
reduces to
se%uence
or
t') and the concepts
(red. seq.) are then defined as
in 2.2.2. e, 8,
denote reduction
sequences.
The concepts of strict reduction
sequence and standard reduction
sequence
(std red. seq.) are also defined as in 2°2°2° t ~ t' (std)
~def t ~ t'
by a std red.
seq. t
t >I t'(std)
~def t > I t t
by a std red.
seqo (of length
~def t > t'
for some
in
t
is normalizable
t
is standard
(~td) normalizable
Proposition°
6°4.2.
t'
=deft
Every normal
I).
NF.
> t v (std) for some
t'
in
NF.
closed term of ground type has one of the
forms : 0 °, 01 , 02 , SoU , S1u, S2u with
normal.
u
In particular,
a closed normal
term of type
0
is a
numeral. Proof°
This is proved simultaneously
the number of symbols in ground type. length is possibly
the term.
for all ground types by induction on So let
t
be a closed normal
If its length is I, then it must be
> I , then it must have the form Rirsutlt2...t n
by induction hypothesis,
(n~O) u
with
u
has the form
S.u, l
normal. 0j
or
0 ° , 01 with
or
u
term of
02 .
normal,
If its or
But in the latter case, S.v, j
so
R.rsu I
is not
normal. 6°4.3°
* f
Uniqueness
of normal form°
Note that in chapters I - ] Z " CTY~" is used for "closed terms", " C T " for "Church's thesis". This was denoted by t _~' t' in chapter lie
and
405
Theorem.
If
Proof.
t > t'
and
t ~t"
and
The proof of t h e o r e m 2°2.23
t', t"
are in
NF,
then
(using 2 . 2 . 2 0 - 2.2.22)
t' ~ t" .
extends to the
present case. 6.4.4.
Definition
For each type type
T
of (standard) T , the set
def t ~ CT o
t E CO
is the least
(i)
CTCCT T
is defined by i n d u c t i o n on
and then p r o c e e d i n g
CI
computability.
from
p
and
~
is std n e r m a l i z a b l e .
subset
X
of
CT I
t~O I
(std)
then
t EX,
tSS1u
(std)
with
u
subset
Y
t~O 2
(std)
then
tEY,
(ii) if
t~S2u
(std)
with
u
For
t~CT(p)~:
tE
terms of
T = 0, I
and
2 ,
~t E CT I :
and
CT 2
if
(i)
such that
normal
of
(closed)
(p)¢ .
t
if
is the least
to
and
(ii) if C2
of computable
T : i.e. first for
and
(~sE C o ) ( U S 6 X ) ,
such that
then
t6 X o
then
tEY.
~ t 6 CT 2 :
and
normal
C(p)¢ ~def t
and
(~sE C 1 ) ( u s E Y )
is sta n o r m a l i z a b l e
,
and
(~s~ C p ) ( t s ~ C ).
Notes. (a) C o m p u t a b i l i t y definition
of
is defined here only for closed terms,
Comp"
in 2.2.5.
(b) The above concept of c o m p u t a b i l i t y to standard r e d u c t i o n s concepts tions
(as w i t h
of c o m p u t a b i l i t y
(as with
(c) Statements
Comp'
Comp"
relative
and
in 2.2.5).
resp.
of
computability")
CT
for
T = I or
2
be called :
C "
(i)
or
T
" i n d u c t i o n on
can often be proved of
CT .
This will
t E C " T
Lemma.
s>t (std) and
t~C
--
~s~%. T
(ii)
Conversely,
(iii)
t E CT ~ t
is std n o r m a l i z a b l e .
s>t
(std)
and
sE C
(iv)
t E C(~)T ,
SE C
--
Proof.
or a r b i t r a r y reduc-
in 2.2.5).
to the inductive d e f i n i t i o n
" i n d u c t i o n on
corresponds
W e could also define
to strict reductions,
Comp
about elements
("standard
by i n d u c t i o n c o r r e s p o n d i n g
6.4.5.
as o p p o s e d to the
(i) Induction
i n d u c t i o n on
t E C
T
= t~
C . T
= is6 Cw" on
T 9 and w i t h i n that, for
T= I
and
2 ,
on
sE C
o
T
(ii)
I n d u c t i o n on
• , and for
T = I
and
2 ~ induction
o
T
(iii) Immediate f r e m the d e f i n i t i o n for (iv)
T= I
and
2 )o
F r o m the d e f i n i t i o n of
C(~)T
of
CT
(and by i n d u c t i o n on
t E C
406
Note.
The argument for (ii) breaks down if we consider computability
relative to strict or arbitrary 6.4.6.
Lemmao
Proof°
For
Every constant of
0 °, So, Ro, Z
and
the remaining constants of (a)
01
and
02
(b)
SI
and
$2.
(1)
(rather than standard) T2 ~,
of type
T
reductions.
is in
CT o
the proof is as in 2.2.6.
Now consider
T2 .
are easily seen to be in Consider
SI . SI
CI
and
is normal,
C2
so
resp.
$16 C((0)1) 1
iff :
(v,,c C(o)l)(Sl~ Cl).
SO let
Let
u 6 C(0) I . Then
u'
be the
NF
u
of
is std normalizable
u.
Then
u ' 6 C(0) 1
(6.4.5 (iii)).
(6.4.5
(ii)).
So
(Vs~ Co)(U,s ~ c 1) . Hence
SlU6 C 1
This proves
(by definition of
The proof for
S2
and superscripts (c)
R1, ~
(where
and
p
(2)
is exactly parallel
R2, ~ °
Consider
is the type of
First note that their
(replacing throughout
subscripts
RI, T . RI, ~
is normal,
so
RI,
E Cp
RI, T ) iff :
C1)(Rlrst ~ %)
r
NF's resp. Suppose
and Then
t>01
r
and
s , by induction on
s
are std normalizableo
r' 6 CT, s'6 C
(std).
Then
(ii) Suppose
(3)
Let
r'
and
s'
be
R 1 r s t > R 1 r ' s ' O I (std)
>I R1rst6 CT
t 6 CI .
.
6C
Then
(std)).
~ ~ ((O)T)((O)I)T o
(2) is proved for fixed
So
SlU>SlU'
" 0 " and " I " by "I " and " 2 " resp.).
(ffr~ %)(vs ~ c )(fft~
where
(i)
C I , since
(I).
r' T
(std)
.
(6.4.5 (i)). t S SIu (std),
with
U
normal and
(~v~ Co)(UVC ci). R 1 r s t ~ R 1 r ' s ' ( S 1 u ) (std)
(4)
>1 s ' ( ~ ( n ( ~ l r ' s ' ) ) ~ ) ~
(std).
Now we show that
(5)
Z(n(R1r's'))u
Let
v 6 C O.
Let
v'
6 C(o)~.
(We must show
be the
NF
of
v.
Z ( H ( R 1 r ' s ' ) ) u v £ C T .)
v
is std normalizable.
407
Then
v' 6 C O , so
uv' 6 C I
R1rs(uv' ) £ C T and hence
by (3), and
by induction hypothesis for (~),
R r's'(uv')6 C . I T Z(n(R1r's'))uv Z Z(n(R1r's'))uv'
Also
(std)
(std)
>I R1r's'(~v') 6 C , T
E(n(R1r,s,))uv ~
so
SO (5)
%.
follows, by definition
(since also
Therefore the term shown in (4) is in u 6 C(0)I ),
and hence so is
The argument for 6.4.7.
Theorem.
(since also
is normal). and
s' £ C
This proves (2).
is again exactly parallel.
Every closed term of
C T = CT T Proof.
R2~ r
CT
R1rst.
Z(E(R1r's'))u
for all
~2
is computable,
i.e.
T.
From 6.4.5 (iv) and 6.4.6.
6.4.8.
Corollary.
Proof.
From 6.4.5 (iii) and 6.4.7.
6.4.9.
Note.
ever~ term of
Every closed term of
~2
is std normalizableo
Theorem 6.4.7 does not (apparently) yield immediately that ~2
is normalizable.
However, this will follow (6.5.13)
from theorem 6.5.11 below. 6.4.10. Definition of We define for each its computability
ItIC. t6 CT i
(i = I
(by 6.4.7).
or
2) an ordinal
Itlc
by virtue of
The definition is by induction on
t 6 C. i
(i= I and 2) :
(i)
if
tZ0 i
(std) the~
Itlc= 0,
and
(ii) if
t>S.u
(std)
u
then
with
normal,
ItIc = sup{luVlC+ I : v 6 Ci_l}°
(Not_~e. Since the only normal closed terms of type 0 are numerals clause (ii) (ii') if where
(for the case
tZSIU
(std)
with
n =def S O ~o. SoO° .) n times -
_
(6.4.2),
i = 1 ) can be changed to : u
normal, then
ItlC = sup{lu~Ic+ I : n6 N},
408
§ 5.
Strong
In this ~2"
computability.
section we prove
However,
of
~2
cf~
6.4°9)°
(not n e c e s s a r i l y
6.5.1.
closed)
computability
elsewhere,
are
of all closed
except
(standard)
terms
of
to show that all terms
normalizable
(from 6 . 5 . 1 3 ;
Definitions.°
t ~ t' t
the strong
this will not be used
(strongly)
is strongly
=def every r e d u c t i o n
normalizable
N o w we define, closed
terms
for each
of type
T .
t E SC o ~def t E CT o
and
SC I
is the least
(i)
if
(ii)
if
then
~def
subset
• ,
t X
the set
CT I
then
some n o r m a l
u,
SC T C CT T
contains
t'
from
t
of strongly on
is finite.
computable
• :
normalizable.
such that
tE X,
t~SlU
t
sequence
is by i n d u c t i o n
is strongly of
from
reduction
The d e f i n i t i o n
t~O I (strongly) for
sequence
every
~t E CT I :
and
(strongly)
(ffsE SCo)(USE X) ,
and
t E X.
SC 2
is
(i)
if
(ii)
if
then
the least
subset
Y
t ~0 2 (strongly) for
of
CT 2
then
tE Y ,
u, t~Syu
some n o r m a l
such that
~ t E CT 2 :
and
(strongly)
(fist S%)(uss Y),
and
t E Y.
,inany,
SO %el
U IS%, ~ 2 } .
Note.
Statements
proved
by i n d u c t i o n
about
This will be called : 6.5.2.
Lemma.
Proof.
Induction
6.5.3.
Lemma.
Immediate
6.5.4.
Lemma.
of
SC for T = I and 2 can often be T to the i n d u c t i v e d e f i n i t i o n of SC T °
"induction
s~t
Proof.
elements
corresponding
on
on
and
sE SC T
T ,
and for
t E SC(~)T
and
~ every
or
"induction
on
t E SC "
T = I and 2 , i n d u c t i o n
on
s E SC
T
T
t E SC T
of
T
t s E SC T "
sE S C
from definition
t E SC.
SC "
SC(~)T .
reduction
sequence
from
t
contains
a term
1
in
SC.
(i=O,
I
or
2).
1
Proof. For For
For
i= 0 :
i = I or 2 : ~
:
Suppose
~
immediate. follows
immediately
every r e d u c t i o n
by i n d u c t i o n
sequence
from
t
on
t E SC.. l c o n t a i n s a term in
SC.. I
Let
s
be a term in
SCi
contained
in (say)
the
std r e d u c t i o n
sequence
409
from
t.
W e prove
t 6 SC.
by i n d u c t i o n on
s6 SC
i
(i)
If
e
contains
s'
a term in
$C i ,
is strongly normal,
By u n i q u e n e s s o f
NF
so
of
Thus
t > 0 i (strongly),
(ii)
I f f o r some normal
e
say e
t
he any reduction
ends in a normal (6o4.5),
and so
sequence from
t.
s' ° term.
this term must be
0i .
t 6 SC..
--
1
u
s>S.u
'
1
then by the same argument, Note.
(strongly),
1
t > S.u (strongly). --
6.5.5.
: 1
s ~ O 1 (strongly) : let
E v e r y type
and
(ffvESCi_I)(uv6SCi):
So
t ~ SC i °
l
T 6 T ~2
can be put (uniquely)
in the form :
(TI)(~ 2) ... (Tn)i where
i=O,
remark
(iii).)
I or 2
NOW
suppose
and
T ~ (T1)
n~O
.
o..
(Tn)i
t c (~l)°"(~n)i
and
tjs~.j
(l_<jSn),
for
TIO.O Tn , (TI) .o. (Tn)i ,
t
for
t l ° . O tn ,
t~
for
tt I o.. t n ,
t6 •
for
(t I 6 71) & o . . & (t n 6 Tn) ,
for
(t lcsc 1)&...&(t n~s%n)` t E T m (~)i ,
then
t £ SC T * (Vt6~= SC )(every r e d u c t i o n = Next,
affecting
if
t 6 (~)i t
and
some term in ~6 • ,
means a r e d u c t i o n
ttl"''tn ( i ° e . f o r each
>1 t t ~ o o . t ~ k
and
--
sequence from
it=
contains
SCi) o
then a r e d u c t i o n
sequence f r o m
tt
no.t
sequence of the f o r m :
>1 t t ~ k ) ° ' ° t ( kn )
h...
1< j < n, --
1.6.2,
~e ~ i l l use the n o t a t i o n :
for
if
cf.
Then
=~
SO by lemma 6.5.4, (I)
TmT°
t6
(%)i
~cSC
ana
and
T :
)°°.(vt nCs%n)(ttl...t nssoi).
t s s% ~ (vt l s s c If
(Proof by i n d u c t i o n on
t!k) > J
1
%(k+t) j
>1 ' ' ° or
t ( k ) ~ t(k+1) ), 3
J
also w r i t t e n tt ~I t~' >I "'" >1 t~ (k) >1 °°°~
with
t > t (k)
Note that if (2)
t1,°o.,t n
{ a reduction
are all strongly normalizable,
sequence from
tt=
not affecting
then t
must be finite°
41o
6 5.6.
Definition.
•
I or 2;
and
Note that
_
6.5.7.
0 (p)~ ~
0T
~", 0 T , is defined for each 0 T is the given constant for
A "zero of type _
(cf. 2.2. 7 (ii)) by induction H
on
T:
T T = 0,
p~ 0 p "
is normal.
Lemma.
(a)
t ~ SO T = t
(b)
0 r c so
is strongly normalizableo
° r
Proof.
(a) and (b) are proved
I_~f T = 0, I O_~r 2 : t 6 SC
for
T
simu!taneousl ~ by induction
(a) is immediate
on
T o
T = 0 , and proved by induction
for
on
T = I or 2 .
(b) is immediate.
If
~ : (~1)''°(~n)i'
(a) Suppose
nat:
t~ SC T , and let
e:
t ~ t ° > I tl ml t2>I "'"
be any reduction
sequence
e' : to0
¢I
from
~1 tl0
~1
t o Then
~1 t20
T1
~t
"°"
T1
is
a
reduction
hypothesis
sequence
So by induction (b) Suppose
hypothesis
~ E SC T
0r ~
not affecting
0r t
has the form :
H0rltlt2"
6.5.8.
nemma.
(~),
S.t l
hypothesis
(k+1) st term is
If
t 6 (~)i t
so is
for (~),
@.
t I .... ,t n
are
any reduction
sequence
from
So any reduction
sequence
from
SC i
0~It~ k)
>
I
°°
t (k) n
>
1 "'"
by j.uduction hypothesis
for
0 T 6 SCr °
and
~ 6 SC r
is finite.
then any reduction
sequence
=
(a) and (2) of 6.5.5.
For all
(i = I or 2)
t
in
CTCo~I
or
CT:I~2 \
2
has the f o r m :
Sit >I Sit' >I S i t" >I where
is finite ; hence
must be finite.
\
from
@'
>I 00 rl t (k)t(k) .. t(k) I 2 " n
net affecting From 6.5.7
6.5.9.
for
So (by (I) of 6.5.5)
Corollary.
Proof.
0T
k ; and this
tt=
SC(T2)...(Tn)i, , by induction
, which is
So (by (2) of 6.5.5)
tn >1 . . . . .
(k) and 6.5.3.
from
tO
Then by induction
strongly normaliza~le.
for some
from
for (b) and 6°5.3.
°''
t >I tv >I t" >1 °°°
Proof.
By inspection
6.5.10o
Lemma.
of the contraction
The constants
of
~2
are
rules° SC o
/
9
any reduction
sequence
411
Proof.
Again,
Now consider
for
(a)
01
and
02
(b)
SI
and
S2 .
(1)
0 o, So, He, H
the remaining
and
constants
are easily Consider
Z ,
the proof is as in 2.2.19o
of
~2"
seen to be
SC.
SI . SI
is normal,
s 1~ sC((o)1)
so
iff
(~t c SC(o)t)(stt ~ s % ) .
SO suppose
t E SC(o)I. >1Slt'
Sit where
t
Every reduction sequence from
>1Slt"
t
Then
Sit ~ S1u (strongly).
is
Further,
SN,
>1 "'"
by 6.5.7
(a).
Let
u E SC(o)I , by 6.5.2, so Sit E SC I
The proof for (c)
has the form:
>1 t' >1 t" >1 "'" ' by 6.5 " 9 °
Also
Hence
$11
RI, T
and
(2)
(by definition S2
u
be its
NF.
(~v ~ SCo)(UV E SC1) .
of
SC I ),
proving
(I).
is parallel.
R 2 ,T"
Consider
R 1 ,T"
R 1 ~T
is normal,
so
R I ,T
SC
is
(~r ~ s%)(~s ~ sco)(~t c SCl)(H~rst ~ S%)
(where
~ ~ ffO)T)((O)l)~).
(2) is proved
for fixed
(i)
Suppose
t~O 1 (strongly).
Let
~ E SCT,
where
Then by 6.5~8, is finite. finite.
r
and
on
(where
(by 6.5.7
So any reduction
So
r~r',
sequence
from
(R1rst)~ , not affecting
(a)) any reduction sequence
from
(3)
s S s ' , ~ S ~ ' ) , with
sequence
(R1rst)~
r't'
R1rst E SO T , by (I) of 6.5.5.
(ii) Suppose
t~S1u
t E SC I :
• ~ (~)i.
any reduction
Also
s , by induction
(strongly)
where
E SCi
Thus u
from
r
or
R1rst , s
is
has the form :
(Hlrst)~ >1 "'" >1 (Rlr's'01)~ ' ~I r't' >1
" ' "
(by 6.5.2 and 6 . 5 . 3 ) .
(2) is proved is normal
in this case.
and
(Vv ~ sco)(~v ~ s o l ) .
Let
~ E SC T
Then (by the same argument)
any reduction
sequence
from
(Hlrst) ~ h~s the form, (4)
(Rlrst)~ >I "°° >1 (R1r's'(SlU))~)
(where
rZr',
sZs',
~£!'
>1 s ' ( Z ( H ( R 1 r ' s ' ) ) u ) u
)"
Now we show that
(5) Let
iff
Z(H(Rlr's'))u v C SCo .
~ SC(o)~
(We must show
Z(H(Hlr's'))uv
E SC T .)
>1 °°"
412 Vs E
SC
,
Z(n(R1r,s'))uv
any reduction sequence from
~
has
t h e form (by
6.5.8 a~d 6.5.7 (a) again) :
z(~(R1r's'))U~>l
>1Z(n(Rlr"s"))uv'z'
" ' "
>I n(~lr"s")~'(uv')~' >1
(6)
"'" >l Z ( R 1 r " s ' " )
" ' "
v"(uV'")~"
>I R1r"'s'"(uv'")s" >I "°" (where
r' ~ r" ~ r'",
s~s'~
S~
~
~
Sm~
V
~
V I,
V I
>
V ~I
V !
~
V ~t~
'')
Now
(A)"
S T
uv E SC I
Therefore
by (3), and
so i s
R1rs(uv ) ~ SC T
R1r'"s'"(uv"')
,
by induction hypothesis for
and so i s t h e l a s t
termshov~a i n
(6).
Hence (5) is proved, by (I) of 6.5.5. So the last term shown in (4) is So (again by (I) of 6.5°5) The proof for
R2, ~
SC
(since also
and
u
are
R1rs% E SCT,
6.5.11o
Theorem°
From 6.5.3 and 6.5.10.
6.5.12.
Corollar 7.
All closed terms of
6.5.13.
proving (2) again in this case.
All closed terms of
~2
Corollary°
All terms of
~2
are
~2
SC.
are strongly normalizable.
are strongly nermalizable (and hence
normalizable)°
Proof.
......
Let
variables
T1
TI Tk t ~ t[x. , ..., x k ]
T~
x I ,...~x k
8:
.
be a term of
T 2 , with (only) free
Consider any reduction sequence from
t :
t ~ t o >1 tl >I t2 >1 " ° " T1
where
Tk
ti E ti[x I ~ ..., x k ].
Let
t*i = ti[O ~I, "" ., 0 ~k] " Then
t*o >1 t*I >1 t2* >1 °'" is a reduction sequence from the closed term 6.5.12.
SC).
is parallel°
Proof.
std
s'
1
Therefore so is
e°
t*o ' and so is finite, by
413
§ 6.
Models of
~2 ; modelling
= = = = = = = ~ =
6.6.1.
~2
Definition.
which satisfies
A model of
~2
~2
> , T E T 2 , VAIN,
M
, the function
prets application between terms, of type
Val M
• , ValN(C ) ~ M T M T , then
can be extended to
Notation.
(cf. 2.4.1)
=~
T
interprets
is called normal
CT
by:
Val(st)
~2
of
range over elements of
o M1
range over elements of
M2 .
~I
~2
For
f E M(~)T
For
t ~ CT,
and
Well-founded is a w f
tM
~
or just
is (isomorphic to)
(2)
MI
is the least subset
01E
(it)
if
f
~ M(O)I
and
fx
t
for
for
model of
~, X
~2
as in the
Ap~'~(f,x) .
ValM(t ) .
ApN , Val M
and
=M"
if it is a model of
~2
and
conditions :
the set of natural numbers. of
N I compatible with = N * and satisfying :
(~nE Mo)(fnc X)
is the least subset
02 E Y ,
(ii)
if
f E
Y
and
just state that
FI, TI I
From now on we only consider If
N
E X.
M 2 compatible with =M
(~GE M1)(f~E Y)
The above conditions
(T = I or 2)
of
Slf
then
and satisfying :
and
M(1)2
versions of the axioms
6.6.3.
E
X , and
(i)
Note.
Val(t)) .
are often dropped.
(second-order)
%
M2
is the identity
models.
(I)
(3)
=M
M
is often dropped from
(well-founded)
satisfies the following
(i)
we write
and subscripts
the subscript
N
x E ~,
we write
Type superscripts
6°6.2°
If
(6.3o3), e.g.
range over elements of
I
inter-
(as ih 2.4.1).
= Ap(Val(s),
~, ~,
or
as
= , and for each constant
is its interpretation. ~
a, b, ..., m, n, ...
Also,
are interpreted
ApMa,T : ~(~)v × M ~ g T
We will often use the same notation in discussing
(meta-)language
2,
is a structure
=M>
when the variables of type
ranging over the domain
relation on each
I~D2(@) .
==
M = <<MT>TE~2 , <ApM
C
in
= = ~ = = = = = =
is a
wf
model of
and wf
TI 2
then N
~2
satisfies
(6.3.5
models of
the second-order
(d)).
~2"
then statements
can be proved by induction corresponding
conditions ; this will be called "induction on ~(~i~. ~C X and ~ = ~ ~ ~E X )
Y.
S2f E
N " T
about elements of
MT
to the above inductive Similarly functions
414
and relations
can be defined on
For example, ordinal
M by induction on T canonically with each ~
we associate
I~TIM,
defined by induction on
I011M = o,
MT
M
° T
E MT
(~ = I or 2)
(and using remark 6.3.6
Is1flM = supIIfnIM+ I :
an
(c)) :
n E No} ;
Io2tM=o, Is2flM=suptlf t +l, ItlM
We write 6.6.4.
IVal(t) lM.
for
Four examples of well-founded J2'
Four
HR02'
h~02
examples of
Many definitions
and
wf
models of
CTNF 2 .
models are discussed in this section.
and proofs concerning
proceed by induction on
~2 :
T
wf
models
M
of
~2
and, within this, by induction on
M
will for T
T = I and 2. (a)
#2'
the full set-theoretical
This extends the model of
model.
E-HA w
in 2.4°6°
The domains
M
T
are defined
(in set theory) by: o ~I
is the least set
X
such that :
then
S1f E X o
(i)
01 E X,
(it)
if
f : Mo ~ X
Here
01
is (say) the empty set,
No
to
and
X , and
S1f
the definition of N2
02 6 Y ,
(it)
if
N(~)T Ap (b)
ranges over arbitrary functions from
can be taken as
f
Y
such that :
f: M I ~ Y
then
SIf6Y.
Val
HRO2
,and
the t h e o r y Sets ~N. o V i ~ 0.l
02 .
HR0
can be formalized
Z~D2(O) ,
V Tc N
in
~2 of
i~n
MT "
~-~w I~D2(0 ) .
are defined by induction on
i = I or 2
of finite type o__nn
I DD2(O ) . (2.4.8). (The sets
were d e f i n e d i n 6 . 2 . 2 . )
V
for
to
recursive operations
Interpre,t, ing
extends the model
given below,
~
are defined in an obvious way.
HRO 2 , the hereditarily
GI
Similar remarks apply to
and
is the set of all functions from
and
itself°
N2 :
is the least set
(i)
f
~ :
Its construction, 01
and
0 2 , and
415
{e:
V(a)T m
Vxe V
We define the domains NT
is defined as
VT)}.
Sy(Texy & UyE
M
of our model simply by : ~
V T X {~J,
(In 2.4.8,
but this is an unimportant difference.)
_ x ~ V(a)T , y E V~ : Ap ~,T-(x,y), =def
For
~ VT "
{x~(y)
Equality is interpreted as identity in each
VT .
Val
for
Val(C) ):
and
[H]
is defined as follows (writing now [C]
[o °] ~[o 1] ~[o 2 ] ~ o . [Za,T]
and
[Si]m
[Hp,~,T]
Ax(x+ I),
[Ro,T]
are the numbers
Ax ( 3 . 5x)
the n~mber
is
[RI,T] m [RI]
and
[R]
Ax(32.5 x)
resp. for
resp. defined in 2.4.8. i = 0 , I, 2
resp.
defined in 2.4.8.
is a numeral, found by use of the recursion theorem,
HA t,.....~ ~[R1]}(x,y,0)
(1)
[Z]
such that:
x,
b l[~}(~,y,3-5 ~) ~ {yt(t[z]t(~[n]J(~[R~]l(x,y)))~,~). Similarly, 3o5 z .
[22,7] ~ JR2]
is defined to satisfy (I) with
~ ~ ((0)T)((0)I)T) , by induction on
Hence where
(IER~II(~,y,~)) z E V I (m 01).
Z~2(0 ) ffVp([R1] ) p
is the type of
Similarly,
[R2]
duction on
V 2 ~ 0 2 ).
~2(0)
RI, r
is (provably in
So for all constants
C
of type
i~D2(O ) ) in the appropriate
t 6 CT m , writing
[t]
(by in-
for (the numeral of)
~2(o) ffvT([t]) since (2) holds for the constants,
and further,
~2(0) pv(o)~(=) ~ v(y) ~ v (Ixl(y)), and for any numbers
a, b, c
the formula
Sy[T1(~ , b, y) & U(y) = ~] if true, is provable, For
P
pV ([el).
Hence, more generally, for all
(cf. 2.4.8),
V
T ,
Val(t): (2)
replacing
Then
~2(o)pv(~)~v(y)~v1(~)-v @here
32.5 z
even in
m = I , (2) becomes:
HA
(1.5.10).
416
(3)
CT I, ~2(O) WOI([t])
Zt ~
This is used in proving theorem 6.6.8. (c)
HE0 2 , th___~ehereditarily effective extensional operati~s over We describe briefly a
wf
model
HE0 2
Define a ~inary relation
IT
on
for all
N
extending T
01 and
0~.
ITE0 (2.4.11).
as follows.
lo(x'Y) mdef X=yo 11
is inductively defined by :
(i)
II(0,0) ,
(ii) VnZy,y'(Teny&Te'ny' & I1(Uy,Uy'))-- II(3.5e, 3.5e'). 12
is inductively defined by :
(i) (ii)
12(0,0), Vn,n'[llnn'--~y,y'(Teny& Te'n'y' & 12(Uy,Uy')) ] -- 12(32.5 s, 32.5e').
I(¢)T(e'e') ~def Vx'x'[I~ x x ' ' ~ y ' y ' ( T e x y ~ Te'x'y' & IT(Uy,Uy'))] o Finally, WT ~def {x : ITxx} o (This, in fact, extends the definition of Now
M = HE0 2
on each
is defined with
M r , and
Val(C)
MT
IT
and
defined as for
HR0 2
Now write Then (d)
0~ =
I ~ (II,I2)
G~ ~ W I ~I'
but
and ~
in ~
~
IT C .
I D2(I ) , with the defin-
~. ~ W2 o
G2"
CTNF 2 , the term model of This extends the model
in 2.4.11.)
for each constant
This construction can be formalized in a theory ing formulas for
WT
W T , equality interpreted as
~2 °
C~F
(: "closed terms in normal form") of
[_ ~m
in 2.5.1. For each
t 6 CT,
For each
T ~ N
Val(C) ~ C
let
{
be its (unique) normal form.
is the set of normal closed terms of type
T for each constant
• .
C.
Ap(s,t) ~ st o So for each
t6 CT,
Proposition. Proof.
is
wf.
By computability of closed terms of type I and 2 . More precisely
(with Let
CTNF 2
Val(t) ~ {.
N = CTNF2) : X~M I
be such that
~ X,
and
(i)
0
(ii)
[u6 M(O)1[([n6 MO)(~-n 6 X) ~ Slu6 X]
We show (Zt6 CTI)(t6 X ) Similarly for M 2 .
by induction on
.
t6 C I .
417 Proposition. Proof.
For
M = CTNF2,
By i n d u c t i o n on
Remark.
NI
t C CT I or CT 2 : and
T h e o r e m 2.5.5 can be extended
is a v e r s i o n of
HRO 2
ItlM = ItIc.
M2 .
in w h i c h
to
CTNF 2
wf
models of
~ 2 : i.e.,
can be embedded.
there
The proof extends
that of theorem 2.5.5. This also gives an a l t e r n a t i v e closed terms of 6.6.5.
~2
Extensionality :
The concepts
proof of u n i q u e n e s s
of normal f o r m for the
(as in 2.5.6)° some geperal remarks.
of homomorphism,
embedding and submodel
(for
wf
models of
~2 ) are defined as in 2.4.3. The r e l a t i o n as follows
~
by i n d u c t i o n on = T
means -
0
:
=M ' m ~ n
T -----I :
of e x t e n s i o n a l
equivalence
in
(extending the d e f i n i t i o n in 2.4.4). T , and, for
and -
~0
m=
n
N
is defined
T = I or 2 , by i n d u c t i o n on
MT .
MT
(Below,
=M *')
.
01
~S1f
~2 ~ S 2 f f~
models
is taken to be c o m p a t i b l e with
~
"~ = 2 : 2 ~ 0 2
• -(p)~:
wf
It is defined on each
~
M(0)1[~= %g and (~n~ M0)(fn ~n)]o
2=
02 '
~.g6 M(1)2[~2:S2 g and ( ~ e M 1 ) ( f ~ g ~ ) ] ° -:
(~=~ Mp)(f~ ~gx).
More definitions°
(a)
is p r e - e x t e n s i o n a l
if for all
~, T :
(~f ~ M(~)~)(~xw ~ ~) (x ~y ~ f~ ~fy) . (b)
~
Notes.
is extensional
if for all
(I) This is the same as the d e f i n i t i o n
(2) We always have : x = N y = x ~ y w.r.t.
(b)
If
(c)
If
on
(a)
M
is p r e - e x t e n s i o n a l
N
N
is pre-extensional, as
of "extensional
is a congruence
model" in 2.4.1.
relation
iff
~
is a congruence
relation
then
N
becomes
is extensional,
then
M
In each case, by i n d u c t i o n on
can be embedded in T , and for
T
x~y,
extensional
when
=M
is
~.
M (i.e.
~
Ap°
re-defined
Proof.
, i.e.
=~o
Proposition. wor.t.
T :
x=Mx~ , y=Myt
= x' ~ y ' )
#2 "
T = I or 2 , i n d u c t i o n
418
Note that
J9
and
(corollary below), Theorem.
If
submodel
gE.
Proof. I'
is a
wf
mdef
is pre-extensional
model of
~2'
then it has a pre-extensional
We just mention here the definition of
~
~"
is the least, subset
(i)
(02,02)
(ii)
~,8[I~(~,~)
~ Y,
The definition Note.
CTNF 2
is not pre-extensional.
T = I and 2 :
I~(~,~) I~
are extensional, HR0 2
As for 2.4.5.
for
T
M
HE0 2 and
ME
o_~f ~2 XM2
M
T
Corollary.
(compatible with =M )
such that
~d = ( f ~ , g S ) E Y] = (S2f,S2g)
of the domains
is pre-extensional
striction of on each
Y
~,
E Y.
and the proof,
then proceed as in 2.4.5.
if equality is interpreted
=~ ; it is extensional
in it as the reas I T!
if equality is re-interpreted
o CTNF 2
is pre-extensional°
Proof.
For
M = CTNF2,
its pre-extensional
of
M,
since it includes
the constants and is closed under application.
6.6.6.
Some distinctions
between
In looking for significant
wf
say
find it useful to compare properties T
models of
distinctions
and iterated inductive definitions,
submodel
must be the whole
and
~2"
between theories of non-iterated
I~D (A)
of
~1
~E
wf
for
~ = I and 2 , we may
models of the corresponding
. For example, we notice the following differences between trees of type
I and 2 in
wf
models
(a) Relation between For
t ~ CT I ,
ItlN
N ItlC
of
~2 :
and
ItaM
is an invariant
for of
t
i__nn CT I
and
~2
for any
wf
fact
Jtl =Itlc
(I)
(Proof by induction on However,
for
ItI
t E CT 2 ,
ItlN
varies with
N,
and
Z Itlc
(by induction on
Itic
t E C I .)
t E C 2 ).
was defined in 6.4.10,
and
ItI
in 6.6.3.
CT 2 . model
M ; in
419
sup{Itl~
In fact, the o r d i n a l
countable but non-recursive for
: t 6 CT2J M = HR02
i s uncountable f o r
and
M:J2,
HE02 , and recursive for
= CTNF 2 • (b) Consider the pre-extensional submodel ~
is the whole of
of
~
T
Remark.
for
Mr
for
ME
of
~
r = 0 and I , but (in general) a proper subset
We can also make a distinction between HR0~
IDD2(A)
different (2.4.12). ~
HR0 E
and
I~D3(A ) HR02 ) and HE0
are
However, their three ground types are the same, for :
and
@! resp. for the domains of i I (i = I, 2) , we see that 0 E~ =
and
(the pre-extensional submodel of
HE02 . They are, of course, different since (even)
i
The domain
T = 2.
Compare the models
writing
(6.6.5)°
~~ =
01 ,
and
0 E2 =
HR0~
O~
However, if we define, by analogy, a theory E and HEO 3 , and domains 03, 03 and ~
and
HE02
of type
02o
~3 ' with models
HR03 , HR0~
of type 3 respectively, then
E or since
0 E3 ~ ~ 3 '
rability of 6.6.7.
W4
but
and
~ W~,
is incomparable with
Definition of the ordinal
=
It]c
for any
the "ordinal of
I~21 o
~t 6 CT I ,
Let us call this ordinal just Now define
(Cf. the incompa-
2.4.12.)
We saw in 6.6.6 (a) that
]tim
03 "
wf
Mo
Itl .
~2 " as
I~21 ~def sup{I%l : t 6 CTI} . 6°6.8. Proof.
Theorem. I~21 ~ II~D21 • Two proofs are given here.
(a) Usin~ the formalization of the model
HR02
i__nn Iv~D2(O) :
From (3) of 6.6.4 (b), we obtain:
(b) Using the. f o r m a l i z a t i o n The definition
o f computability i_~n I~D2(C) :
of the computability predicates
computability of any given closed term of ized in a theory , and
CT
~2
C
(in 6°4.4) and proof of T (theorem 6.4.7) can be formal-
ID2(C ) , with the defining formulas for
defined arithmetically from
CI
and
fact, this also holds for strong computability°)
C2
C ~ (CI,C2)
for each
So for all
T .
t E CT~,
(In
in
42o
~2(c) b %(~'), where
%~
is the godelnumber of
t
in the theory.
Further, it is easy to show (by induction on
t6 C I ) that
I ~7[A1 = Itl for
t E CT I , where
C ~ (CI,C2).
A ~ (AI,Ar)
is the pair of defining formulas for
Hence
1~21 ! I~2(c)I ! I~=l" Remarks.
(a) The reverse inequality is proved in 6.7.9.
(b) The above result shows that
I~21 < w1"
421
§ 7-
Functional
6.7.1.
interpretation
Introduction.
interpretation
realizability
interpretation
and
[~IP1(~1,t )
as
I~D2(A) o
In order to obtain the reverse inequality
we define a functional Q2 t
of
of
~
and
of
IDD2(A )
(chapter I I I , §
Z~2P2(~2,t )
to 6.6.8,
which extends the modified 4) by translating
resp., where
PI
and
QI t
P2
are
two new binary predicate symbols adjoined to the theory ~ 2 " Intuitively, i i Pi(~ ,a) means: " a is put into Qi at stage i , , or " ~ realizes
Q.a"
(i= 1, 2) .
1
SO I~2(A )
i s i n t e r p r e t e d i n a theory
by the predicate on
PI
and
P2
~-~2~'
and appropriate
which i s
22
augmented
axioms for them (which depend
A m (AI,A2)), and also axioms of extensionalityo
Definitions Let
of the theories
~2
{2[P]
is
£2
prime formulas:
and
~2[P]
T2[P], ~2[' ~ - ~ 2 '
be the language of
augmented by the predicate s= t
(as for
sE I, t E 0
P2(s,t)
for
s E 2, t E 0.
in the language
Z2
i : 8i & m : n
for all
Vf (p)~
symbols
PI
and
P2'
~2[P] , i.e., including
-- p i ( i , m )
with
the induction
-- Pi(~i,n)
PI
or ~ :
and
P2 :
(i : I, 2) .
together with the axioms for
~ ~ 52' ~2[ P]
with the extensionality
E-~2~.
~2[P] , and equality axioms for
~2 ~ is the theory T2[P] (given below in 6.7.4).
Finally, for
and
, and also:
for
is
EXTp, a :
%)
P1(s,t)
axioms for all formulas of
[ - ~ 2 [P]
~2 "
~-~
PI
and
is the theory H
P2
to~ether
axioms :
g(P)°[wP(fx=gx)
-- f = g ]
p,~ E T 2 .
I D2(A )
will be interpreted
in
~'~2~"
ality axioms are discussed in 6.7.10. Definition
of the translation
with each formula
F
of
of
~Q]
(The reasons for the extension-
Actually we only need
I~D2(A)
into
EXT0, I .)
£2[P].
(the language of
~2(A))
we associate
a formula
x= m~ r F of £2[P] ° The definition is by induction on the comple} F , and extends the definition for HA (3.4.2, 3.4.5 (A)) by furth~ defining:
ity of
r~ (vii) Another
of
~Q]
i mrQi way
of
, its
t m pi(.it ) saying
the~same
(i= 1, 2) . thing
m r - translation :
is
that
we define~
for
each
formula
422
(1)
F° ~ ~ F o ( ~ )
extending the definition in 3.4.2 (see 3.4.3~ Notational
convention),
by
further defining mr (vii)
(Qit) ° ~ ~ i P i ( ~ i , t )
(i = I, 2) o
O,,ut,line of this section (9 7). Theorem 6.7.5 is the central result of this section.
From it (or rather
from corollary 6.7.7) and theorem 6.7.8, we immediately obtain (as theorem 6.7.9) the inequality:
This, combined with the reverse inequality (6.6.8), yields (as corollary 6.7.10) the main result of this chapter:
the characterization
of
IIDD21 as
1 21 = 122I" In order to prove theorem 6.7.5 we first need lemma 6.7.3, which provides a normal form for the translations of formulas state the axioms for
PI
and
P2
(AI,A2) 6 ~ .
We can then
(6.7.4).
However, in order to prove lemma 6.7.3 and theorem 6.7.5, and even to state the axioms for their inverses (in
PI ~-~2
and )°
P2 ' we need certain pairing functions and These are defined in 6.7.2.
Finally, notes 6.7.6 and 6.7.11 discuss, respectively, simultaneous to simple transfinite recursion in
~-~2
the reduction of
' and the rSle of the
extensionality axioms. Notation.
In this section, the relation
~
between formulas denotes provable
e,,quivalence i__nn E - T 2 [ P ] o S u g g e s t i o n t o t,,h,,e r e a d e r .
Readers who o n l y want a rough i d e a o f t h e methods
of this section may skip 6.7.3, 6.7.4 and the proof of 6.7.5, and instead look at § 8 (particularly 6.8.1, 6.8.2 and 6.8.7).
§
8 is to consider functional interpretations
Although the main aim of
of classical systems
ID:(O)
(v = I and 2) , the subsections indicated also consider briefly the modified realizability interpretation of the corresponding intuitionistic I D r ( 0 ) , which is e a s i e r t o d e s c r i b e than the g e n e r a l case o f
(any)
A ~
6.7.2.
systems
ID2(A )
for
~o
Pairing functions and their inverses in
~ - ~2 °
In order to prove lemma 6.7. 5 and theorem 6.7.5, and to state the axioms for
PI
and
P2
(6.7.4), we must show that certain pairing functions and
their (left) inverses (i.e. satisfying can be defined in
~-~2"
(2), but not necessarily
(3) of 1.6.16),
(Below, "inverse" means left inverse.)
423
Notation.
(i) The
X-notation,
used below, is to be understood as defined
in 1.6.8 (as stated previously). (ii) We indicate that fT or f E T .
f
is a functicn(al)
or term of type
•
by writing
(a) Embeddin~s o_~f ~round types° First we define embeddings
u01 , u12 , u02 ,
and inverses ("u"
for up, " d "
~-
do1 , d12 , do2
for down), with
{ u010°
FR, TR 1
and
= 01 ,
Uol(Son)
dij 6 (j)i,
such that
,
do2
do1(fO ) + I,
{ d1202
s2~u12f(dol~),
and
= 0° ,
{ dol(Stf)
= 02 '
~12(slf)
TR 2 ) :
do101
Sl~k(uo1n)
I u1201 u02
and
dij uij xi = xi.
~2 b
[The definitions are (using
Final!y~
uij E (i)j
= O° '
d12(S2f)
=
Sl~kdIJ(Uolk)-
are the composite functions :
u02 = u12 o u01
,
do2 = do1 o d12 .]
(b) Pairing of two objects of (the same) ground type
as one object of the
same $~pe. For each ground type
i
there is a pairing function
D
6 (i)(i)i
and
I
inverses
D~, DUI
such that :
,, i i i - ~2 ~ D~Dixiyi = xi & DiDi X y = y " [The definitions are :
i=o:
DO,
D~, D~
i: I :
DI~ ~ = Sjf, fO e
can be taken as where
f E (0)I
J' Jl
and
J2
is defined by
of 1.3.9, B. FR :
= ~,
{ f ( S o n ) = ~. The inverses are defined by
D~OI
=
t D~(slf) i= 2 :
01 (say), fo ° ,
D2~2~2 = S2f , where f01
= 2
f(Slg ) = 92 .
TR I :
D~O 1 = 019 t D~(Slf ) el (where f ~ (I)2
is defined by
TR I :
1 - SoOO )o
4-24
The inverses
are defined by
D~O 2
= O2
{D~(%f) (O) Pairing
~,T 6 T 2
and
D"
D~0 2
= 02
D~(%f)
= f(Uoll).
For
Functions
we can define
such that
~-~2 [(i)
f01 1
]
at all types.
For all D'
TR 2 :
~ D'Dx~yT
a pairing
(dropping = x
a
subscripts
& D"Dx~y T = y
~, T
both ~round
types:
D..
6 (i)(j)k,
D~
function
put
. 6 (k)i
T
D~, T
with inverses
~, v ) :
.
i=~,
j = T, k = max(i,j) .
and
D?
. 6 (k)j
are defined
as f o l l o w s : If
i = j ,
then the definitions
If
i/ j ,
suppose
i < j .
are as in (b).
(The case
j< i
is symmetrical.)
Then by
definition : D
(ii)
x l y J = Dj(uijxl,yJ ) , l,j
D! z j z,j
= d..D!z j , z~ j
D[' .z j z,j
= D'~zj . j
I__n_nt h fi ~eneral
= ~ (i, j
ground
(~)...
~m TI Tn = XXl "°'Xm Yl '''Yn Di,j(fxl'''Xm)(gYl'°'Yn)
defined
functions
objects
in general,
in the obvious needed
T ~ O, I
scripts
and
~ ).
< ' >T 6 (T)(T)T
(c).)
way
(cf.
1.6.17).]
for lemma 6.7.~.
6.7.3,
of the same type
by part
(for certain
(L)j
Then by definition :
In the proof of lemme single
" ~ ('I)"'"
~I
D~,Tf g
(d) Pairing
suppose
(=)i,
types).
T with inverses
case :
we must code pairs
of objects
The types for which
with inverses
T
we define
~,
w~ ~ (T)T
a pairing such that
function (dropping
T ) :
Fq,r, T ~ 0 or I :
T
are:
they are given by
as
T6 T (This is not given, ~2" this is needed are:
(0)~
For each such
[The definitions
of type
T , for certain
DT, D T' and D~
of (b).
sub-
425
For
• ~ (0)~ :
g(O)~>(o) ~
d e f i n e d as the function
is
h 6 (0)~,
where h(2n)
= fn ,
{h(2n+1)
"{O)"
and
~.
=
-'io),
are d e f i n e d i n the obvious way.]
(e) Special pairing functions f~r the axioms for In order to state the axioms for
PI
and
P2
PI
and
P2 "
(6.7.4) we need the
following :
(i)
a coding of a pair of functions of type function of type ( 0 ) I :
(0)1
and
(0)0
as one
f(o)l = w~, w 0I
with inverses
~-~2 (ii)
such that
I b ~I<~,
0 h> = g ~ ~ l < g ,
a coding of a t r i p l e
h> = h ;
2
(0)(1)t
and
(0)(1)0
(1)2 :
f(I)2 =
(0)(1)2,
of functions of type
as one function of type
1
n2' ~2' ~
go(°)(1)°>
such that
2
~-~2 ~ "2 = ~2 1 & n2
go > = g t go > = go °
The actual form of the definition of
(~f)(n)
~II
(given below), viz.
= f(2n) , is used in the proofs of theorem 6.7.5
pretation of
Q1.2 ) and theorem 6.7.8 (b).
the d e f i n i t i o n
W~ , viz.
of
(for the interpretation of
(w~f)(n,~)
(for the inter-
Similarly, the actual form of
= fDo,1(3n,~)
,
i s used i n 6 . 7 . 5
Q2.2 ).
[The definitions are : For (i):
= f
f(2n) = g(n), f(2n+1) = u01h(n),
where
(-~f)(n) = f(2n), For (ii) : First, from
~(3n,~) ~(3n+1, ~) ~(3n+2, ~) and then define
(~f)(n)
g2' gl
and
= g2(n,~), = u1291(n,~), =
Uo2go(n,~),
f 6 (I)2
by
= d01f(2n+1). go '
define
c (o)(I)2
by
426
f~ = ¢(D$,1~, D,,o,1~)° Then
(with
(~f)(n,~)
= f%,l(3n,~),
(~f)(n,~)
= d12f DO,l(3n+l, ~),
('~f)(n,~)
= do2f %,1(3n+2, ~)
%,I' D,0,1
and
6.7.3.
Lemma.
(a) Zf
AI(X,a ) E ~I
then
D"0,1
define~ in (e)(i)).]
as
Normal forms for translations (where
a
AE~.
is a list of the free no. variables of
A I ),
A1(Q1,a)°
xg(O)lh (°)° Vn~l(n, g, where
of
VnB 1
~)
h,
corresponds to the matrix
Fo(X)
of (I) of 6.7.1, and
BI
has
the form : C(n, h, a) --, D(n, h, a) . V. E ( n , h) ~ where : C, D~ E
are formulas of
~
Pl(gs, ts)
augmented by the variable
(but with equality only between type
and
(b) I f then
D
and
E
are
qf
in
s E
possibly
Tm 0 ,
t ~ CT(0)O
(quantifier
A2(X, Y , a ) E ~2
primitive
containing
the
free
variables
n, h,
a ;
(where
a
is a list
o f the free no. variables of A1),
A2(QJ' Q2' a)° ~
Vn,~B 2
is the matrix,
C, D 9 E
contain
and
Vn,~ B2(n, ~, g2' g1' go' a) B2
PI
has the form : .
V. E(n,~,g O) & P2(g2rs,tr)
positively only, and not 0
has number quantification only ;
D
and
E
t E CT(o)o
are
P2
,
(and equality
terms) ;
C
r E Tmo,
Proof.
hence
(representing a primitive recursive function).
only between type
and
and
h ;
C(n,~,go,a) ~: D(n,~,gl,go,a) where
free)
h (0)0
terms) ;
has quantification ever numbers only ;
~g~0)(1)2g~O)(1)Ig~O)(1)O where
0
C
recursive
,
qf ;
s ~ Tm I , possibly containing the free vars.
n,~,go,a ;
(representing a primitive recursive function).
By induction on the inductive definitions of
~I
and
¢2
resp.
(cf. the proof of lemma 4°5 of Kreisel and Troelstra 1970, from which the idea for this lemma came).
~e give some cases.
427
(a) If
AI
shown,
is
s=t
(Q1s) ° ~ X~PI(~,s)
Q1s ' then the form of
or
which is O.K.,
or it may be padded
~ ~(0)I
VnP1(gs,ts)
A °I
is simpler
to the required
, with
than that
form :
e.g.
take
t = ~.k.
New suppose
v~[c(~, h, a)
A° ~ ~h
D(n, h, a) . V.
-,
(A') ° ~ Zgh V n [ C ' ( n , h , a ) --: D ' ( n , h, a ) . Then
(A&A') ° ~ ~gg'hh'Vn[C(r~)
-: D ( n ~ ) ~ D ' ( n ~ b ) , v .
E(n, h) ~ P l ( g s , t s ) ] ,
v.E'(n,
h) & P l ( g s ' , t ' s ' ) ]
.
& C'(r~',a)
E(n~) ~ E ' ( ~ ' )
~ Pl(gs,ts)
~ P~(g,s,,t,s,)]
So put
C"(m,n,h,h',a) -= m=0 ~ C ( ~ , h , a ) . Vo m/0 ~ C ' ( n , h ' , a ) , D " ( m , n , h , h ' , a ) -= m=O & D ( n , h , a ) . Vo m/O & D ' ( n , h ' , a ) E"(m,n,h,h')
m m=0 & E ( n , h ) . V. m/0 & E ' ( n , h ' )
,
,
and define s" --- <sg(m),
~(m).s
. sg(m)°
s'>,
(I) t" -- ~ . [ ~ ( ~ , x ) . t ( ~ " ~ ) so
s" =
and
t"s" =
Then
(A&A')
+ ~(~,x)ot,(~,~).t,(~"~)],
<0,s>
if
m= 0 ,
<1,s'>
if
m/O
ts
if
m= 0 ,
t's'
if
m/ 0 o
,
~
Zg"hh' V m n [ C " ( m , n , h , h , , a ) - ~ : D " ( m , : n , h , l o ' , a ) . v . E ( m , n , h , h ' ) @ ~ P l ( g " s " , t " s " ) ] (taking
g"
as : g"x =
g(~"x)
'g,(~"~) so that,
conversely,
gx = g"
which can be put in the required ly as in 6.7.2 Next,
if
~'x= 0 ,
if
~,~/o, and
g'x = g"<1,x>
form by pairing
m,n
), and
(d).
(A vA') ° -~
~mch[m=O ^ Vn{C--- D .Vo E & P l ( g s , t~) ~ .V v.m/o ^ V n { C - - D roe & P 1 ( g s , t s ) } ] NOw define
and Then
s",t"
C'"(m, n,h,a)
= m:0&C
.V° m / O & C '
,
D'"(m, n , h , a )
- m=O&D.Vo m/O&D' ,
E'"(m, n , h )
- m=0&E
.V. m/O&E' ,
as in (I) again.
( A v A ' ) ° ~ Zmgh Vn[C .... : D'".v.E'"
& Pl(gs",t"s")]
,
h,h'
respective-
428
which again can be put in the required form by coding function of type Next, let So
m
and
h
as one
(0)0 , which is easy.
C I E ~ O.
C OI m c 1
Then
(by 3.4.4 (i)).
(CI--A) ° ~ ~ g h E C 1 - - ~ I C ~ : D .V. E & P 1 ( g s , t s ) I ]
~hV~[(c1& c)~, ~ .v.
z~P1(gs,ts)],
which again has the required form.
Next, where and
(VmA)° ~ ~ghVmn[C(n,hm,a)~: D(n,hm,a ) .V. E(n,hm)&Pl(gmS,ts) ] ,
hm gm
is defined by:
hm(k ) = h(<m,k>)
similarly.
So define
s"' ~ <m, s) ,
and
t'" ~ ~k t(~"k)
(so that and
t'"s'" = ts ),
and we have the required form (after pairing
m
n ). Finally,
(~mA) ° ~ ~mghVn[C~: D .V. E & P I ( g s , t s ) ] ,
which has the required form (after pairing
m
and
(b) The general procedure here is as in (a).
n)o
We omit details except for
the one interesting case : suppose o A 2 ~ ~g2glgOVn,~B2(n,~,g2,gl,go,a) B 2 ~ C~: D or. E & P 2 ( g 2 r s , t r ) ,
where
and suppose
AI ~ ~1 °
We want to show : (A I ~ A 2 ) °
First define a
I
functional
~(o I)
: xko I
~(slf)
: f.
(So for any
~ / 0 1 , ~(~)
has the required form. @ E
(1)((0)1)
by
TR1 :
(say), is the sequence of immediate predecessors of
~.)
Now by part (A) : o ~ AI
SO
ag(O)lh(O)O C'(g,h)
(say)
~2~(0)1 C'(w~f, wif)
(see
~ c,,(~)
(say).
(e)(i))
6.7.2
( A I ~ A 2 ) ° ~ ~g2glg0Vn,~,~[C"(~)~B2(n,~,g2,~,g1,~,go,~,a)] ,
where
gi,8
is defined by : gi,~(k,v) = gi(k, <~,V>) .
So (AI--A2)° ~ ~g2glgOVn,~,8[C"(~) & C(n,~,go,8) " : D(n,~,gl,8,g0,8,a ) .V. E(n,~,g0,~) Finally~ replace (after pairing
s
by
and
&
--
e2(g2,srs,tr)] .
s' _ <~,s> , and again we have the required form
~ )°
429
6.7.4.
The axioms for
Let
BI
and
B2
ing formulas in Axioms for
P1.1)
PI
and
P2
can now be stated.
be as in lemma 6.7.3, when
AI
and
PI"
PI.2)
vnBl(n, g, h, a) ~ PI(Sl , a) ~ Vn~1(n, ~1If , ~Io f, a) .
P1(Slf, a)
Axioms for
P2 °
P2.1)
m P2(O 1, a)
P2.2)
Vn,~ B2(n , ~, g2,gl,g0, a) -- P2($2
P2.3) (where
P2(S2f,a) -- Vn, ~ B2(n , =, 2 2 f, w2f 1 , w2f, o a) g~ ( 0 ) I , hE (0)0, gi C ( 0 ) ( 1 ) i ) .
6.7.5.
Theorem.
such that
those of Proof.
If
ID2(A ) b F ,
E-22~tmrF
consider the axioms for (Remember, solution for
H~
(in
~Q])
m r - realizability
QI
and
are exactly
, the proof proceeds as
(3.4.5)°
It remains to
Q2"
tmr F m Fo(~) , where
F ° ~ ~_Fo(~) .
~
is called a
F~
QI.1 : By lemma 6.7.3 (a),
(Q1.1) °
~x Vgh[Vn ~1(n,g,h,a)
Q2.1:
tmrF
F)° For the axioms and rules of
X E ((0)1)((0)0)1.
setting
of terms
then there is a sequence
(and the free variables of
in the soundness theorem for
with
are the defin-
m P1(01,a)
PI-D
~2
A2
I~D2(A) .
X = ~ghS1
~ Pt(Xgh, a)]
A solution is easily obtained i n and using axiom
By lemma 6.7.3 (b),
E-T2 ~
by
PI.2.
(Q2.1) ° ~
EXVg2glg0[Vn,~B2(n,~,g2,gl,gO)
~ P2(Xg2glgo, a)]
with X E ((0)(1)2)((0)(1)1)((0)(1)0)2. A solution i s again e a s i l y obtained, this time by setting X = ~g2glgOS2 and using axiom P2o2 . QI.2 : Let
(1)
F(a)
be any formula of
F(a) ° ~ ~ 3 o ( ~ , a )
~Q],
with
~ ~x~F~(x~,a),
where the sequence of variables
x
is represented as a single variable
=
by means of the pairing functions of 6.7.2 (c) (cf. 3.4.9, "variant I"). Then, parallel to the proof of lemma 6.7.3
AI(F,a)
~ ~(°)~h(°)°Vn{C
(a), we can prove that
~, ~ .v. ~ ~ F~(gs,ts) }
x
T
430
with
C, D, E, s, t
as i n lemma 6 . 7 . 3
(a)
(but now
g6 ( O ) T ) .
Va[AI(F,a ) ~F(a)]
So
D .v.~&F~(gs,ts) l - F~(Xagh, ~)]
~Vag(°)~h(°)°[VnIC-:
Va[Qla--F(a)]
and
So to solve and
(QI.2) °
So define by
TR I
on
=
y(O1,a)
(2)
{Y(%f,a)
This function TI I
Y
we must find
Y
~
(suppressing
X
as a function of
as an argument of
~, a
(4)
Y ) :
0T
is then proved in PI.1
and
E - ~2 ~
to satisfy
E-T2~)
(QI.2) °
PI.3 ; i.e. assuming that for
Vag(°)Th(°)°[Vn{C--: D .V. E ~ F ~ ( g s , t s )
we prove (in
by use of
X :
} -- Fo(Xagh,a)] ,
:
V~,a[P1(~,a) -- Fo(Y~a,~)] TI I
For
on
(5) where
~,
~ = 01 ,
Now suppose
Now
[-~2['
X(a, ~kY(f(2k),tk), "~f) *
and the axioms
(5)
by
in
X o
as follows°
(4) follows trivially from P1(S1f,a) . Then by
PI.1.
PI.3 :
Vn{C' --: D' .V. E ' & P 1 ( ( ~ f ) s ' , t s ' ) } C'~ D', E', s I (n~f)s' = f(2s')
Pl((~f)s',ts'
are
C, D, E, s
by definition,
resp. with
h
replaced by
~f
°
so
) ~ P1(f(2sl),ts')
which implies, from (4) by induction h~pothesi.s,
F~(¥(f(2s'),ts'),ts').
So (5) implies
Vn{C, -: D'.v. ~' A F~(Y(f(2s,),ts,),ts,)}, which implies,
by (3), taking
g = kkY(f(2k),tk)
and
h = w~f :
F~(X(a, ~k¥(f(2k),tk), ~f), a), which implies, by (2), F(Y(Slf,~), a) O
°
* Remember,
f(2k) = (w f)(k) , and
It may be seen more clearly how
w~ Y
Y(Slf) = ~a.X(a, k k ( Y o f ) ( 2 k ) ( t k ) , n ~ f )
arid ~I
were defined in 6.7.2 (e)(i)o
is defined by .
TR I
if we write
431
Q2.2 : The argument here follows a similar pattern, but using of
TR 2
instead
TR I . [The following are some of the details.
F(a)
of
~Q] ,
Taking again, for any formula
F(a) O m ~.xVF~(xW,a) , we prove ( p a r a l l e l
to the p r o o f of
6.7.3 (b)): A2(Q1,F,a) ~ ~2glg0V~, ~IC--: D .v. S &F~(g2r~,tr) l with
C, D, E, r, s, t
as in lemma 6.7.3 (b),
gl E (0)(I)I,
go E (0)(I)0,
but now g2 ~ (0)(I)~ so
V a [ A 2 ( Q 1 , F , a ) -- F ( a ) ] ° ~XVag2glgo[Vn~{C--: D . v . E & F ~ ( g 2 r s , t r
and
)
~ Fo(Xag2glg0,a)] ,
Va[Q2a--F(a)] °
8o we must find So define by
Y
as a function of
TR 2
on
y(O2,a)
2
a, ~
2
and
X.
(suppressing the argument
X ) :
= 0T
I~(s2f,a) = X(a, ~k~Y(f%,1(3k,~),tk), ~IIf , ~f) (Remember,
6.7.2
fD0,1(3k,~) = ( ~ f ) ( k , ~
(e)(ii),
and
DO, 1
Then this function
Y
in 6.7.2
TI2, P2.1
6.7.6.
For the solution of
and
with
x~
solution of
o ~2
defined in
P2. 3 .
to satisfy
(Q1.2) ° , the functional RI, T , where
T
Y
(in (2)
is the type of the
in (I) of 6.7.5, obtained by the use of the pairing functions
of 6.7.2 (c). •~Fo(~,a )
and
(Q2.2) ° by use D e t a i l s f o r t h i s are o m i t t e d . ]
E-T2P)
of 6.7.5) was defined from the constant variable
2 ~ ~2'
(c).)
is proved (in
of the axioms
Note.
,
Alternatively, we could have kept
F(a) °
in the form
here, and then used simultaneous recursion on type (QI.2) ° .
I to define a
This could then be reduced to (simple)
TR I
(i.e.
(2) of 6.7.5) by means of these same pairing functions (cf. 1.6.16). Similar remarks apply to 6.7.7.
Corollary.
closed term
t
(Q2.2)° .
For any number
of type
a,
I such that
Proof.
Immediate from theorem 6.7.5.
6.7.8.
Theorem.
P2
(a) In any
wf
if
I~D2(A) ~ QI ~, then there is a
E-~2~
model
M
PI (t'~) "
of
~ - ~2 ' relations
Can be defined so as to satisfy the axioms for
PI
and
PI
and
P2"
N
(b) Further, if
1
PI
is any relation in
M
which satisfies the axioms for
432
Proof.
(a) Let
M
be a w f
pM 1 c M 1XM 0 by induction and
on
P2.1 - ~
MI
model
and and
of
E-~2.
We can define relations
N P2 c M 2 X N 0 M2
respectively,
so as to satisfy axioms
PI. I - 3
(and the equality axioms for PI and P2 )° M PI can be defined inductively by the two clauses
For example, M
I
(i)
Va mP1(O ,a) ,
(ii)
Va[P~(Slf,a)
(where,
M I ~if, o a)] vnB1(n, ~If,
N
of course,
BI
is the interpretation
taken to be compatible Notes.
recursively
on
could be changed
(ii)'
M P2
and
01 X N M do not define PI
2 ° ) The axioms above
=N )' and similarly
M N = HR02 ' PI
I o ) Taking
(partial)
with
and
However,
for
cannot
in M , and M P2 °
in general
and
M
PI
is
be defined
For example,
clause
(ii)
to :
to (ii). N PI in
any relation
also satisfies
BI
02 × N o M P2 uniquely.
Va[P~(Slf,a ) ~ f is of the form M Vn B1(n,~,h,a) ]
This is not equivalent
of
M
which does
and
satisfy
the axioms
the conclusion
of part
(b) of the theorem,
that,
to the proof of lemma 6.7.3
for
PI
to which we now
turn. (b) First note that if
parallel
A I ( X , a ) 6 ~I '
(~)
then
m~(°)°w[c 4:
with
C, D, E9 t, s
Now suppose for any i.e.
that
~6M I
AI(X,a )
PI.1 , PI.3 :
(2) Put
In
~/01
and
,
M
a 6 M o , if of
so
P1(~,a),
6.2.1) on
~ = S1f
1 , h = ui f . g = wlf
E~L+ ACoI + IP~+EXT X
).
Then
then
a6 q l , v " dE M i
aEQ 1
where
v
and
< I~IM ' lalA1_
= I~IM.
(or on the ordinal
for some
M I= VnB1(n,~f,°~f, a)
variable
* to
as in 6.7.3 (a) (cf. Kreisel and Troelstra 1970 , lemma 4.5). N PI satisfies the axioms for P I " We must show that
(using the notation
By
itself is equivalent
D.v. E~X(ts)]_
The proof is by induction M So suppose P1(~,a) o By
(a), one can prove
v).
f.
.
(2) becomes
(in the language
of
EL
augmented
by the predicate
433
(3)
M l= VnEC--, D .v. ~&P1(gs,ts)]
( a g a i n i n the n o t a t i o n of 6.7.3 ( a ) ) . ~ther, since gs = (~f)s = f(2s),
P~(gs,ts) ~ P~(f(2s),ts) which implies by induction hypothesis that tsE QI so
ts 6
U
QI ~'
and
ItslA1 ~ If(2s) IM < ISIflM,
where
v = ISIflM.
so (3) = V n [ C - , D . v . ~ ( t s
A I ( ~ %,W a) = 6.7.9. Proof.
a
E
U %,~)]
~
b<~
by (I)
Ql,v
Theorem. I~21 ~ I~I Immediate from corollary 6.7.7 and theorem 6.7.8.
So we finally have our characterization 6.7.10. Corollary. Proof.
I~21
of
II,~D21 = 1~21"
From theorems 6.7.9 and 6.6.8°
6.7.11. A note on the extensionality axioms. The axioms
EXT p,a are used to prove the characteristic properties of the pairing functions and their inverses in 6.7.2. These are used in two places: (i)
The pairing functions of 6.7.2 (c) are used in the proof of theorem
6.7.5, for the interpretation of axioms taneous to simple recursion on types
QI.2
I and
and
2
Q2.2 , to reduce simul-
(see note 6.7.6).
(it) The pairing functions of 6.7.2 (d), (e) are used in the proof of lemma 6.7.3 and in the axioms for Ad (i).
The extensionality
PI
and
P2
(6.7.4).
axioms could be avoided here, by either (a) ex-
tending the type structure to include products o f types, and thus reducing simultaneous to simple
TR I
and
TR 2
(cf. 1.6o16,(B));
(b) introducing constants for simultaneous
TR I
and
TR 2
o__rr(presumably) (of. 1.6.16,(A)) •
However, it is not known whether either construction results in an expansion of
~2
(as with
N-H~W:
of. 1.8.2 for (a), 1.7.7 for (b)).
Ad (ii). A careful examination really needed here is
shows that the only e~tensionality axiom I EXTo~ I . This is used in proving that d12u12~ =
I
434
(6.7.2
(~))
by
~I I
occur in formulas
statements of the form Even the axiom theory of trees,
~I .
on
(The point is that paired functions
"through their values"
only,
(~')x =fx, rather than
EXTo, I
often
so that one only has to prove ~' = f .)
could be avoided by considering
a more complicated
so as to avoid the codings of trees in the axioms for
P2"
435
§ 8.
Functional
interpretations
of classical
I DI(O ) and
systems
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
6.8. I.
Introduction
and definitions°
For convenience,
~v(O)
~nd
attention is restricted
~(0)
(v = 1, O
with
in this section (§ 8) to theories
0 = ( 0 1 , ~2)
as defined i n 6 . 2 . 2 .
This is no real restriction
in the study of classical
proof that 0 1 is complete I plete ~I in @I (Richter
1965), it can be shown that for any
monotonic
~II
(provably in, say,
ean be interpreted
I DD2(0 ) .
===
in
systems,
since from the
(Rogers 1967, chapter 16) and
HAS O + E X T ) ,
02 A
each finite subsystem of
is comwhich is I DC(A)
IDC(@) . (Feferman 1970 gives the proof for v= I , I H I set.) Further, it follows that IIDCl =
and with another complete
It should be clear that (by a simplification the analogue of 6.7o10 for
v= I
of all the preceding work)
holds, i.eo
I~11 = I~II The q u e s t i o n we want to consider now i s whether
I zDe~
I~1
as , HA (in
40])
for
~= 1
and 2 . ID2(0 ) is now convenientl,y axiomatized ing axioms for
~I
and
+ the follow-
02 :
0 I. Is) 01o 01 . Ib)
Vn~y(Teny
01 . 2)
F(O) & Ve[VnZy(Teny
02 . Is) 02 . tb) @2" 2)
020 Vn{01n-~ '~y(Teny & 0 2 ( U y ) ) } - ~ ~2(32.5 e) F(O) & Ve[~n{01n -~ ~y(Teny Ec F(Uy)) } -~ F(325e)] -~ Va(02a-*F(a)) ,
where
F(a)
~c ~1(Uy)) ~ 01(3.5 e)
is any formula of
IDDI(0 ) is (of course)
~(o)
is
~(o)
I~D2(0 ) i s
0I. 3) 02. 3)
HA
Ec F(Uy)) -~ F(3o5e)] -* Va(01a-*F(a))
40] (in
. 401]
) + the axioms for
with classical 1o~ie
01
only.
(~ = I , O o
I D2(O ) + the axioms :
v~(~ 01n-. 01n) Vn(m-~ 02n-* 02n ) ,
an~ T2~(O) ~ote.
~(0)
(i.e.
'
is
~I(0)+
01.3.
can easily be interpreted in
I DD+(O ) by the
m m translation
of Io10.2)o
So from now on we will consider
ID+(O)
~ V
instead of
~(0)
43g
0utline of § 8. In 6 o 8 . 2 - 5, we will consider functional interpretations of I~D~(0) ;
I~DI(0)
and
in 6.8.6 we give a historical survey of other (known) methods of
characterizing
I~II
and
IIDII ; and in 6 . 8 . 7 - 1 1
of functional interpretations of
I D2(0 ) and
we turn to the problem
IDa(0) .
First we need some definitions. The axioms for
PI
and
P2
in the corresponding theory
~2~
of trees
are now more conveniently given in the form : Axioms for
PI :
PI.1')
P1(Ol,a) ~ - ~ a = O
PI.2')
Vn[T(e,n,hn) & P1(gn,U(hn))] ~ Pl(S1, 3.5 e)
P1.3,)
Pl(Stf,a)
with
g E (0)I
Axioms fer
1
~ a = 3.5 (a)~ ~ V n [ ~ ( ( a ) 2 , n , ~ f ~ ) and
h E (0)0 ; and
< , ),
~ Pl(~lfn,
~
:(~fn))]
as defined in 6.7.2 (e)(i)o
P2 :
P2o1')
P2(02,a) ~-~ a = 0
P2.2')
V~,n[Plam ~ T(e,n,hn~)& P2(gn~ , U(hn~))]~ P2(S2, ~2.5e)
P2.3')
Pf(S2f,a) ~ a = 32.5 (a)2 & 72
o~
a V ~ , n [ P l ~ ~ T ( ( a ) 2 , n , ~o2fn~)a p2 ( 2fn~ ' g ~ (0)(t)2 an~ h ~ ( 0 ) ( 1 ) 0 , and p a i r i n g f u n e t i e n s
with
inverses Remark.
~' ~2 o
< , >
and
defined similarly to 6.7.2 (e)(ii)o
It may help to clarify these axioms, if we re-state them in a
simpler (but less accurate~) form : for
PI : P1(O1,a) ~
a= 0 ,
{PI(S1 f, 3.5 e) ~ for
P2 : 'P2 (02'a) ~ a=O , IP2(S2f, 32.5 e) ~ V¢,n[Plam ~ P2(f~,
In this form, a ~
VnPl(fn , {e}(n)) ,
PI ~
just means that
~
fe}(n))] .
is extensionally the same tree as
01 : However, the axioms for
PI
and
Pf' as actually given, have to take
i~to account (for the functional interpretation of hidden in
lel(n).
(Also, in the axioms for
must be taken as a function of
n
P2'
as well as
IDf(0) ) the quantifier it turns out that
f
~.)
Note definitions.
~he theories
~I' ~21
the corresponding theories types.
and
~
~IK
are defined as the re~trictiens of
~2 ' etc., with
0
and
I
as the only ground
437
The "ordinal of
is defined (as with
~I"
a closed term of
I~II ~def sup{Itl : t The set-theoretical model of denoted by
#I
~nd
By "functional", #2"
~I
~v
closed terms of Now let
(v = I or 2)
is
(in fact, any object) in
#v
be a ~inite or infinite)
Then
generated from (the denotations in
~v(Y1,~2 .... )
sequence of functionals in
means the system (or collection) ~v
prevloue definition of
~I' ~2' °'"
in
is the
~v(W1,~2,...) o
is empty, this is consistent with the
If~l (6.6.7).
Functional interpretations
I~1
~vT , and
I~(~I,~2 .... )I
supremum of the ordinals of the functionals of type I So when the sequence
of functionals in
of) the constants of
L' ~2, .... by ~epeated) application ~ and
6.8.2-6.8.5.
denoted by the
~v-
~I' Y2' °~"
proof that
~I
~' Y' ~I' "'"
~.
I DI(O ) and. I~DI(0) ;
of
~ I~11"
Consider first the modified realizability interpretation of
obtained by extending the (01t)° ~ ~ P 1 ~ t
m r - translation
o
of
HA
IDI(0)
(5.4.2 - 3) by defining
(as in 6.7oi).
Now consider the translations of the axioms for
01 :
G I. la
is translated as
~PI(~,O) , which is solved in
= 01
(and using axiom
PI.1' ).
(I)
~2 )
we will mean, net only the formal theory
Jr'
01 o lb.
for
J2 o
we will mean a functional
of trees, but also the collection of functionals in
6.8.2.
~2
(corresponding to
Functionals are denoted by
Further, by
also
of type I}o
~I
#I"
Functionals in or
~2 ) by :
E-~I[
by taking
The translation proceeds to the stage ZghVn{T(e,n,hn)~
P1(gn,U(hn)) } ~ ~ P I ( ~ ,
3.5 e)
and then to
~X Vgh[VnlT(e,n,hn) & Pl(gn,U(hn)) } -- Pl(Xgh, 3.5e)] which is solved by taking
X = ~ghS1
(and using
PI.2' ).
01" 2.
The translation is solved (as in the general case, theorem 6.7.5) by
use of
TR I ; and the fact that this does give a solution is proved in
-~1~
by use
of
TIt, P1.1'
and
PI°5 ' .
Thus we obtain a functional interpretation so we can prove (cf. theorem 6°7.9) :
(We omit of
details.)
I21(0 ) in
E - ~I['
and
438
Now for an interpretation The first
of
I~D;(0) ,
step in the translation
of
we must also consider
01.5
01.5.
gives:
V~[mm~Pl(~,n ) -- ~ P 1 ( ~ , n ) ] , which becomes :
~V~(~P1(~,n However~
a solution
) ~ F1(Xn,x) ] .
for
X
would mean a functional
~ ~ (0)I
such that
(in ~I): n601 Then
SI~
-- P1(~n,n) •
would be a tree of type
1 ~ with ordinal
at least
wI ,
so that
1~1(')1 Z ~1 > I~11" So the ordinal Remark.
bound has been "spoilt".
This argument
shows that
01o5
is independent
of
I DI(0 ) .
(See
also remark 6.8.8.) 6.8.3.
Now let us try rather a Dialectica
~-~I~'
extending
The Dialectica
that of
~
in chapter
translation
d(i) - d(vi).
It remains
with the
m r - translation,
d(vii)
(~It) D ~ ~P1~t.
FD
quantifier-free lation of Next, G I. 3
axioms
B
of
see 3.5.4).
01 . lb.
n
as
TB
of the axioms
of
PI '
for all
to solve the trans-
for
of
PI "
61 .
(GI.3) D
,
X = kn~.~.
so this is again
solved by taking
proceeds 9 as with the
mr - translation,
01 .
=
~
to stage
(I)
and then to :
must also be "pulled
solved for tl
function
This will imply the decidability
~,XVg,h[T(e,Ngh,h(Ngh)) i.e.
as in 5.5.2~
) -- P1(Xn~,n)]
(01.1a) D ~ ~PI~O
of 6.8.2,
in
this is defined,
a term
~2[P] , as in 1.6.14,
solved by taking
The translation
is defined
Suppose
the characteristic
the translations
~XVn,~[m~PI(~,n
~I" la :
(01t) D .
~ - ~1~
now offers no problems:
which is easily
I~D~(0)
5. F
(so that we can construct
let us consider
of
by :
formulas
PL 10 :
III,§
of a formula
to define
Now first we must adjoin to with appropriate
interpretation
X
as before,
( non-constructive
& P1(g(Ngh),U(h(Ngh)) ) -- P1(Xgh,
out" as a function
i.e. by taking
and discontinuous)
of
g
and
h.
This is
X = IghS1 , and for
function
of
g
and
h ,
3.5e)],
N
as a
say a total
439
"least number operator", 0 I. 2
or in fact any number
gives no trouble : its translation is solved, as in the modified
realizability
i~terpretatien,
To sum up:
ID~(~)
by the use of
selection operator,
(1)
f(o)%
i.e. a constant
= o ~
f(~f)
function of
axioms
= o.
function of
P I . 1 ' - PI.3'
functional
~(fn
PI
= o) ~
does n o t a f f e c t we w i l l
f(~f)
=
112~1 = I~II Definition.
below).
A functional
(TI)(r2)...(Tn)0
(~l)(T2)...(~n)l
form Note.
~
provides a
since (I) implies :
~I
of any functional
51'
result.)
i.e.
I~(~)1
Hence we o b t a i n
b
satisfying
= I~11-
(~n
the result
#I n~O
is type - 0 - valued if its type has the form .
It is type- 1 -valued
if its type has the
.
Every functional
The functional
~
then the so that
First we need:
in
for some
to
bound o f
a more g e n e r a l
(6.8.5
The point is that
TR I
o.
the ordinal
obtain
such a functional,
of number quantificati0n,
We will show that the adjunction
(1)
~ - ~I
can actually be defined by
are derivable.
interpretation
in the theory
PI ' and also a number
~ E ((0)O)0 , with the axiom
In fact, if we adjoin to the theory characteristic
TR I .
admits a Dialectica interpretation
- ~I~ ' augmented by the characteristic
fact
selection operator.
#I
in
is either type - O - values or type- I - valued.
is type- O - v a l u e d .
So is any characteristic
function,
e.g. of equality at any type. 6.8.4.
Theorem°
ale in
#I ' then
If
~I' Y2' "'"
1~I(~1,~2 .... Proof. T
in
I~I
#I '
so
The idea of the proof is this. ~*
we define a binary relation maj
for
~f(P)~,g(P)~,xP,y p
2° )
~majl
~ ~ I~l ~ majorizes
or
MT
is the domain of objects of type
of majT
With each functional
~
~I
it.
which "majorizes"
on each
M
with of
~E M I ° ~I(~I,~2,...)
More precisely,
with the following properties
maj~ ) :
I° )
We say " x
r
is the ordinal canonically associated
we associate a functional
(writing
function-
)1 = 1511"
We will work in #I ' and
is any sequence of type- O - v a l u e d
(f maj g
and
x maj y ~ fx maj gy)
l~I • y "
for
x maj y.
Then we show (lemmas 3 and 4 below)
that if
is one of the constants of
440
~I ' or one of majorizes
YI,Y2,... ,
It follows,
by property
~ I ( Y I , Y 2 .... ) , (lemma 5). This
then there is a functional
of
~I
which
~. I ° above,
then there is a functional
From this and property
"majorizing"
technique
(section VI and appendix). technique,
that if
as applied
of
~
is any functional
~I
which majorizes
2 ° , the theorem follows.
is a modification
(See 6.8.6
to models
of
of
of one used by Howard
(b) below.
Another
1965
example
of this
YA w , is given in the appendix
of this
volume.) We now define = 0 4
the relation
~m,n(m maJ0 n) .
maJT
(Notel
of
by induction
on
Any number majorizes
T= I o
The definition
~ mail
~
(i)
01 mail
(ii) (iii)
~m,n(fm mail gn) = Slf maJl Slg , ~m(fm mail ~) = S l f maJl ~.
Note.
~ mail
T. any other.)
is by induction
on
o6 M I :
01 ,
8 = I~I ~
I~ 1
(by induction on
~
l~I ) ,
or
but not con-
versely.
T= (p)~ . Note.
f maj(p)a g ~def ~x p ' y p ( x majp y ~ fx maj~ gT) .
It is clear from this that property
Now we define such that lemma
in
~I
a "generalized
I (below)
holds.
supremum"
The definition
Sup 0
~ kf(°)°O°
(say),
SuPl
~ SI
(the given constant),
Sup(p)~
~f(o)(p)~x~
of
By induction
SUPT 6 ((0)T)T is by induction
for all on
on
T.
x) ~ For
SupJ maj
x].
T = I , we use clause
(iii) of the definition
mail. Next we define
for all induction
T ,
in
~I
a "~eneralized
such that lemma
on
2 (below)
maximum"
holds.
functional
The definition
Max
6 (T)(T)T
is again by
T=
Nax0(m,n ) = 0. Nax1(~,B )
is defined by
TR I
on
Max1(~,o) ~: { ~axl(~,%g ) = slx~ M a x 1 ( = , ~ ) .
~ax(~)~(f,~) Lemma
2.
T ,
• :
Sup~ ~(fn~)
~emma ~° Sf(°)~Vx~[Xm(fm~ ~ maj Proof.
I ° holds.
o ~x ~ M a x ( f x , g ~ ) .
VxT ,y T ,ZT[X maj z or y maj z = ~axT(x,y ) maj z] .
Proof° induction on T . For T = I , use induction S=(~ mail O I ) (proved by induction on ~)o
on
y E MI ,
and lemma :
441
Lemma 3-
If
~
is any type - 0 - valued functional in
majorized by a functional of Proof.
Suppose
is
J1 ' then
~I "
Y 6 (TI).°.(Tn)0.
Then
T1 kx I ...~xnn0° maj Y. T
(The whole point is that Lemma 4.
~n°(0 ° m a j n ° ) !)
For every constant of
~1 ' there is a functional of
~I
which
majorizes it. (Note o
This is not trivial.
Proof.
Consider the constants in turn.
Zp,~, T . (Proof. so
E p , a , T maj Zp,~, T . Suppose
y* maj y,
Not every functional majorizes itself.)
x,x* ~ (p)(g)T,
y,y* E (p)q,
z* maj z . Then, by property 2 ° ,
x*z*(y*z*) maj xz(yz) , i.e. H
T maj H
z,z* E P,
Similarly,
0° o
0 ° maj 0 ° , by definition of
01 .
01 maj 01 , by clause (i) of the definition of
S0 .
S 0 maj S O
S1 .
S 1 maj S I .
since
x* maj x,
y'z* maj yz , and
Ex*y*z* maj Zxyz .)
H .
'
with
x'z* maj xz,
T o maJo -
~m,n(Som maj Son )
maj I •
°
Proof :
f maJ(0) I g --~m,n(fm mail gn)
by definition,
S1f maj I S1g, by clause (ii) of the definition of Ro, T . Define
Now take
R*o ~ R*O,T'
x,x* E T
and
mail
of the same type as
y,y* E (0)(T)T , with
Ro,T , by :
x* maj z
and
y* maj y °
Then Vn(Rox*y*n maj Roxyn ) by induction on
n.
(But this does not imply that
Rox*Y* maj RoXY ~)
Then by lemma I, ~ ( S u p T ( R o X * Y *) maj Ro~V~), so
km. SupT(Rox*Y* ) maj RoxY ,
and hence RI, T o Define
R*o maj R ° . R~,T ' of the same type as
R I,T , by
TR I
(with variables
442
*** R1x Y
@* =
~*01
where
{~*(Slg)
No~
take
y* maj y
= ~x~[Y*(~*o
~,x* ~ ~
~nd
g,g),
y,y* ~
Sup~(~*o g)] •
((o)~)((o)I)~
~*
* * * = R1x y
as above.
maj ~ = ~*~ m&j ~ The proof is by induction (i)
If
(ii)
If
x* maj x
and
on
definition
of
~,
~ = 01 , ~ = 01 ; then = S1f , ~ = Slg,
and
or on the inductiv~ ~*~ = x*,
(1)
~* o f maj
(2)
f maj g
so
Y*(@*o f,f)
maj y(~ o g,g) ,
~*(slf)
~(slg)
If
~
maj ~(gn) , by induction
maj
and
maj
~m(fm maj
Also
~
~) :
~*(fm)
by lemma
and so
~*(S1f ) maj @~
by lemma 2o
by induction
~* maj ~ ; and so
hypothesis.
I,
R~ maj R1 .
3 and 4, and property
I ° of
maj , we immediately
If each
Hence
~. is t y p e - 0 - valued, then every functional i is majorized by some functional of T I C
obtain: of
the theorem follows.
6.8.5.
Coronary.
Proof.
From the discussion
in 6.8.3 and theorem 6.8.4.
6.8.6.
Historical
other methods
(a) Firstly,
(1) (where
so
by le~m~ 2,
SupT( ~*o f) maj @~
TI(YI,T2,... )
~.
by (1) and (2).
So
Lemma 5.
¢*e maj
by definition,
= $If
From lemmas
so
h~pothesis,
by definition.
og
then
Hence
= x,
= IZ11"
survey:
it is known
of characterizing
that
~ ( o ) _< ~1(0) ~
maj I :
~m,n(fm maj gn):
Vm,n ~*(fm)
Hence
Then
.
then
(iii)
, with
°
¢ = R1xY ,
Let
,
= x*
means p r o o f - t h e o r e t i c a l
reducibility),
so t h a t
l~11
= ISll ).
which gives another proof of 6.8.5 (using
IIDDII o
443
The reduction (I) is highly non-trivial. PP. 3 4 5 - 6 ,
It is outlined in Kreisel 1968 A,
and also here, for convenience.
The steps are :
(ii)
where
BI 0
can be taken as
(iii)
BIQF , BI D
or
BI~
(iv)
(1.9.20) .
Step (i) is accomplished by an explicit definition of ~
(i.e. saying
that there are no infinite descending sequences of a certain kind from elements of (ii)
0~).
is a Dialectiea interpretation (Howard 1968:
(iii) : q f - W E - ~
w+BR 0
is modelled in
ECF
of. 3-5.19).
(Tait 1963 , Kreisel 1968 A,
footnote 35 ; incidentally, this modelling is extended to
BR
at all types
in 2.9.9). (iv) :
by "elimination of choice sequences" (Kreisel and Troelstra 1970 , § 7).
(v)
by a realizability interpretation of
:
where
K IcN
ID~BBI
in a theory
I~DI(KI) ,
is the set of indices of recursive neighbourhood functions re-
presenting continuous type 2 functionals (Kreisel and Troelstra 1970, §§ 3°7 and 3.8.1").
Then
I~DI(KI)
explicit definition of reducibility of where
T
(b) Let
KI
~I(~)
by
K I , which follows from the proof of the many-one
to
~I
is used instead of ~I
can be interpreted directly in
(cf. Ro6ers 1967 , exercises 11- 61 and 16- 27, K I ).
be a theory of functionals of finite type over the countable
ordinals.
There is only one ground type, that of the ordinals, with ordinals
less than
•
acting as natural numbers.
~I
• , and transfinite recursion on the ordinals.
includes constants fo~
0 and
The exact formulation is not
so important here, since Howard 1963 (section VI, appendix I) showed, by a ma0orizing technique (cf. 6.8.4) that various formulations of
~I
(includ-
ing e.g. adjoining characteristic functions of predicates or a functional for bounded supremum, or changing the exact form of the recursion functional) lead to the same value for
I~II'
by the closed terms of
of ground type.
~I
i.e. the supremum of the ordinals denoted
Howard 1963 (section VI) described a Dialectica interpretation of into a quantifier-free version of
IDB m
(say
qf-WE-
Now one can associate an ordinal canonically with each element of hence with each closed term of the "ordinal of terms of type funetionals of
ID~Bw
of type
K,
I DB I
IDB w, cf. 1.9.25).
and so define
K,
and
IID~BWl ,
IDB w'', as the supremum of the ordinals of these closed K.
Then Howard proved, by a majorizing ~rgument between the
ID~Bw
and those of
~I
(in both directions) that
• Op. cir., § 3.8.1, actually refers to primitive recursive indices, but general recursive indices are more convenient here.
444
From this, and the reduction of
IDa(0)
to
ID~BI
(described in part (a)
above), we obtain:
(2)
1=2~1 ~ I~l-
We remark that it can also be shown that
i~IL = I~iI, again by a majorizing argument,
this time between
~I
and
~I
(in both
directions). (c) Feferman
1968 gave a direct proof of (2) (in fact with equality :
I I~Dil : t~1I ) i n the f o l l o w i n g way, For the i n e q u a l i t y ~ , he d e s c r i b e d a f u n c t i o n a l I~D~(A) , for any positive
A,
into
~I o
interpretation
of
This proceeds in three stages,
as follows. Let
OR
be a first-order,
(with decidable
=
and
the axiom for cardinals, and
w,
and defining
quantified,
< )o
intuitienistic
theory of ordinals
(We can take the system of Takeuti
but with (4) below.)
OR
includes
constants for
schemata for certain function constants
cluding (predicative)
transfinite recursion,
1965 without 0
f,g,.°. , in-
functions for bounded quantifi-
cation :
(3)
f(~'~l .....
~k) = 0 ~
and an axiom for " m - u p p e r
(4)
v~< ®~ OR
OR c
is
Can be interpreted
B
for of
bounds" :
with classical
cation as quantification
formula
6 f(~,~,~1,...) = o
OR c
again as
.
by translating number quantifi-
over ordinals bounded by
ordinals less than
HA c
• ~<
logic.
directly in
w,
B ) :
is, by (3), a function constant
(5)
~g(~,Y1 . . . . ,Yk) =0
(~,~,~I .... ) = o ~-~ ~6 w <
m e
as variables
~<
f
•.
Then (writing
and writing the translation of a
for every formula of
a,b,..°
0R
B
of
HA c , there
such that
0R b B*-~ f ( a t . . . . . an) = 0
(a1,..°,a n Now
the free variables of
I2~(A) , for a positive
(further)
translating
(6)
o ~ P ~ a ~ A ( S ~ < ~P~b, a)
and the characteristic
QI t
as
B)o A(X,a) , is interpreted
~P~t,
function of
P
where
P
in
~.~OR c
by
satisfies
can be defined in
OR
by transfinite
445
recursion, using (3) to eliminate the number quantifiers in (Compare this with the use of the acteristic function of
PI
by
~ - o p e r a t o r in 6.8.3 to define the char-
QI.2
transfinite induction on the ordinals.
A(~
as in (5).
TR I .)
New the translation of the schema
(7)
A,
(6.2.2) is proved in
OR
Finally the translation of
by QI.1 ,
P~b,a) ~ ~ P ~ ,
is proved as follows.
Using the positivity of
A(X,a) , we can bring the
hypothesis of (7) to prenex normal form: Q1Cl "'" QnCn ~ I " ' " where
Q.c. Ii
fier-free.
~mA*(c1'''''cn'~1'''''~m'a)
denotes quantification over ordinals
< w,
and
A*
is quanti-
This implies by (3) and repeated use of (4) : Za~ICl "°" QnCn ~ I < ~''" ~ m ( ~ A*(c1'''''Cn'~1'''''~m'a) "
But this is equivalent to
~A(b ~<
~P~b,
a), i.e., by (6), to the desired
conclusion. The second stage of the interpretation consists in interpreting OR.
This is achieved simply by a
~
sup(~,~) = of type
(0)0
~
~I
in
translation.
The third stage is a Dialectica interpretation of Feferman's formulation of
0R c
0RR in
~I"
includes a supremum functional :
~(~),
(0 = type of ordinals), which solves the Dialectica trans-
lation of (4), and a functional of hounded quantification, which takes care
of (}). This proves the inequality (2). The reverse inequality was proved by modelling
~I
in
I~D~)
as a
system of hereditarily hyperarithmetisal operations of finite type over 0 fl interpreting < ,
(so as to be able to define a linear order in
0 I
and
also to account for hounded quantification)~ It was the present authorls (unsuccessful) attempt to extend this method
to
~(o)
that led him to consider a theory of trees.
(d) HOward 1972 considers theories
I~DI(A) , with
A
positive in the sense
of Kreisel and Troelstra 1970 , § 4.4, and gives another proof of
l~1(A) l ! l ~ 1 1 , First,
I DDI(A)
of trees.
as follows. is interpreted in an intuitionistic first-order theory
For this, a normal form theorem for
A
is used, like that of
Kreisel and Troelstra 1970 , § 4°5. Then
~
is Dialectica-interpreted in a theory
(quantifier-free) version of our
~I "
Q1 t
~
which is like a
is translated as
Z~P~t ,
qf
446
where
P
is now not
qf,
but of the form
VnP'(n,~,t),
with its characteristic function definable by
where
P'
is
qf,
TR I .
It is interesting to compare this method with the proof of by means of another Dialectica interpretation of
IIDDII !
~D~(O) , given earlier
(6.8.3-4)° On the one hand, because of the different translation of method does not need non-constructive functionals such as a
Q1t , Howard's ~ - operator.
On the other hand (again because of this different translation), his method applies (apparently) only to intuitionistic clear that the translation of the ordinal bound of
01.3
I~II "
01.3
I~DI(A) , since it is not
(6.8.1) can be solved without affecting
With our definition of
(QIt) D,
the translation
(6.8.3) comes for free, so to speak.
In the same paper Howard also gives a characterization of
I~II
in
terms of Bachmann's notations (see below). (e) Analysis in terms of Bachma~u - Isles notations. We mention that =
t
11
=
in the notation of Bachmann 1950.
The inequality
I 11 !
follows
from Howard 1970 A (or, more simply, Howard 1972), and the reverse inequality from Gerber 1970. I conjecture that, further,
1~21 = ~
(1) (I) w2+1
in Bachmann's notation (i.e.
FI(F2(F3(2,1),I),I)
in that of Isles 1970 ).
Nartin-L~f conjectured this independently (Nartin-Lof 1971). Note.
It is stated, op. cir., that I have proved the above conjecture.
This is not so, although it seems that it could be proved by (in one direction) an ordinal analysis of
22
by means of infinite terms, extending the
method of Howard 1972 , and (in the other) an extension of the method of Gerber 1970. Nartin-L~f 1971 also gives an ordinal characterization of his system of finitely iterated inductive definitions (op. cir.) in terms of Isles notations.
(See 6.9.2.)
447
6.8.7- 6.8. tl. 6.8.7.
Functional interpretations
of
I~D2(0)
and
Consider, first, the modified realizabi!ity
I D2(O ) , defined in 6.7.1. ID2(A ) , for any
A~,
interpretation of
We have already treated the general case of A ~ (AI,A2)
is the pair of defining
(01,02) .
The axioms for axioms for
(~-)
with theorem 6.7.5, but now for convenience we
review briefly the special case where predicates for
I D~(O) o
G I
have been dealt with in 6.8.2.
Now consider the
02 °
02. ta taking
is translated as ~2P2(~2,O) , which is solved in 2 = O 2 , and using axiom P2.1' (6o8.1).
02 . l b .
The translation proceeds to the stage:
(~)
E-~2~
by
~g,hVe, n [ P l ~ n - ~ T ( e , n , h n e ) & P2(gn~,U(hne))] -~ ~2P2(~2 , 32.5e) o
The translation is completed, function
$2(g,h)
of
g,h
and solved in
E - ~2~ ' by taking
(and using axiom
92
as the
P2.2 t )°
02 . 2.
The translation is solved as in the general case (6°7.5) by the use
of
(and the axioms
TR 2
TI2, P2.1' and P2.3' )o
Thus we o b t a i n a functional i n t e r p r e t a t i o n of
I D2(O )
in
E-~2~,
showing (as a particular case of theorem 6.7.9) that
t~2(~)1 ~ t221" However, t h i s i n t e r p r e t a t i o n is unsuitable for D + 01.3 of ~ 1 ( 0 ) ) in 6 . 8 . 2 .
~(o),
as ~ho~ (al~e~y
for the axiom
6.8.8. ~2(0)
Remark.
This argument also shows that
0 I" 3
is independent of
°
6.8.9.
SO now l e t us try a Dialectica interpretation of
Dialectica translation defined by: extending the interpretation of translations of the axioms for (0i" 3)D
(i = 1,2)
(Oit)D ~ ~ i P i ( ~ i , t )
I~D~(O) ~1
and
in 6.8.3).
I D2(0) , (i = 1,2)
w i t h the (i.e.
Consider again the
02 :
now gives no trouble (as shown for
i= I
in 6°8.3).
as
(0 i. la) D (i = 1,2)
is solvedAfor the modified realizability interpretation.
(0 i . 2 ) D
is again solved by
( i = I 2)
We are left with
(Oi.lb)D.
For
TR
i = I , this can be solved as in 6.8.3,
by adjoining a number selection operator
~
(and then using a majorizing
argument for the ordinal analysis). However, for
i = 2 , the situation is more serious.
The translation
proceeds to the stage (I) (in 6.8°7), and is then completed by pulling out
448
n
and
and
~
(as well as
~2 ),
and solving for them as functions
n
this can be done again with the functional
of
g, h
e . For
cannot be done for without
affecting
~
(i.eo
adjoining
the ordinal
bound
For suppose we could find
a suitable
b.
However,
tree selection
this
operator)
of the system of functicnalso
(and
a s functions
n)
of
g, h
and
e :
= ~ ghe ,
(I)
In
satisfying
= N ghe (in
Jy)
[P1oLn -
for all
g, h, e :
T(e,n,hn~) & P2(gn~,U(hn~))]
--
P2(S2, 32.5e) ,
i,e°
(2)
mP2(S2 , 32.5 e) -- [Plain &(mT(e,n,hn~) V mP2(gn~,U(hn~)))] ,
with
~, n
as
(I).
in
Then we could define
(3)
s u c h that
k ~ 01 ~ P t ( ~ k , k )
(proved > ~I '
in 6.8.10 below),
so that
So both functional IIDyl <
is a tree of type
I
with ordinal
bound has been
interpretations
spoilt.
(6.8.7 and 6.8.9)
fail to show that
ITyl , and it is still an open problem whether
1~I [6.8.10.
:
t~21
Derivation of
Let resp.
$I~
and hence
so that again the ordinal
ek
and
k, n ,
(I)
hk, n
(3)
>
of 6°8.9.
be numbers
T(ek,n,hk,n)
& U(hk,n)
delta,
fek}(n ) =
V~,n
Iz2~1 1~21.
or
such that for all
(dk, n = Kronecker
SO
C (0)I
a functional
= 0
if
which depend
primitive
k, n : = 6k, n k~n,
I if
k=n).
8k,nO
Now define g
= ku,~. 0 2 ,
(2)
{
Finally,
define
h k = ~.n,~. hk, n % E (0) I
and (for any
k) .
by
9k = Tghke k o We will now show that
9
satisfies
(3) of 6.8.9.
recursively
on
k,
449
From
(3) Now
P2.2', F2.3'
and
(1) (with
g,~
(2)) :
as in
P2(S2 , 32.5 ek) ~ V~,n[P1~n ~ P2(gn~, 6k,n)] • Vn,~(gn~= 02 ) , so
P2(g n~, 6k, n) * P2(O 2, 8k, n) 8k, n = 0 by P2.1' *k/n. SO (3) becomes: P2(S2, 32.5 ek) ~ V~,n[P1a:n = n / k ] V ~ P1~k
k¢01° So : k e a I = ~P2(S2 , 52.5 ek) (4)
~ P1(~ghkek, Nghkek)
by (2) of 6.8°9°
and mP2(02, 8k,Nghkek )
Now
P2(0 2, 8k,Nghkek ) = 6k,Nghkek ~ 0
by
P2.1'
= Nghke k = k. So i.e.
(4) = P1(Yghkek,k) , k 6 01 = P1(gk)k) o
]
6.8.11. Remark. There i__s.san easy interpretation of IDa(0) into an intuitionistic system I~D2(~mm ) (say) of iterated inductive definitions, namely the m m interpretation (i.e. ' of 1.10.2) ; but the inductive definitions of ID2 ~ m m ) do not even satisfy the condition of positivity (let alone ~) , so this does not seem to help for an ordinal analysis
450
§ 9.
Extensior~to
I Dv(A )
and
= = = ~ = = = = = = = =
t~(A)
for
= = =
v> 2 ,
Equivalences
~ = =
with some subsystems of classical
= = = = ~ = = = = = = =
analysis.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
6.9.1.
The work of this chapter can be extended to systems
inductive definitions trees of the f ~ s t v~w
o
IDa(A)
iterated
I+ v
v
times,
classes,
for
1970 for the definitions
for
For example, we can define a system
v > w o)
~
is the binary predicate
ID (A)
(CI,02,03,...)
of
systems
ID (0)
of all
(cf. Richter
I,
T
for some
of the (classical)
I: x 6 0
of
~V
systems
v> 2 , even (apparently)
(See Feferman
the recursive finite number classes where
a~d corresponding
1965),
and also (it seems) a
n
corresponding
theory
T
of trees of all the finite tree classes,
distinct ground type for each class,
and transfinite
recursion
with a
on each class,
such that
(I)
I~(o)1
where
lID (0)i
y~D (O))
in
terms of
is the supremum of the ordinals of numbers
0 1 , and
T
Further,
= IZJ,
the first recursively
is the supremum of the ordinals of the closed
(i.e. the second tree class).
I DI(O )
inaccessible
1961, or
F
(something like the systems
of i n d u c t i v e l y
defined
sets
of
(0a:
a6 I)
(recursively)
a
and a t h e o r y
inaccessible,
SI
so that
(it seems):
(where the two sides are defined analogously
to (I)).
The interest of these results lies partly in this. lished the proof-theoretical
equivalence
with subsystems
analysis,
of classical
denoting proof-theoretical
(3)
~(o)
where
C
of Richter 1968), and, correspondingly,
of trees of all classes up to the first again
(in
we can define a system I of ordinal notations up to (apparently)
Kreider and Rogers
theory
l~wl
of type I
provably
I + nl-
~2
Feferman
of the classical
for various
v.
1970 estab-
systems
I DD~(A)
For example
equivalence) : CA + BI
o'
is classical second order arithmetic (i.eo ~+EXT), 1 H I - comprehension axiom, and BI 0 can be taken here as
1
~2
is the
(1.9.20) with
Pn ~ Xn
(X
the schema
BI
in Feferman
N o w let
~2
be
induction formula iterated
inductive
~2
a predicate variable)
and
Qn
H I - CA BI D
arbitrary,
or as
1970.
with induction restricted
F(x) ~ Xx
(with
(X
definitions
to the single axiom with
a predicate variable).
For systems of
451
which correspond to
Z~ + H~- CA
and
~2 + H~- CA,
see (resp.) 6.9.2 and
6.9.3 below. Continuing this, we have (it seems) :
(4)
~(o) ~2
(where
IDa(O)
Now if
is
+ ~ - c A + ~io I91(0 )
1122(0)1 = II~.Dv(O)]
knew that
we
with classical logic). for
v: w
and
I,
we could
derive, from (I), (2), (3) and (4), interesting characterizations "ordinals of"
~2 + HII CA + BI 0
of their provable well-orderings):
1~2 + n~- cA + ~iol : l ~ t ,
(6)
I I~2 + ~2-
so the t ~ t h
cA + ~Iol : l~iI
~ 1~(o)1
v>2
whether
or
of (5) and (6) remains an open problem.
The important problem here is to settle (7) for
falsityof
I~l
~ 1~21 (~ I ~ 2 1 ) ,
surely generalize to give proofs of 6.9.2°
of the
(i.e. the suprema
and
However, it is not known for any
1~2(o)t
CA + BI 0
namely,
(~)
(v)
~2 + A2I
and
~=2,
i.e. the truth or
since a proof of this, if true, would
(5) ~nd (6).
One positive result we do have in this direction is the following,
pointed out in Nartin-LSf Let
IDn(A )
iterated
n
1971.
be the (intuitionistic)
times, and let
A m (AI,A2,..o)),
and
I D<w(A )
ID ~ ( A )
theory of inductive definitions
be the union of these theories (for
the same with classical logic.
Firstly one can see, by adapting the arguments in Feferman 1970, that I~D~w(A) ~ 2 (for suitable
I
+ E1- CA
A)°
Next, Takeuti 1967 gave an analysis of his system 42- + H11 _ CA)
SJNN
(essentially
by means of his system of ordinal diagrams of finite order.
Further, Kreisel 1964 formalized the proof of well-foundedness ordinal diagrams in
I D<~(A)
for suitable
of these
A o
Putting this all together, we get
(I)
ID I~®,t : l~<~I
=
t ~ + ~I1 _
CA I = s u p O ( n ) n
where
O(n)
is the supremum of the ordinal diagrams of order
Finally, Levitz 1970 showed that ordinal
sup O(n) = sup ~n'
where
n° ~n
is the
452
FI(F 2..° (Fn(2,1))... I)1) in the notation of Isles 1970°
So
sup n
is another characterization of
n
the ordinal in (I), 6.9°3.
We conclude with a description of a theory ~ - IDa(A) ("W" for I ~2 + ~I- CA o For convenience, we
weak induction), which is equivalent to first repeat the definition of
Let
A(X,Y,x,y)
X,Y~x,y
!D~(A)
be a formula of
its only free variables.
12~(A )
is the theory
HA °
in
given in Feferman 1970o
~]X,Y]
which is positive i_nn X,
Let
Q
~Q]
, together with the axioms:
with
be a new binary predicate symbol.
Q.1) A(Qy, ~ ( u < y & Quv),x,y) ~ Qyx
for any
F(x)
Definition. Q.2,
F(x)
in
~[Q]
W - IDD~(A)
(where
Q
is the unary predicate I Q y x )o Y ~(A) except that in the induction schema
is like
is restricted to being a
W - formula (w.r.t.
that in any occurrence of the prime formula
x),
F(x)
which means
Qst
in
(where
s, t
Qst
is not in the scope of a
are number terms): I °) if any variable quantifier
Vy
or
2 ° ) the variable
y ~y
x
occurs in in
s , then
F(x) ; and
does not occur in
This just means that
F(x)
So
has the form
F(Qsl , Qs2 .... , x) , where
e~ch
s. contains variables only as parameters~ i°e° not quantified in l (and distinct from X)o
[The proof that ~-~m(A)
1 ~2 + ~1-
CA
(for suitable
A)
F
modifies
that in Feferman 1970 (henceforth [Fef]) for (3) of 6.9.1 : (i)
TO model
of a
W-ID
W - formula
(A)
IF(x)
in
~2 + HII _ CA : the point is that the extension HI I CA o
can be proved to exist as a set, by
(it) Conversely, we model
~2 + H1I _ CA
in a theory
W - I~Dm(A) like
IDTw
of [Fell (except for having the weakened induction schema, as described). The proof of [Fef, theorem
2.2.1 (it)] is modified as follows (using the
notation there) : First note that the inductive definition of X
R(s, ~x(Lh(s))) v VySe%(s.) and proof by induction on
Sec~ :
R - securability : X
- SeeR(s)
453
(1)
Vs[R(S,~x(Lh(s)))
where (c~.
F(s) [Fef,
is a
~ - formula ~ . r . t .
(8) ~ d
x ~ Vs(SeCR(S ) ~ F ( s ) ) ,
V VyF(s* ).~F(s)] s,
are both d e r i v a b l e in
~-~®(A).
(9) of § 2 . ~ . )
~e~ ~o~ul~(8)o~
[Fef,
§ 2 . 2 ] ca~ be p ~
i ~ t~e ~orm
x (ioeo
the universal
- ID (A)
func%ion
quantifier
by applying
the induction
~(Lh(s)) = s ~
~yR(~y,~xy)
and the argument
proceeds
is pulled
schema
out),
(I) above
and then proved
to the
in
W - formula
;
as in [Fef]
to prove
(the translation
of)
I ~I - CAo ]
Appendix HEREDITARILY
NAJORIZABLE W.A.
F~NCTIONALS
OF FINITE TYPE
Howard
The purpose of the following is * to show that the Dialectica tion (chapter I I I , §
5) of the simplest nontrivial
axiom of extensionality
(theorem 3.2).
hereditary
majorizability
functional
To accomplish this we introduce and show that if a functional
then it does not satisfy the functional
axiom of extensionality
(§ 2).
is hereditarily
The functional
of
majorizable
over which the variables
is hereditarily
depends
Vy3E3(y )
In contrast with
satisfying the function-
are taken to range.
Indeed,
if the variables
set-theoretic
of a functional
VyE3(y )
implies the axiom of choice for sets of number-theoretic Hence by known results,
interpretation
are
then the
of fttuction-
there are models of Zermelo-
set theory (without the axiom of choice) in which there is no
functional
§ I.
providing the functional
functionals,
existence
Fraenkel
of
strongly on the class of functionals
taken to range over the class of ordinary
als (theorem 4.1).
of the
recursive
(§ 3).
of the next most simple case
is discussed briefly in § 4.
Vy3E3(y )
the notion of
interpretation
Vy2E2(Y) , the existence of a functional
al interpretation
of the
Then we show that every primitive
interpretation
the axiom of extensionality the case of
Vy2E2(Y)
cannot be carried out by use of a primitive recursive
functional
majorizable,
case
interpreta-
satisfying the functional
interpretation
of
Vy3E3(y) °
Extensionality.
Supposing WZI...Z s namely,
X, ~, ZI,~°.,Z
are terms of type
S
to be variables 0 , let
X =e W
(~ZI.o.Zs)(XZI..°Z s = ~Z1...Zs)
** such that
XZI..°Z
denote extensional
(as in 2.7.2).
Now let
S
and
equality, X I ..... X k
References with three numbers refer to this volume outside this appendix. References
such as 4.1, § 2 etc. refer to this appendix.
In this appendix also capitals are used as variables type.
for objects of fini~c
455
be variables
of types
(¢I)(a2) °.. (¢k) 0 ~
¢I,...,~k,
respectively,
A functional
G
of type
and let
~
~
denote
is said to be extensional
if (1.1)
(~1"''Xk)(~1°°'Wk)(Vi[Xi
where
Vi[X i = eWi]
us abbreviate of type
~
=e Wi] -- GXI°°°X ~ = G~1...~k),
denotes the conjunction
(1.1) by
E~(G) °
of
(2.7.2) is here taken in the form
trivial case of the axiom of extensionality
where
~
XI =e ~1'''''Xk =e Wk °
The axiom of extensionality
and
8
type (2)(I)(I) 0
have type satisfies
I
and
Y
Vy~Ea(y) .
is
Let
for functionals
The simplest non-
Vy2E2(Y) ; namely,
has type
2 °
A functional
the Dialectica functional
F
interpretation
of
of (1.2)
if (1.3)
VY~8[Y~/Y~
We will work in
~ ~(FY~)
H~A~
because it simplifies
/ 8(FY~)] o
(1.6.15)o
We use the formalism of typed combinators
the exposition of § 3.
The
X-operator
is assumed to
be defined by the rules of 1.6.8. As a point of methodology we note here that the three theorems of § 2, and all instances of the two theorems remark 3oi)o
Thus the theorems
in particular
(Cfo chapter II) :
and
HEO
based on partial
and
ECF
based on continuous
CTNF,
of § 3, are derivable
the set-theoretical
recursive
model,
function application,
function application, functionals
theorems which deal directly with the functional (namely~
~w
(see
of §§ 2 - } are valid for all models of
and Kleene's general reoursive
of extensionality
in
~:
the models
the models
the term models
HR0 ICF
CTM
Not that the
interpretation
of the axiom
theorems 2.2, 2.3 and 3.2) are of much interest
in the case of nonextensional
functionals :after all, the negation of the
axiom (1.2) of entensionality
implies
pretation of (1o2) trivially. hereditary majorizability intensional
§ 2.
continuous
Hereditarily
A relation
induction on we define
T°
n majm
appear to be of independent
majorizable
between functionals
(F*
F If
and ~
the negation of the functional
is
to mean
inter-
But theorems 2.1 and 3.1 on houndedness
functionals
F* maj F
and
(cf. 2.8.2).
of types
I
and
interest. 2
~nd
(Also~ the
are extensional.)
functionalso hereditarily F
majorizes
of the same type
0 , n > ms
then If
F* c
and is
F)
¢ ° F
(T)p
will now be defined The definition is by
are numbers then
n
F* maj F
and
m:
means
455
VG*G(G* maj G ~ F'G* maj FG) ° majorizable
if there exists a f u n c t i o n a l
The f o l l o w i n g
three remarks
R e m a r k 2.1.
If
F* maj F
R e m a r k 2.2.
If
G*r maj Gr
R e m a r k 2.~°
Suppose
all
We say that a functional such that
is h e r e d i t a r i l y
F* maj F .
are easily verified.
and
G* maj G ,
for
then
0 < r < p ,
HXI.ooX p
* XI,.O. ,X *p, X I , O o . , X p
F*
F
such that
then
0 .
has type
F'G* maj FG o G0@I.. * * .G*P maj GOGI..°G p . * ..X*P _> H X l o . o X p H * XIo
if
X*r maj Xr
for
I < r < p ,
for
then
H* maj H . In the f o l l o w i n g
theorem,
X
and
Z1,...,Zs(r)
r
XrZ1...Zs(r)
has type
T h e o r e m 2oio
Suppose a f u n c t i o n a l
tarily
majorizable.
are v a r i a b l e s
such that
0 .
Let
the set of f u n c t i o n a l s
k X
of type
F
( ~ 1 ) ( ~ 2 ) ... (Sp)O 1 < r < p,
be f i x e d and~ f o r of type
¢
r
let
is herediM
denote
--r
such that r
(VZl°o°Zs(r))(XrZl.°.Zs(r)~k). Then (2.1) ~m(~l e g 1 ) °°. (VXp ~ ~p) (~Xl° ..Xp_<m). Proof.
By a s s u m p t i o n
l e t G*r denote Hence
there exists
F*
(XZl.°°Zs(r)).ko
such that
Then
P _ by remark 2.2. T h e o r e m 2.2.
Thus For
* ..G*P F * G1o
r = I, 2 ,
F* maj F .
N
I < r < p,
by remark 2 . 3 .
o.Xp)
is the required n u m b e r
let
For
(VXr E ~ r ) ( C ~ maj Xr)
m.
denote the set of f u n c t i o n a l s
X
of
--r
type
r
such that
V Z ( X Z ~ I).
Let
F
be a functional
of type
(2)(I)(I)0
such that
(2.2) Then
~m(NYE~2)(V~c~I)EFY~(Xu.O)~m ] o F
does n o t s a t i s f y
the f u n c t i o n a l
interpretation
(1.3)
o f the axiom
of e x t e n s i o n a l i t y o Proof.
It is easy to define a primitive
that, for all
Y ~=
n
Assume
functional
kn.Y
n
such
~,
I
if
(Vu
0
otherwise.
and
a~ = I ,
I
Also it is easy to define a functional
n
recursive
0
if
u<
I
if
u > n°
(1.3) and take
p
kn.~
such that n
n
to be
ku.O.
Denote
~Y~oFY~(ku.O)
by
G°
457
Clearly
Yn(kU.O)
Yn~n = I. But
YnE~2
Hence
= 0 °
Hence and
~n £ ~ I °
2.}. Suppose a
Theorem
Then
F
Hence
GYn~n 2 n
Thus the hypothesis
so (1.3) has been refuted.
majorizable.
~n(GYn~n) /
Yn~n/ 0 ~
~n(GYn~n) / 0 o
0
by (1.3).
But
by the definition
of
%1~
(2°2) has been contradicted,
Qo Eo Do
functional of type
(2)(I)(I)0
is hereditarily
does not satisfy the functional
interpretation
(1.3)
of the axiom of extensionality. Proof°
By theorem 2.1
Remark 2.4. following
By inspeotion
k= I )
and theorem 2.2.
of theorems
2oi - 2.3 we see that actually the
sharp form of theorem 2.3 has been proved.
recursive funotionals if
(for
F* maj F,
kuoY n
and
kno~ n
There are primitive
such that, for all
F
and
F* :
then
[Yd% / ~o ~ ~d(Yd~d~O ) / ~o(Yd~d~O )]' where
~0
is
kUo0
and
d = F*(X~ol)(~u.O)(ku.O)
°
Indeed 9 the above proofs go through for relative hereditary majorization: all that is assumed of the set of functionals
~
of majorizing functionals
being majorized is that
~
tion and contain certain simple primitive Construction
2.1.
Given
Vn(F*n maj Fn) ~ to find
F
and
F*
F*mXI.~X k
§ 3.
Primitive
has type
recursive
0 .
and the set
are closed under applica-
recursive
functionals.
(0)~
such that
H* maj F ° Solution : take o,X k
. .
are variables
We denote this
H*
by
H*
to be
of types such
(F*) + o
functionals.
We indicate extensional By applying universal
X1,
~
of type
H* such that
n F*mXI..oX k 9 where
(XXl...Xk).m~ that
and
equality of
quantifiers
F
and
G
by
to the appropriate
F =
G as in § Io e axioms for h~w we
obtain :
(3oi)
(VXY)[~XY =e X],
(3.2)
(~YZ)[ZXYZ =e XZ(YZ)],
(3°3)
(VXY)[RXYO=eX] ,
(3°4)
(vxYu)[~(su) °e ~(~Yu)u]°
The set
~
of all primitive
recursive
ence %o a given notion of functional. notion
(2.4.6)
functionals
is defined with refer-
In the case of the set-theoretic
there is no problem since in this case the equations
(3.4) pick out functionals
H, Z
and
R
of all appropriate
(3oi) -
types from the
458
supply
in a u n i q u e
Hence we define
way.
P
inductively by the following two
clauses :
(a)
contains H
(b)
zero,
the
successor
of all a p p r o p r i a t e
If
FE~
and
GE~,
then
In the case of the model is to be named associate Z
and
HRO .
If
Theorem
3.1.
and
H =
Proof.
We
clause.
H*
H =
F
Similarly
Similar
associate
pick out an
for the c o n s t a n t s
remarks
of n o n u n i q u e n e s s
do not
clause
uniquely
apply
to the case
cause
to
any trouble
(a) and
(b) :
H C Pc --
recursive
functional
has a p r i m i t i v e
for each of the clauses
H
is obtained (b),
The case
and
does not
does not d e t e r m i n e
add the f o l l o w i n g
then
of clause
2.1o
and
recursive
functional.
for
In the case
by r e m a r k
that if
F,
shall indicate,
ing f u n c t i o n a l
~*G*
sources
primitive
majorizing
H, Z
as to w h i c h
(3.1)
way.
of a p p l i c a t i o n
e
Every
hereditarily
in a u n i q u e
We m e r e l y
--
is some choice
the equation
on associates.
these
paper.
F 6 P
there n:
rCF
operation
Fortunately
in the present (c)
ICF
the o p e r a t i o n
the c o r r e s p o n d i n g of
supply
Also,
and the f u n c t i o n a l s
FCC~.
by the constant
f r o m the
R.
function,
types.
when
H
is
of clause
F* maj F ,
then
H
(a) - (c), h o w the m a j o r i z -
arises
FG
by use
of the given
so we can take
(c) is h a n d l e d
F* maj H .
H*
to be
by the o b s e r v a t i o n
It remains
to treat
the
e
case
of clause
(a) ; namely,
generating
functionals
S*
0
to be
and
(3.1) - (3.2) Similarly,
by remarks
R*
Theorem ality
3.2.
(1.2)
Proof.
This
als.
For
mentioned
(y)
minimal
and
w
maj RXYn
whenever
term model
we can take
(2.2) - (2.3)
n
and
E,
does not have follows
from
~ ,
X* maj X,
0*
and
and equations
respectively. and i n d u c t i o n
Y* maj Y .
Hence
on
we can
2.1.
interpretation
of the a x i o m
of e x t e n s i o n -
functional.
3.1. a Dialectica
the fact
and it has
we can use
consisting
that a model
classes
terms
recursive
of
We HA w
as
function-
of f u n c t i o n a l s
the e x t e n s i o n a l
operations.
of the closed
in itself.
can be a x i o m a t i z e d
by p r i m i t i v e
functionals,
effective
on themselves.
interpretation E_~W
any of various
the s e t - t h e o r e t i c
or the extensional
extensionally
to be
as in c o n s t r u c t i o n
recursive
2°3 and
such a model in § I :
3.1 for each of the
Obviously
By remarks Z*
is no D i a l e c t i c a
for all
theorem
R.
(2.2)- (2.3), equations (3.3)- (3.4)
by a p r i m i t i v e
E-H~
functionals,
act
There
corollary
+ VyE
H*
KXY.(RXY) +
By theorems
Corollary.
~w
HX*Y*n
to be
and
respectively.
we can take
n , we find take
S ,
we must v e r i f y
O, S, H, Z
continuous
can even use the since
these
terms
459
Remark 3.1.
To treat theorem 3.1 by use of
case of a primitive recursive (b) alone.
functional
~
let us consider first the
defined by use of clauses
Inspection of the proof of theorem 3.1 provides a corresponding together with a derivation, case of a primitive well as clauses
in
recursive
(a) -(b),
use of variables.
~,
of the formula
is
For illustration
VX(X=e F ~ F* maj X) o
functionals clauses HAW°
term
F* maj F o
will be described
suppose that clause
in
F*
In the
~w
(c) as
by the
(c) has been applied
Then the formula to be derived in
(This shows, by the way, that the majorizin~
in theorem 3.1 can be chosen from the functionals
(a)- (b) alone.)
F.
f~muctional defined by the use of clause
the functional
only at the last step of the definition° ~
(a) and
Then the functional will be represented by a closed term
generated by
Thus each instance of theorem 3.1 is derivable
A similar remark applies to theorem 3.2.
Also, remark 2.4 and the proof
of theorem 3.2 provide an effective
procedure which when applied to the
definition of
(a) - (c) yields primitive recursive
an
and
~0
F
by use of clauses
in
Yn 9
such that
Thus if a functional primitive recursive
H
agrees with a primitive reeursive
arguments,
then
H
functional
at all
does net satisfy the functional
inter-
pretation of the axiom (1.2) of extensienalityo Remark 3.2. pretation
There exists a functional
(I.3) of the axiom
VYE2(Y )
recursive in the sense of Kleene that and
Y, ~ I ,
and
~n
is
Namely,
Y~ = Y~,
successively
for
o~a% ~n o
Since
such that F
If
~ - recursive
Exactly
of extensionality
take
the instructions FY~B
to be
of types
for calculating
0.
is extensional,
If
Y ~ % Y~, FY~
the required
2, I
FY~
are
examine
~n
to be the least n
exists.
F
1959A~P.
satisfying
(1.3) when
Y, ~
continuous functionals.
94, the functional
F
and
Hence, by
is itself continuous
(which is also easy to see directly). If
Y, ~
operations, F
and
~
are understood
to range over (extensional)
effective
then clearly the above definition yields an effective
satisfying
(1.3)o
n
Thus
(Kleene 1959, pc 45)°
to range over extensional
theorem 4 of Kleene
which is general
functionals
n = 0,I,2,.°° , and take Y
inter-
1959 (Cfo 2.8°2), where it is understood
the same definition yields
are understood
satisfying the functional
range over all set-theoretic
respectively°
as follows. and
~
F
operation
460
§ 4.
Discussion
The a x i o m (4.1)
of
VyE3(y ) o
V y E 3 ( y ) is
w~(v~[x~=
where
X
and
functional
W
Y~] - Yx = ~ )
have type
interpretation
(4.2)
V~X~[YX~ ~
If
Y, X
and
W
,
2 , and
Y
has type
are t a k e n to range over e x t e n s i o n a l
(4.2) : we m e r e l y generalize
the d e f i n i t i o n
the fact that the extensional dense base
continuous
(2.6.16).
dense base for the f u n c t i o n a l s h0°
If
YX % Y W ,
take
These considerations arbitrary
(ioeo,
(2)0)°
The
-- X(FYX~) / W(FYXW)] .
als, then there exists a Kleene general recursive
recursively
3
of (4.1) is
Let
of
to be
obviously
hk,
generalize
~ , where all v a r i a b l e s
be a r e c u r s i v e l y
YX = "#W,
where
function-
satisfying
of a given type have a
ho,h1,o.~,hn,°O, If
F
given in r e m a r k 3.2, using
functionals
of type I .
FYXW
F
continuous
functional
take
FYX~
to be
k = minn(Xhn%Whn)
to the case of
VyE
o
(y)
for
are taken to range over extensional
con-
tinuous f~uctionalso In the f o l l o w i n g theoretic
T h e o r e m 4.1o set theory,
~M~ M
Let
for all
~
I
~N
and
are taken to range over set-
H
~_~
is
of type 3
ZF
satisfying
Then c l e a r l y
I
or
0
in
of type I ,
~.
That is to say:
as to whether
or not.
set
theory
ZF
On the other hand,
that
Define
every nonempty
of the axiom of choice.
Z~F
of functions
function of
according
V ~ ( X ~ = O)
~ME M f o r
it is show~ that there are models of
can be defined in
~
~o
denote Zermelo-Fraenkel
of the real numbers.
(4.2) we can construct,
be defined by the c o n d i t i o n
a c c o r d i n g as to w h e t h e r
.
F
defined on all sets
for all n o n e m p t y
t e n d e d by t h e a d d i t i o n
4.I.
W
denote the c h a r a c t e r i s t i c
Let
FH(~)(X~.O) Let
~
of type I ,
or not. or
Y, X
F r o m a functional a function
such that Proof.
theorem,
functionals.
¢~
~
is in
HX
is
0
to be
~o
and l e t
ZFC
In Rosser
denote
t969,
pp.
Z~F
ex-
1 1 3 - 115,
in w h i c h there is no w e l l - o r d e r i n g a well-ordering
of the real numbers
with the help of the choice function
¢
of t h e o r e m
Hence we obtain :
Corollary
Io
There are models of
fying the f u n c t i o n a l Similarly,
~
interpretation
in w h i c h there is no functional (4.2) of
VyE3(Y ) .
from the fact that there are models of
formula well-orders
the real numbers
(Rosser
Z3~C
in w h i c h no
1969, p. 89), we obtain :
satis-
461
Corollary
2.
a formula of
There are models of ZF
satisfies
Of course the existence pretation of
VyE3(y )
From corollary
Corollary functional
3.
in which no functional
of a functional
follows immediately
2 and Kleene
interpretation
definable by
(4.2) of
VyE3(Y ) .
satisfying the functional
inter-
from the axiom of choice.
1959, p. 32 we obtain :
There are models of
satisfies
ZFC
the functional
the functional
ZFC
in which no Kleene general recursive
interpretation
(4.2) of
VyE3(Y) o
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to
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D. Gabbay 1969 Applications of trees to intermediate logics, I. ONR Technical Report No. 31. 1969A Mentague type semantics for nonclassical logics, I. Airforce Office Scientific Research. Scientific Report No. 4. I.L. Gal, J°B. Rosser, D. Scott 1958 Generalization of a lemma of G. Rose° J.SoL. 23, pp. 137- 138o R.0. Gandy 1962 Effective operations and recursive functionals (abstract). J.S.L. 27, pp. 378 - 379. 1967 Computable functionals of finite type, I, in : J.N. Crossley (editor), Sets, ~odels and Recursion Theory. Amsterdam (North-Holland), pp. 202- 242. G. Gentzen 1933 Ueber das Verhaltnis zwischen intuitionistischer und klassischer Arithmetik. Galley proof, ~athematische Annalen (received 15th Narch 1933). First published in English translation in : The collected papers of Gerhard Gentzen, M.E. Szabo (editor), PP. 5 3 - 67. Amsterdam (North-Holland). 1935 Untersuchungen uber das logische Schliesseo Nathematische Zeitschrift 39 (1935), PP. 1 7 6 - 2 1 0 , 4 0 5 - 4 3 1 . English translation in : The collected papers of Gerhard Gentzen, pp. 6 8 - 131 (cf. under Gentzen 1933). 1938 Neue Fassung des ~iderspruchsfreiheitsbeweises f~r die reine Zahlentheorie. Forschungen zur Logik und zur Grundlegung der exakten ~issenschaften, New series No. 4- Leipzig (Hirzel), pp. 19 - 44. English translation in : The collected papers of Gerhard Gentzen, pp. 252- 286 (cf° under Gentzen 1933). Ho Gerber 1970 Brouwer's bar theorem and a system of ordinal notations. IPT, pp. 327- 338. J.Y. Girard 1971 Une extension de l'interpr@tation de Godel ~ l'analyse, et son application g l'@limination des coupures dans l'analyse et la th@orie des types. 0slo Proc.~ pp. 6 3 - 92 . 1972 Interpr@tation fonctionnelle et @limination des coupures de l'arithm@tique d'ordre sup@rieur: Th~se de doctorat d'4tat, Universit@ Paris VII. A Quelques r@sultats sur les interpr@tations fonotionnelles, in : Cambro Proc.
465
K. G~del 1933 Zur intuitionistisohen Arithmetik und Zahlentheorie, in : Ergebnisse eines mathematischen Kolloquiums, Heft 4 (for 1951 - 1932, appeared in 1933), Pp. 34- 38. Translated into English in : The undecidable, M. Davis (editor), PP. 7 5 - 81, under the title : "On intuitionistic arithmetic and number theory"° For corrections of the translation see review in J.S.L. 31 (1966), pp. 4 8 4 - 494. 1958 Ueber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12 (1958), pp. 280- 287. N. Goodman 1968 Intuitionistic arithmetic as a theory of constructions. Thesis, Stanford University. 1970 A theory of constructions equivalent to arithmetic, in : IPT, pp. 101 - 120. 1972 A simplification of combinatory logic. J.SoL. 37, PP. 225 - 246. A The arithmetic theory of constructions. TO appear° B The faithfulness of the interpretation of arithmetic in the theory of constructions. To appear. C The inductive theory of constructions° To appear° D The first-order content of the inductive theory of constructions° To appear. E The theory of the Godel functionalso To appear° N° Goodman, J° ~yhill 1972 The formalization of Bishop's constructive mathematics, in : F.Wo Lawvere (editor), Toposes, Algebraic Geometry and Logic. Lecture Notes in Nathematics, Vol. 274. Berlin - Heidelberg- New York (Springer-Verlag), pp. 8 3 - 96° So Gorneman 1971 A logic stronger than intuitionismo J.S°L° 36, pp° 2 4 9 - 261. A. Grzecorczyk 1964 Recursive objects in all finite types. Fundamenta Nathematicae 54, PP. 73 - 93° Yo Hanatani 1966 D@monstration de l ' w - n o n - c o n t r a d i c t i o n de l'arithm@tique~ Annals of the Japan Association for Philosophy of Science 3, ppo 105 - 114. R. Harrop 1956 On disjunctions and existential statements in intuitionistic of lo@iCo Nath. Annalen 132, pp. 347 - 361o 1960 Concerning formulas of the types A ~ B v C , A ~ (Ex)Bx in intuitionistic formal systems. J.S.L. 25, pp° 2 7 - 72.
systems
466
A. Heyting 1931 Die intuitionistische Grundlegung der Mathematik° Erkenntnis 2, pp. 106- 115o 1934 Mathematisehe Grundlagenforschung, Intuitionismus, Beweistheorie° Ergebnisse der Mathematik und ihrer Grenzgebiete, Berlin. Expanded translation in French as: Les fendements des math@matiqueso Intuitionnisme~ Th@orie de la d@monstration. Paris- Lcuvain (Gauthier-Villars, Nauwelaers), 1955. 1956 Intuitionism, an introduction° Amsterdam (North-Holland Publo Co°). Second, revised edition 1966o Third, revised edition 1972 . 1956A La conception intuitionniste de la logiqueo Les @tudes philosophiques 11, pp. 226- 253o S. Hinata 1967 Calculability of primitive recursive functionals of finite type. Science reports of the Tokyo Kyoiku Daigaku, A, 9~ PP. 218 235. S. Hinata and S. Tugu@ 1969 A note on continuous functionalso Annals of the Japan Association for Philosophy of Science 3, pp. 138 145. -
J.R. Hindley~ B. Leroher and J°P. Seldin 1972 Introduction to combinatory logic. Cambridge (Cambridge University Press)° W.A° Howard 1963 (i) The axiom of choice (Z~ -AC01 ) , bar induction and bar reoursion (Section II, with four appendices) ; (ii) Transfinite induction and transfinite recursion (Section IV, with one appendix)° Stanford report on the foundations of analysis. Mimeographed. Stanford University. 1968 Functional interpretation of bar induction by bar recursion. Compositio Mathematica 20, pp. 107 - 124o 1970 Assignment of ordinals to terms for primitive recursive funetionals of finite type, in : IPT, pp. 4 4 3 - 4 5 8 . 1970A Assignment of ordinals to terms for type 0 bar recursive functionals (abstract). J°SoL. 35, P. 554° 1972 A system of abstract constructive ordinals° J.S.L. 37, PP. 355- 374. A The formulae-as-types notion of construction° 1969, unpublished manuscript. B Hereditarily majorizable functionals of finite type. Appendix, this volume° W.A° Howard and C. Kreisel 1966 Transfinite induction and bar induction of types zero and one, and the r61e of continuity in intuitionistic analysis° J.S.L° 31, pp. 325- 358° R. Hull 1969 Counterexamples in intuitionistic analysis using Kripke's schema° Zeitschrift fir mathematische Logik und Grundlagen der Mathematik 15, pp. 241- 246. D. Isles 1970 Regular ordinals and normal forms. IPT, pp. 339- 361.
467
So Jaskowski 1936 Reoherches sur le syst~me de la iogique intuitionnisteo Acres du congr~s international de philosophie scientifique, VI° Philosophie des math@matiqueso Aetualit@s soientifiques et industriel!es, 393° Paris (Hermann & Cie), ppo 5 8 - 6 1 . H.R. Jervell 1971 A normal form in first order arithmetic, in : Oslo Proc., pp. 9 3 - IO8° D.H.J. de Jongh 1968 Investigations on the intuitionistic propositional calculus: Thesis, University of Wisconsin° 1970 The maximality of the intuitionistic predicate calculus with respect to Heyting's arithmetic (abstract). J.S.L. 35, P. 606° 197OA A characterization of the intuitionistic propositional calculus° IPT, pp. 211 - 217 . 1971 Disjunction and existence under implication in intuitionistic arithmetic (abstract)° J.S.L. 36, p. 588. A The maximality of the intuitionistic predicate calculus with respect to Hsyting's arithmetic. To appear in Compositio Nathematicao B Formulas in one propositional variable in intuitionistic arithmetic° Report (1973) Zathematical Institute, University of Amsterdam. D.HoJ. de Jongh and A.So Troelstra 1966 On the connection of partially ordered sets with some pseudo-Boolean algebras° Indagationes Mathematicae 28, pp. 317- 329° S.C. Kleene 1944 On the forms of the predicates in the theory of constructive ordinals° American Journal of }~thematics 66, pp. 4 1 - 58° 1945 On the interpretation of intuitionistic number theory° J.S.L. 10, pp. 109- 124. 1952 Introduction $o metamathematicso Amsterdam (North-Holland Publ. Co.) ~ Groningen (Po Noordhoff)~ and New York, Toronto (D° van Nostrand Cemp°). 1955 On the forms of predicates in the theory of constructive ordinals (second paper). American Journal of Mathematics 77, PP. 405 - 428. 1957 Realizability, in : Summaries of talks presented at the Summer Institute of Symbolic Logic in 1957 at Cornell University, pp. 100- 104o ~eprinted in : A. Heyting (editor) Constructivity in Mathematics. Amsterdam (North-Holland Publ. Co.) 1959, pp. 285- 289. 1959 Recurs£ve functionals and quantifiers of finite types I. Trans. Amero Math. Soc° 91, pp. I - 52. 1959A Countable functionals, in : A. Heyting (editor) Constructivity in Mathematics, pp. 81 - 100o 1960 Realizability and Shanin's algorithm for the constructive deciphering of mathematical sentences° Logique et Analyse 3, PP. 154- 155o 1962 Disjunction and existence under implication in elementary intuitionistic formalisms. J°S°L° 27, pp. 11- 18. 1963 An addendum° J°S°L. 28, pp. 154- 156.
46s (S.C. Kleene continued) 1963A Recursive functionals and quantifiers of finite types II. Trans. Amer. ~ t h ° Soco 108, pp. 106- 142. 1965 Classical extensions of intuitionistic mathematics, in : Y. Bar-Hillel (editor), Proceedings of the 1964 International Congress Amsterdam (North-Holland Publ. Co.), pp. 31- 44~ 1965A Logical calculus and realizability. Acta Philosophia Fennica 18, pp. 71 - 80° 1968 Constructive functions, in : "The foundations of intuitionistic mathematics" in: LNPS III, pp. 137- 144. 1969 Formalized recursive functionals and formalized realizability. Memoirs of the American Mathematical Society, Nr 89~ S.C. Kleene and RoE. Vesley 1965 The foundations of intuitionistic mathematics, especially in relation to recursive functions° Amsterdam (North-Holland Publ. CO.)o D.Lo Kreider and H. Rogers jr 1961 Constructive versions of ordinal number classes. TranSo Amer. Math. Soc. 100, pp. 325 - 369. Go Kreisel 1951 On the interpretation of non-finitist proofs, Part I. J.S.L. 16, pp. 241 - 267° 1958 Mathematical significance of consistency proofs° JoS.L. 23, pp. 155 - 182. 1958A The non-derivability of ~(x)A(x) ~ (Ex) ~ A ( x ) , A primitive recursive, in intuitionistic formal systems (abstract). JoSoL. 23, pp. 4 5 6 - 457. 1958B Constructive mathematics. Notes of a course given at Stanford University, 1958- 1959. (Mimeographed°) 1959 Interpretation of analysis by means of constructive functionals of finite type, in: Ao Heyting (editor), Constructivity in mathematics. Amsterdam (North-Holland Publ. Co.), pp. 101- 128. 1959A Reflection principle for subsystems of Heyting's (first order) arithmetic (H) (abstract). J.S.L° 24, p° 322o 1959B Inessential extensions of Heyting's arithmetic by means of functionals of finite type (abstract). J.SoLo 24, p. 284. 1959C Proof by transfinite induction and definition by transfinite induction in quantifier-free systems (abstract). J°SoL° 24, ppo 322- 323. 1959D Inessential extensions of intuitionistic analysis by functienals of finite type (abstract). J.SoL. 24, pp. 284- 285. 1962 On weak completeness of intuitionistic predicate logic. JoS.L. 27, pp. 139- 158. 1962A Proof theoretic results on intuitienistic higher order arithmetic (abstract). J.S.L° 27, po 380. 1962B Consequences of Brouwer's bar theorem (abstract). J.S.Lo 27, pp. 3 8 0 - 381. 19628 Proof theoretic results on intuitionistic first order arithmetic (HA) (abstract), J.SoL° 27, pp~ 379- 380. 1962D Foundations of intuitionistic iogie~ in : E. Nagel, Po Suppes, A. Tarski (editors), L ~ S , Stanford ~tanford University Press), pp. 198- 210.
469
(G. Kreisel continued) 1963 The subformula property and reflection principles (abstract). J.S.L. 28, pp. 305 - 306. 1963A Reflection principle for Heyting's arithmetic (abstract). J.S.L. 28, pp. 306- 307. 1963B Reflection principles and w ~ consistency (abstract). J.SoL. 28, pp. 307- 308. 1963C Generalized inductive definitions. Section III in the Stanford Report on the foundations of analysis° Mimeographed. Stanford University° 1964 Review. Zentrablatt fur Mathematik 106, pp. 237- 238. 1965 Mathematical logic, in : T.Lo Saaty (editor), Lectures on modern mathematics (Vol. III). New York (Wiley and Sons), pp. 95 - 105. 1968 Functions, ordinals, species, in: LI~S IIIo 1968A A survey of proof theory. J°SoL° 33, PP. 321- 388. 1968B Lawless sequences of natural numbers. Compositie Mathematica 20, pp. 222- 248. 1969 Course notes on functional interpretations. Autumn quarter. Mimeographed. Stanford University° 1970 Church's thesis: a kind of reducibilit~ axiom for constructive mathematics. Oslo Proc., pp. 121- 150. Review : Zentralblatt f~r Mathematik 199 (1971), pp. 300 - 301o 1971 A survey of proof theory II. 0slo Proc°, pp. 109- 170. 1971A Classes of functions of finite type ; uniformity properties. ~imeographed handwritten course notes. Winter 1971 - 1972. Stanford University. 1972 Which number theoretic problema can be solved in recursive progressions on H - paths through O ? J.S.L. 37, pPo 311- 334° G° Kreisel, D. Lacombe, J.R. Shoenfield 1959 Partial recursive functionals and effective operations, in : A. Heyting (editor), Constructity in mathematics° Amsterdam (North-Holland Publo Co.), pp. 290- 297~ G. Kreisel and Ao Levy 1968 Reflection principles and their use for establishing the complexity of axiomatic systems° Zeitschrift f~r N&th. Logik 14, pp. 97- 142o Go Kreisel and H. Putnam 1957 Eine Unableitbarkeitsbeweismethode f~r den intuitionistischen Aussagenkalk~lo Archiv fur mathema~ische Logik und Grundlagenforschung 3, PP. 35 - 47G. Kreisel and A.So Troelstra 1970 Formal systems for some branches of intuitionistic analysis. Annals of mathematical Logic I, pp. 229- 387. S° Kripke 1963 "Flexible" predicates of formal number theory. Prec. Amer° Math° Soc. 13, pp. 6 4 7 - 650. 1965 Semantical analysis of intuitionistic logic, I, in : J.N. Crossley, N.A.E. Dummett (editors), Formal systems and recursive functions. Amsterdam (North-Bolland Publ. Co.), pp. 92- 130.
470
S. Kuroda 1951 Intuitionistische Untersueh~ngen der formalistischen Logik. Nagoya ~ t h . Journal 2, pp. 35 - 47. He Lauchli 1970 An abstract notion of realizability for which intuitionistic p r e d i c a ~ calculus is complete, in : IPT, pp. 2 2 7 - 234. H. Levitz 1970 On the relationship between Takeuti's ordinal diagrams Schutte's system of ordinal notations Z(n) , in : IPT, pp. 3 7 7 - 4 0 5 .
O(n)
and
E.G.K° Lopez-Escobar 1968 An w - r u l e in intuitionistic number theory. TR 6 8 - 63, Technical report, Department of Mathematics, University of Maryland. H. Luckhardt Extensionale Funktionalinterpretation der klassischen Analysis. 1970 Ein Widerspruohfreiheitsnachweis. Habilitationsschrift, Marburg/Lahno English translation in Luckhardt 1973. Anhang : Ueber das bar-rekursive Modell der klassischen Analysis 1971 und die allgemeine Barinduktion uber Spezieso Manuscript, English translation in Luckhardt 1973. Extensional Godel functional interpretation. A consistency proof 1973 of classical analysis. Springer Lecture Notes Vol° 306. Berlin, Heidelberg, New York (Spri~er)o P. Martin-Lof 1971 Hauptsatz for the intuitionistie theory of iterated inductive definit$ons. Os~o Prec., pp. 179- 216. 1971A Hauptsatz for the intuitionistie theory of species. 0slo Prec., pp. 217- 233° 1971B On the strength of intuitionistie reasoning. Report 1971, No. 5 of the Mathematical Institute, University of Stockholm. To appear in L ~ S IVo 1971C A theory of types° Report 1971 , No. 3 of the Mathematical Institute, University of Stockholm. 1972 Infinite terms and a system of natural deduction. Compositio Mathematica 24, pp. 93- 103. 1972A About models for intuitionistic type theories and the notion of definitional equality. Report No. 4, ~971~of the Mathematical Institute, University of Stockholm. Yu.V. Matijasevich 1970 Diofantovost pereohislimikh mrozhestvo Doklady ANSSSR 191 , ppo 279- 282; improved English translation in: Soviet Mathematics Doklady 11, pp. }54- 357. 1971 Diophantine representation of reeursively enumerable predicates. Oslo Prec., pp. 171 - 177. Go Mints 1969 Imbedding operations associated with Kripke's "semantics", in : AoO. Slisenko (editor), Studies in constructive mathematics and mathematical logic, Part Io New York (Consultants Bureau).
471
JoR. Mesehovakis 1967 Disjunction and existence in formalized intuitionistic analysis, in : J.N. Crossley (editor), Sets, models and reoursion theory~ Amsterdam (North-Holland Publ° Co.)o 1969 A survey of intuitionistie logic. Lecture Notes, Spring 1969, University of Bristol. 1971 Can there be no non-recursive functions ? JoSoLo 36, pp. 309- 315o A A topological interpretation of second-order intuitionistic arithmetic° To appear in Compositio Mathematicao A. Nostowski 1961 A generalization of the incompleteness theorem. Fundamenta Nathematicae 49, pp° 2 0 5 - 232. 1966 Thirty years of foundational studies° New York (Barnes and Noble). J. Myhill 1967 Notes towards an axiomatization of intuitionistic analysis° Logique et Analyse 35, pp. 280- 279° 1968 Formal systems of intuitionistic analysis I, in : LNPS Ill, pp. 161 - 178. 1970 Formal systems of intuitionistic analysts II : the theory of species, in: IPT, pp. 151- 162. 1972 An absolutely independent set of Z~ sentences. Zeitschr. fir Math. Logik und Crundlo der Mathematik 18, pp. 107- 109o D. Nelson 1947 Recursive functions and intuitionistic number theory° Trans. Amer. Math. Soco 61, pp. 507- 568. I. Nishimura 1960 On formulas in one variable in intuitionistic propositional calculus. J°S.Lo 25, pp. 527- 331. H. 0sswald 1972 Vollstandigkeit und Schnittelimination in der intuitionistischen Typenlogik. Nanuscripta Mathematica 6, pp. 17- 31o (Preprint under the title : Ein Berechnungsverfahren fur die Zulassigkeit der Schnittregel im Kalkll yon Schltte fir die intuitionistische Typenlogiko) 1973 Ein syntaktischer Beweis fir die Zulassigkeit der Schnittregel fir die intuitionistische Typenlogik. Manuscripta Mathematica 8, pp. 243- 249. Ch° Parsons 1970 On a number theoretic choice schema and its relation to induction° IPT, pp. 4 5 9 - 473. 1972 On n - quantifier induction. JoSoL. 57, PP. 4 6 6 - 4 8 2 . W. Pohlers 1973 Ein starker Normalisationssatz fur die intuitisnistische Typentheorie. Nanuscripta Mathematica 8, pp. 371 - 387. D. Prawitz 1965 Natural deduction, a proof-theoretical study. Stockholm, Ooteborg, Uppsala (Almqvist & ~iksell)o 1970 Some results for intuitionistic logic with second order quantification rules. IPT, pp. 259- 269.
472
(D. Prawitz continued) 1971 Ideas and results in proof theory° 0slo Proc., pp. 235 - 307. A Towards a foundation of a general proof theory° L ~ S IVo W. Richter 1965 Extensions of the constructive ordinals° JoS.L. 30, pp. 193- 211o 1968 Constructively accessible ordinal numbers° JoS.L. 33, pP. 4 3 - 5 5 . ToT. Robinson 1965 Interpretations of Kleene's metamathematical predicate intuitionistic arithmetic. J.S.L. 30, pp. 140- 154.
FIA
in
H. Rogers jr 1958 Godel numberings of partial recursive functions. J.SoL. 23, pp. 331 - 341. 1967 Theory of recursive functions and effective computability° New York etc. (McGraw-Hill). C.Fo Rose 1953 Propositional calculus and realizability. Trans. Amero Math. Soc. 75, ppo I - 19o J.B. Rosser 1935 A mathematical logic without variables. Annals of Mathematics (2) 36, pp. 127- 150, Duke N~thematical Journal I, pp. 328- 355. 1969 Simplified independence proofs. New York (Academic Press). L.Eo Sanchis 1967 Functionals defined by recursion. Notre Dame Journal of Formal Logic 8, ppo 161- 174. B. Scarpellini 1969 Some applications of Gentzen's second consistency proof. Nath. Annalen 181, pp. 325- 344° 1970 On cut elimination in intuitionistic systems of analysis. IPT, ppo 271 - 285. 1970A A model of intuitionistic analysis. Ccmmentarii Mathematici Helvetici 45, ppo 4 4 0 - 471o 1971 Proof theory and intuitionistic systems° Lecture Notes in Mathematics, Vol. 212. Berlin, Heidelberg, New York (Springer-Verlag). 1971A A model for bar recursion of higher types. Compositio Mathematica 23, pp. 123- 153. 1972 Znduction and transfinite induction in intuitionistic systems. Annals of Math. Logic @ , pp. 173- 227. 1972A A formally constructive model for bar recurslon of higher types° Zeitschr. f~r Math. Logik und Grundl. der )~thematik 18, pp. 321 - 383. 1973 On bar induction of higher types for decidable predicates° Annals of Math. Logic 5, pP. 7 7 - 1 6 3 o A Disjunctive properties of intuitionistic systems° To appear.
473
D.So Scott 1957 Completeness proofs for the intuitionistic sentential calculus, in : Summaries of talks presented at the Summer Institute of Symbolic Logic in 1957 at Cornell University° 1968 Extending the topological interpretation to intuitionistic analysis° Compositie Mathematica 20, pp. 194- 210. 1970 Extending the topological interpretation to intuitionistic analysis, II, in : IPT, pp. 2 3 5 - 255. Ho Schwichtenberg A On impredicative definitions of primitive recursive functionals. ~reprint 1972.) J.Ro Shoenfield 1967 Mathematical logic. Reading, Menlo Park, London, Don Mills (Ont.) (Addison-Wesley). C. Smo~ynski A Some remarks on measures of complexity and unboundedness theorems (unpublished). B Peano's arithmetic is an essentially unbounded extension of Heyting's arithmetic (unpublished). C PA/HA is essentially unbounded (unpublished). C. Spector 1961 Inductively defined sets of natural numbers, in : Infinitistic methods. Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 2 - 9 September 1959o Oxford, London, New York, Paris (Pergamon Press), Warszawa (Pa~stwowe Wydawnictwo Naukowe)~ pp° 97 - 102. 1962 Provably recursive functionals of analysis : a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, in: J.CoEo Dekker (editor), Proceedings of Symposia in Pure Mathematics V. Providence (R.I.) (American Nathematical Society), pp° I - 27 . S. Stenlund 1971 Introduction to combinatory logic. Filosofiska Studier Nr 11. Uppsala (Philosophical Soco and Dept. of Philosophy, University of Uppsala) o 1972 Combinators, X - terms, and proof theory. Dordrecht (D° Reidel). W.W° Tait 1959 A characterization of ordinal recursive functions (abstract). J°SoL. 24, p. 325. 1963 A second order theory of functionals of higher type, with two appendices. Appendix A : Intentional functionalso Appendix B : An interpretation of functionals by convertible terms (reworked in Tait 1967)o Appeared as: Section V, with two appendices, in : Stanford report on the foundations of analysis. Mimeographed° Stanford University. 1965 Infinitely long terms of transfinite type, in : J° Crossley and M°A°E. Dummett (editors), Formal systems and recursive functions. Amsterdam (North-Holland Publ. Co.), pp. 176- 185o 1965A Functionals defined by transfinite recursion. J.S.Lo 30, pp° 155- 174. 1967 Intensional interpretations of functionals of finite type I. J.S°L. 32, pp. 198- 212. 1968 Constructive reasoning, in : LNPS III, pp. 185- 199.
474
(W.W° Tait, continued) 1971 Normal form theorem for barrecursive Oslo Proc., pp. 3 5 3 - 367.
functions
of finite
type,
in :
G. Takeuti 1965 A formalization of the theory of ordinal numbers. JoS.L. 30, pp. 2 9 5 - 3 1 7 . 1967 Consistency proofs of subsystems of classical analysis. Annals of Mathematics 86, pp. 2 9 9 - 348. R.H. Thomason 1968 On the strong semantical calculus. J°SoL. 33, pp. 1 - 7.
completeness
of the intuitionistic
R.R. Tompkins 1968 On Kleene's recursive realizability as an interpretation intuitionistic elementary number theory° Notre Dame Journal of Formal Logic 9, PP. 2 8 9 - 293.
predicate
for
AoS. Troelstra 1968 The theory of choice sequences, in : L ~ S III, pp. 201 - 223° Some errata : page 221, line 12, replace " A ~ " by " A~E a " ; lines -6, -9, -10 replace " e n ~I'' by " f n ~ 1 ". 1969 Principles of intuitionismo Lecture Notes in Mathematics, Vol. 95. Berlin- Heidelberg-New York (Springer-Verlag)° Some corrections : page 41, line
13, replace
line
15 must read:
= en~1
line
16, replace
it stands. 1970,
K
defined
1971
with
Let Then
Yo > Xo
(c) If
Yo = Xo'°°''Yi Yi+1 - X i + l '
eog.
~ u v = w.
F[u),
F[u,v) ;
~u ~u
by
> ;
~ Yn) ; (10) is not correct
developed
and should read :
u = <Xo,...,Xn>,
~u
Yi+1
> xi+1
F[u,v)
we take
; ~u v =
(d) In all other cases we take
is an order-isomorphism F[u),
is a mapping
~uV=W ;
~u v = < Y o - X c , y1,...,ym>
= xi'
Page 79, the
v = ,
we t a k e
v~u
as
in Kreisel-Troelstra
schemata with parameters.
Yi+2'''''Ym > ;
maps
<
=
A en/0
Page 51 : formula
(a) If
we take
(b) If
<0,...,0,
N.
the techniques
14.2.4 is misstated
as follows.
w = (ixZ1)(n+l) o
170
by
See instead
§ 5 for dealing
definition
1969A
An(lth(n) + k
when restricted
onto elements
of
to species
WO o
Notes on the intuitionistic theory of sequences, I. Indagationes ~athematicae 31, pp. 430 - 440. Notes on the intuitionistic theory of sequences, III. Indagationes Nathematicae 32, pp. 2 4 5 - 252. Notions of realizability for intuitionistic arithmetic and intuitionistic arithmetic in all finite types, in: Oslo Proc., pp. 3 6 9 - 405° Some errata are listed in Troelstra 1972A° Some further errata : p. 372 , line -5, read: "~-~" ; page 381, line -8, add formula number (1) at the end of the line ; page 383, line 11, replace "3.16" by "3.15" ; page 384, line 8, replace " V" by " & " ; page 596, line 6, replace " IP " by " I P R " ; page 397, line -4, add " ) " at end ! page 398, line 4, replace " Vx " by " Vx E V ¢ " ;
475
(A.S. Troelstra, continued) page 398, line 6, replace " H R E " by " H E 0 " ; line -3, read "8.2" for "7.2"° 1971A Computability of terms and notions of realizability for intuitionistic analysis. Report 71 - 02 of the Department ef Y~thematics, University of Amsterdam (mimeographed). 1972 Review of Yasugi 1963. J°S.L. 37, P. 404. 1972A Self review of Troelstra 1971. Zentralblatt fur ~ t h e m a t i k 227, Nr 02015. A Notes on intuitionistic second order arithmetic. To appear in : Cambr. Proc. Correction : in the proof of 2.7, under "Extensions", the analysis should not be applied to a path, but to a spine as defined in § 4.2 of this volume. (The reason is that our permutative reductions did not reckon with permutation with UP, I P - r u l e s . ) The change does not affect the applications. A preprint of this paper was issued as Report 71 - 05 of the Department of Mathematics, University of Amsterdam (1971) . R.E. Vesley 1970 A palatable substitute for Kripke's schema, IPT, pp. 197 - 207. 1972 Choice sequences and ~[arkov's principle° Compositio Mathematica 24, pp. 33- 53.
in:
M° Yasugi 1963 Intuitionistic analysis and G~del's interpretation. Journal of the Math. Society of Japan 15, pp. 101 - 112. Review with corrections in : J.S.L. 37 (1972), P. 104. J.l. Zucker 1971 Proof theoretic studies of systems of iterated inductive and subsystems of analysis. Thesis, Stanford University (mimeographed).
definitions
INDEX
Notions and notations are listed only when they have more than local significance in %he text.
The references given refer to definitions or
special conventions regarding the notion or notations listed. I.
List of symbols
A)
Formal s~s~ems~ arranged Erimarily in a lphabetio order
For some general conventions see 1.1.2 (viii). based on intuitionistic logic,
Hc
For any formal system
indicates the corresponding classical
system,
~
1.9.1o
I~(A)
E-H~ w
1.6.12
I])B
E-~
1.6.12
IDB I
KE-HA m
1.8.4
6.2.2 1.9,18
1.9.18
IDI(Q )
Pj
5.3.1
qf-~
6.8.1
1,5.9
qf-~_m ®
1.6.15
qf-l-~
~
~-zD~ ~ Io9.2.5
12~(Q) 6.8.1
qf-N
~-~1
6.8.1
z~(Q)
6.8.1
qf-~-~®
~®
E~-T 2
6.7.1
ID2(Q )
6.2.2, 6.8.1
qf - ~E - ~ m
1.6.13
1.6.13 1.8.2
~-Z1F
6.8.1
~D~(Q) 6.8.1
~1 6.3.6 (b)
E-T2P
6.7.1
I~(Q)
E2
E - T2EP ]
6.7.1
I-~
6.8.1
m 1.6.11
H~ 1.3.2-6, 1.3.8
X I - ~ . ~ 2.4.18
EAw
1.6.15
In..t-H~ w
HAS 1.9..5, 1.9.9 HAS o 1.9.3-4
I n ~ - ID~ ~
ICF
2.6.26
ICF-
2.6.26
I~D2(A )
6.2.2
1o9.25
1.9.25
HR0 2.4.10 HR0- 2.4.10 N- HA~ 1.6.3-7
§ 6.5
T2EP ]
6.7.1
~2 P _ 6.7.t ~-t~ m 1.6.12
WE- ZDBw 1.9.25 Z2 6.9.1 Z2
6.9.1,
N - HAm 1.8.2
B)
Schemata and rules~ arranged primarily in alphabetic order
AC
3.4.8
ACI
3.6.10
AC00
1.9.18, 3.4.7
AC01
1.9.18, 3-4.7
AC~, T
3.4.7
ACa, ~
3.6.10
1.6.13
BI M
1.9.20
ACA
1.9.4
BIQF
ACR
3.7.1
BR
1.9.20
BI
3.5.19
BR
1.9.26
BI D
1.9.20
CA
1.9.4
1.9.26
477
C-N
1.9.19
IP
~- CON
1.9.10
CON(A)
5.2.17
w- CON(A) CR
5.2.17
3.7.1
CR 0 CT CT 0
1.11.7,
3.7.1
1.11.7
P2.1- 3
3.4o18, 3.6.18
6.7.4
P1 .I' - 3'
6o8.1 6.8.1
IP w
3.4.7
P2.1' - 3'
IP o
1.11.6
PCA
IP~o
3.6.18
PL I- 9
IP~
3.5.10
PL 10- 13
IPoC
3.1.11
QI - 4
1.9.4 1.1.3 1.1.4
1.1.3
D I. 11.3 DC I 1.9.22
IPpR 1.11.6 c 1.11.6 IPpR IPR 3.1.15, 3.7.1
QI.1 , QI.2 Q2.1, Q2.2
Ql.la, Ql.lb, QI.2
6.8.1
DNS
IPR'
3.7.1
Q2.1a, Q2.1b, Q2o2
6.8.1
IPR C
3.1. 7
QF-AC
3.7.1
QF - AC00
DP
1.11. 7
1.11.6
IP I
1.11.4 1.11.2,
3.7.1
EBI D
1.9.21
IPR w
ECR 0
3.7.1
IPR 'w
3.2.14
KI - 3
ECT 0 ED ED' ES EXT
1.11.2, 3.7.1 1.11.2, 3.7.1
2.7.2, 6.7. I 1.6.12
E X T - R' FI FR
1.6.12
6.3.5 6.3.5
RF(A)
1.9.2, 5.2.17
RFN(A)
1.9.2, 5.2.17
RFN'(A)
5.2.17
I~C
2.6.3
T .< 3.4.22
NP
5..4.3 (end)
TI(<)
MR w
1.9.2
1.11.5 1o11.5
TI 1 TZ 2
5.8.1
TR I
6.3.5
TR 2
6.3.5
1.11.5 3.8.1
UP
6.3.5 6.3.5
3.2.31
CCR
3.7.9
NRpR 1.11.5
WC-N
IE 0
2.3.1
ES
3.9.11
WCR
1.11.7
IE 1
2.3.6
NUC
2.6.4
WCT
3.4.15.
I
1.3.3,
For
&l,
1.3.6
vlr~
see 1,1.7 ; for
P1.1 - 3
Vll, V21 ,
-~I, V2E,
VI, XI,
1.9.19
6.7.4
ZI, /%'1' &Er' XE
see 4.5.2.
&El'
3.6.10
1.9.10
Rule- BR¢ 3.5.19 $I- 9 2.8.2
N~
3.3.9
1.9.10, 3.6.10 T '''''~,
1.9'.18
REC
3.9.11
~MpR
2.6.26
3.6.10
3.6.18 3.5.10
}£PR M~
2.4.10
G*I - 5
RDC I
1.11.5
]~1 Nw
1.9.24
GI-5 cc
M
1.9.18 3.9.9
KLSR n
1.9.5
QF-AC
2.6.15
KLS n
1.1.3
EXT¢,T EXT-R FAN
KLS
3.7.1
6.2.2 6.2.2
-~E,
VE,
ZE
478
C)
Syntact,,ical v a r i a b l e s
There are
are
made
many
local
by a d d i n g
( i n o r d e r of app,e a r a n c e )
deviations
sub-
in the use
or s u p e r - s c r i p t s
of v a r i a b l e s .
reserved
Often
for v a r i a b l e s
new variables of a c e r t a i n
category. x
y,
z, u,
a
b,
c
v, w
A
B, c, ...
t
s
1.1.2
1.1.2
1.1.2
(it);
(it) ;
cf.
m
x
y
, z , u , v
t, T
cf.
1.6.5,
1.3.9 D;
6.3.3
of. 5.2.3
, w
1.6.3
1.6.5
~
1.6.5
1.6.5
=~, =~, ~
1.6.5, 6.5.5
X n, yn, Z n 1.9. 3 1 1 1 1 1 1 x , y , z , u , v , w ~, y,
e, f,
...
e',
H,
n',
Z~
Z',
a, b, ~,
;
1.6.5
s~ , t ~
~,
6.3.3
5.2.3
~, ~, ~, v, x__ ~ , z , ~ , y , ~ s
6.2.1,
1.1.2 (it) (iii) , 6.2.1.
~, 2, ~, ~, ~, ~, ~ n
cf.
6.2.1
e",
cf.
e 1,
f', f", ...
H",
H 1,
Z",
...,
...~,
...
~2
6.3.3 1.9.25 ;
cf.
6.2.1
4.1.2
ZO,
e, k, m~ n
~, ~ 1
2,
1.9.10
1.9.10;
ZI,
o..
4.1.2
6.2.1
6.3.3 6.3.3
f(o)1 f(1)2 6.3.3 r,
s, t, u, v,
®,
®'
~,
~
...
~',
D)
Other
The
symbols
ones
tl,
t~
6.3.3
6.4. I
are
6.8.1.
symbols are
grouped
primarily
listed
in o r d e r
of a p p e a r a n c e ,
some
very
together.
-=def
1.1.2
(i)
t[x], t[x,y] 1.1.2 ( i i i ) 1.1.2 (vi) *--> 1 . 1 . 2 (vi)
---
1.1.2
(i)
[x/t]s
&, V, 2, V, ~, ~ 1.1.2 ( i ) ~, ~, ~, ~, ~, _c 1 . 1 . 2 ( i )
1.1.2
(vii)
similar
479
~) 1.1.2 (x) ~P], ~ ) [ p ] 1.1.2 (x) ~, ~x], ~(x,Y] 6.2.1
:t 1.3.10 s~D 1.5.3, 2.3.11 T 1.6.2
~2, ~2[ P] 6.7.1
~'
~[Q], ~%], ~[~], 401] Fm(H), Fm H
1.t.2
Thm(H), Thm H
~-A, ~ A
6.2.2
~0 1.8.9
(x), o f . 5.1.6
I. 1.2 (x)
1.1.2 (x)
&l, Vlr, vll,-~I, VI, vE,-~E, VE, ~E 1.1.7
1.6.16
TK
1.9.25
TS
1.9.27
~2 6.3.1 ~I
1.1.7
(~)~ 1.6.2 =m 1.6.3
V2i, V2E 4.5.2 kl, ~E 4.5.2 A l 1.1. 7
H ,T 1.6.3 E 1.6.3 R 1.6.3 ; of. 6.3.1
0
1.3.2, 1.6.2, 1.6.3 1.3.2, 1.6.3 (cf. 4.1.1) = 1.3.2, 1.6.3 Ini 1.3.4
tlt2..ot n 1.6.5 V__xA, ~ 1.6.5 t E ~ 1.6.5 st 1.6.5, 6.5.5 = =
prd 1.3.9 A, 1.7.2 -- 1.3.9 A, 1.7.2 sg 1.3.9 A
Vxy,= = ~=_y_, % x y 1.6.5 Kxm.t 1.6.8, 1.8.4 Xx.t 1.6.8 =
Ix-yl
~o
S
1.3.9A
1.6.11
max 1.3.9 A rain 1.3.9 A
TA G
J' it' J2 1.3o9 B, 1.8.7 Vx(tl,...,tn) 1.3.9 C
D'm,T, D"m,m, D T 1.6.16, 6.7.2 D', D", D 1.6.16, 6.7.2
j~(t)
Di, D!i '
1.3.9 c
<Xo,..o,Xu> 1.3.9 c
1.3.9 C
1.6.13 1.6.16
X T
D:' for i
I=o,I,2:6.7.2
>, >, <, < 1.7.2 R 0'|i , . . . ~O'rl , R 1.7.5 P= 1.7.6 J 1.7.11 =
*
1.3. 9 C
lth(t)
1.3.9 C
_<, <, _~, >" 1.5.9 C (of. 2.2.2,2.2.29,
P 1.8.5 i pmji 4.1.4) mpj , ~pro=, mpj,
(n) i 1.3.9 C tl(n) 1.3.9 C
j , jk, 1 1.8.7 ~2' V2 1.9.3
ProofH(x,y) 1.5.9 D PrH(x) 1.3.9 D rA~ 1.3.9 D
kXl...Xn.A(x I .... ,Xn) 1.9. 9 R 1.9.10 (cf. 1.6.3, 6.3.1) ~D, ~x 1.9.11
'-A(:~I,
. .
., X n) -~ 1.3. 9 D
1~}~(~), {~}(y) t ~ t'
1.3.10
1.3.1o
~ ~ n
1.9.11
,i1~, ,i2~,, i(~,~) (~)x
1"9"11
1.9.11
48o
V ~, ~,
2.4.8, 2.9.5 (cf. 2.6.2, 2.6.22) ~' 2.4.10 I 2.4.11, 2.4.19, 2.9.5 (cf. 2.6.5, 2.6.22) HEO 2.4.11 HEO 2 2.9. 7
~ ~o' 1.9.12 ~I~
1.9.12
~(~) 1.9.12 Vu(C~1 . . . . ,~u) 1-9.12 • t(~1,...,%)1.9.13 ~(~1 . . . . , ~ ) 1.9.13 ,U
Ji ~
HE02
1.9.13
[~]H~o, [t]~o
1.9.13
k?l
A°x, Alx
X - HRO
1.9.17
A°~, AI~ 1.9.17 ~o . ~o' 1.9.21 <~o,..., ~u>l < >I
L~(~)
1.9.21
1.9.21
I 1.9.25 I, I' 1.9.27 ~I' ~2' ~3 1"9"25 ~. 1.9.25 B 1.9.26 [C] 1.9.26 ST, DT 1.9.27 0~ = 0 m 1.9.27,2.2.7(ii),2.2.16 , 6.5.6 '
6.6.4
1.10.2
% contr %'
2.2.2,2.2.28,2.3.1, 2.3.7,2.3.8,6.4.1
% <1 t' % >'1%'
t < t', t >" t'
2.2.2, 2.2.29, 6.4.1 2.2.2, 2.2.29, 6.4.1
[D], [ D ' ] , [D"] CTN 2.5. I CTNF 2.5.1
CTNF 2
mmo
2.6.3 2.6.4
muc zl 2.6.5 (of. 2.6.22, 2.4.11) ECF, ECF(U~) 2.6.5 [ t ] I C F , [t]ECF E(V) 2.6.14 Wr
2.6.22 2.6.23 2.6.26
r I c
APN
ira
]nA
tqA %(~) mrIA ~1 A
2.4.~, 2.5.5
ir
ICE*, ECF*
t~pA
[~], [~], [~], Is], [~]
2.6.11
ICFr(LL), ECFr(lL), ICF r, ECFr vr
SC, S C 2.2.13, 2.2.~0, 6o5.1 SC* 2.2.31
6.6.4
6.6.4
icm, icF(~) 2.6.2 v CI 2.6.2 (of. 2°4.8, 2.6.22)
s Ic
HRO 2
2.4.19
CT~ 2.5.1, 2.5.2 CTNF 2.5.1, 2.5.2 CTM, 2.5.2 CTNF' 2.5.2
Comp, Comps, Comp', CompS, Comp", Comps 2.2.5, 2.3.7
, Ap 2.4.2, 6.6.1 2.4.4 ~E 2.4.4 HR0 2.4 •8 HR02 2.9.7
2.4.15
2.4.18
5.1.2, 5.1.12 3.1.t3 }.I.16 3°2.2 3.2.3, 3.2.29 3.2.3
5.2.11, 5.3.8 3.3.2 3.3.2
2.6.22
481
tmrpA 3.4.3 tmrA 3.4.3, 6.7.1 tm~qA 3.4.3
DK' --(D 5.1.2 K 5.1.3 V N, CN, F~I 5.1.6 Fm M, Sn M 5.1.6
2°-Ao~ 3.4.3, 6.7.1
(~ ~i) 5.1o9
A1, AI x 3.4.3 C 3.4.22 R< 3.4.22
:~' 5.1.9
5.2.7
DA 3.4.27 AD, AD 3.5.2, 3.5.21, 6.8.3, 6.8. 9 A , A~ 3.5.17 ^
rO, l"I, F2, rn, ran, r'pr 5.6.3 tpA, tppA 3.9.2 s 4.1.1 IEA], [ A ] , H, Z 4.1.2 Z A A
[3
[,;
[~] [~] 4.1.2
etc.
4.4.1
(o)
laI,, 6.2.1 N
6.2.1
b O, bl, b2, b
6.2.2
lI~2(*)l ' 1221'
I~I(A)I
Iz~(*)l, I~1
(~:1,2)
6.2.2
, 6.2.2
(a)2 6.2.2 Tm , T m 6.3.1 O0T 01 , 0 2 6.3.1 S0, S I , S 2
4.1. 5
Rule(II) 4.1.8 (a), 4.4.1 (a) Con(~) 4.1.8 (~), 4.4.1 Pre~$ss(~) 4.1.8 (~), 4.4.1 Rapp(~) 4.1.8 (c), 4.4.1 Param(II) 4.1.8 (0), 4.4.1 Ass(~) 4.1.8 (d), 4.4.1 Term(H) 4.1.8 (e), 4.4.1 Subst(a,t,H) 4.1.8 (f), 4.4.1 Sub(A,l~,Ii,,~) 4.1.8 (g), 4.4.1 PR~(~) 4.1.8 (h), 4.4.1 Prdi(n) 4 . 1 . 8 (h), 4.4.1 sv(n) 4.1.9, 4.3.2 2 e a l ( ~ ' , ~ ) 4.4.1 (b) ~ea(~,,~)
~a, W 5.3.9 (Zi _~)* 5.6.1, 5.6.7 I~I 5.6.26 Q1.v 6.2.1
ql' Q2' Q 6.2.2
I~(¢), Z(t) 4.1.2 n oontr ~, 4.1. II >I 1I,, H <1 1I, 4.1.4 II > I~,, ~ < II, 4.1.4 ~ > I I , , I~ < I~, 4.1.4
/~p, ,~C, /~A, '~S, '~O
~- 5.2.11
6.3.1
RO,T' RI,~' R2, ~' R O' R 1, R 2 C T , CT 6.4.1
6.3.1
NF 6.4.1 CT
6.4.4
Itlo 6.4.1o
t_~ t '
(strongly)
6.5.1
6.5.5 M 6.6.1 M 6°6.1 Val~, Val =M 6.6.1
= 6.6. I
Itl~ 8.6.3 J2
6.6.4
I~l 6.6.2 I~21 6.6.2
Vl(n,m) 4.4.1 (~) sva(~, n) 4.5.~
PI' P2 6.7.1 6.7.1
II- 5.1.2
u01, u12, Uo2
6.7.2
482
do1, d12, do2 6.7.2 w' Iv" 6.7 •2 < , >, n', ~" I wO1 , Trl, '
6.7.2 6.7.2
rr2
2'
,nl
2' 11"20 6.7.2
IT II ~I
6.8.1 6.8.1
I£,~('h,'~2 .... )1
(,~= 1,2)
6.8.1
6.8.7 QI' Q2
6.2.1, 6.2.2.
II. List of notions abstraction operator 1.1.2 (ix) abstraction operator, defined - 1.6.8 absurdity 1.1.2 (i) Aczel slash 5.1.12 admissible rule I. 11° I almost negative (formula) 3.2.9 applicative set 2.1.1 arithmetical comprehension 1.9.4 assumption 1.1.7 assumption class 1.1.7 bar induction I. 9.20 bar recursion I. 9- 26 basic rule 1.3.6 basis (for au applicative set) 2.1.1 bracketing conventions 1.1.2 (v) Cartesian product type 1.6.16 Church's rule 1.11.7 Church's thesis 1.11. 7 closed assumption 1.1. 7 closed deduction 1.1.7 compact functional 2.8.6 completeness 5.1.6 composition 1.3.4 comprehension schema 1.9.4 computability 2.2.5, 2.9.2, 2.9.4 , 2.9.6, 6.4.4 computable, see computability concatenation 1.3.9 C conclusion (of a deduction) 1.1. 7 conservative extension 1.2.2 conservative over 1.2.2, 3.6.4 continuity I. 9.19 contractible subterm (in HA ) 1.5.3 contraction (of terms) 1.5.3, 2.2.2, 2.2.28, 2.3.1, 2.9.2 contraction (of a deduction) 4.1.3, 4.3.1 contraction (&r-' &l-' V-, Vr- , Vl- , ~-, vE-, ~E-) 4.1.3 contraction (~&-, JkV-, ~.-~-, ~.V-, A ~ - ) 4.1.3
contraction contraction contraction contraction
(~-) (VEs-, (IP-) (V2-,
4.1.3 ~Es-) 4oi.3 4.3.1 ~-) 4.5.2
deducibility 1.1.2 (viii) defined abstraction operator 1.6.8 defining axioms (for constants of finite type) 1.6.7 defining axioms (for primitive recursive functions) 1.3.4 definitional extension 1.2.4 de Jongh's theorem 5.3.2 dependence (of a formula occurrence on assumptions) 1.1.7 derivable from null assumptions 1.11.1 derived rule 1.11.1 diagonal sequence 5.3.8 A Dialectica interpretable (M-) 3.5.5 Dialectica interpretable (~,M-) 3.5.5 Dialectica interpretation 3.5°2, 6.8.9 Dialectica translation 3.5.2, 6.8.3 Diller- Nahm variant 3.5.17 discharged assumption 1.1.7 disjunction property 1.11.2 effective operations 2.6.14 E - IP - part 4.3.4 elimination part 4.2.4, 4.2.6, 4.3.4 elimination rule 1.1. 7 embedding (for models) 2.4.3 E - part 4.2.4, 4.2.6, 4.3.4 equational calculus 1.6.14 E - rule 1.1. 7 expansion 1.2.3 explicit definability property 1.11.2 extended bar induction 1.9.21 extended Church's thesis 3.2.14 extensionality 1.9.5, 1.6.12 extensionality axiom (for species) 2.7.2 extensionality rule 1.6.12 extensional equality 2.7.2 extensional model 2.4.1 extension theorem 5.3.6
483
fan functional 2.6.4 +2.6.6 fan theorem 1.9.24 finite tree theorem 5.3.4 floating product topology 2.7.10 force, to- (~ forces A) 5.1.2 formula occurrence 1.1.7 functional (in ~ or ~ ) 6.8.1 generalized inductive definition 1.9.2 Girard's functional s 1.9.27 godelnumbers I. 3.9 D Godel - Rosser - Mostowski - Kripke Nyhill theorem 5.3.11 godelsentence 1.3.9 D Godel' s system 1.1.4 Harrop formula 1.10.5 hereditarily continuous functionals (ECF) 2.6.5, 2.9.8 hereditarily effective operations (HEO) 2.4.11, 2.9.5, 2.9.7 hereditary extensional equality 2.7.2 hereditary extensionality, axiom of - 2.7.2 hereditarily majorizable functionals 2.8.6, Appendix hereditarily recursive operations
(HRO) 2 . 4 . 8 , 2 . 9 . 5 ,
2.9.7
Hilbert- Bernays completeness theorem 5.6.9 homomorphdsm 2.4.3 Impredicative comprehension 1.9.4 independence-of-premiss schema 1.11.6 independent sentence 1.3.9 D induction contraction 4.1.3 induction lemma 1.7.10 induction on V 6.4.4 induction on C T 6.4.4 induction on M T 6.6.3 induction on SC T 6.5.1 induction on t 6 CT 6.4.4 induction on t 6 SC v 6.5.1 induction reduction 4.1.3 induction rule 1.3.5, 1.6.15 (ii) induction schema 1.3.3 inductive definition (in ~ ) § 1.4 inductive premiss 1.3.6 I- part 4.2.4, 4.2.6, 4.3.4 immediate simplification 4.1.3 I- rule 1.1.7 intensional continuous functionals (ICF) 2.6.2, 2.9.8 intensional equality 1.6.10 introduction part 4.2.4, 4.2.6, 4.3.4 introduction rule 1.1.7 inverses (left-) 6.7.2 iterator 1.7.11
Jaskowski sequence 5.3.8 B Jongh's theorem, de - 5.3.2 Kleene's primitive recursive functionals 2.8.2 Kleene stroke 3.1.2 Kleene's system 1.1.6 Kripke model 5.1.2 language 1.1.2 (viii) leftmost minimal redex 2.2.2 length (of a sequence) 1.3. 9 C length (of a reduction tree) 2.2.17, 4.1.4 local reflection principle 1.9.2 majorizing technique 6.8.4, 6.8.6 (b), Appendix major premiss 1.1.7, 1.3.6 Marker' s rule 1.11.5 Marker' s schema 1.11.5 maximal formula (occurrence) 4.1.4 maximal segment 4 . 1 . 4 minimum part 4.2.4, 4.2.6 minor premiss 1.1.7, 1.3.6 model (for N- ~,~,W) 2.4.1 modified Jaskowski sequence 5.3.8 C modified realizability § 3.4, 6.7. I modified realizability predicate 3 . 4 . 2 , 3.4.4, 3.4.27 modulus-of-continuity functional
2.6.3 me dulus-of-uniform-con tinui ty functional 2.6.4 natural deduction system 1.1. 7 negative formula 1.10.6 node (of a Kripke model) 5.1.2 non-logical axioms 1.2.1 normal (term) 2.2.2, 2.2.29, 6.4.1 normal (deduction) 4.1.4 normal form (of a term) 2.2.2, 2.2.29,
6.4.1 normal form (of a deduction) 4.1.4 normal form theorem 4.1.5 normalizable 6.4.1 normal model 2.4.1, 5.1.22 number selection operator 6.8.3 numeral 1.3.9 D, 5.2.3 numerical type 1.8.9 open assumption origin 5.1.25
1.1. 7
pairing 1.3.9 B, 1.6.16- 17, 1.8.2, 2.4.19, 2.6.25, 6.7.2 parameter 1.1.7 partial reflection principle 1.5.6 partial truth definition 1.5.4
484
path 4.2.2 permutative contraction 4.1.3 permutative reduction 4.1.3 p - functor 1.9.12 p- term 1.3.10 pms 5.1.2 positive (in x) 6.2.2 positivity (in X) 6.2.2 predicate calculus with equality, intuitionistic - 1.2.1 predicative comprehension 1.9o4 primitive recursive functions 1.3.4 product topology 2.7. 9 product type 1.8.2, 1.6.16 proof-predicates 1.3.9 D proper contraction 4.1.3 propositional model structure 5.1.2 provability predicate 1.3.9 D pure type 1.8.5
reduction sequence from tt not affecting t 6.5.5 = reduction tree (of a term) 2.2.17 reduction tree (of a deduction) 4.1.4 redundant parameter 4.1.3 result-extracting function 1.3.9 A rossersentence 1.3°9 D
saturated (M-) 5. I. 7 segment 4.1.4 sequence coding 1.3.9 C sequent calculus 1.1.13 K - set 2. 1.4 Shoenfield' s variant 3.5.18 simultaneous reeursion 1.6.16, 1.7.5, 1.7.7 s-m-n theorem 1.3.10, 1.9o15 Spector' s system 1.1.3 spine 4.2.3, 4.3.3 qms 5.1.2 s.p.p. 1.10.5 qusautificational model structure 5.1.2 standard computability 2.2.5, 2.3.1, quantifier-free systems 1.5.8 2.3.7 standard normalizable 6.4.1 standard reduction (sequence) 2.2.2, realizable (mr-] 3.4.3, 3.4.27 realizable ~ - 5 3.4.3 2.2.29, 2.3.1, 6.4..I realizabiiity (P -) 3.2.2 std reduction sequence 6.4.1 realizability (~-) 3.9.2 strict computability 2.2.5 realizability predicate 3.3.2 strictly normal deduction 4.1.4 strictly normal form 4.1.4 realizability I ) 3.3.2 strictly positive part 1.10.5 r realizability 3.2°3 strict reduction (sequence) 2.2.2, realizability I~I-I_ 3.3.2 2.2.29, 2.3.1, 6.4.1 strong computability 2.2.13, 2.2.30, realizability ~ 3.3.2 realizable (~- ~ - , H- m~-) 3.4.10 6.5.1 realizable (M mr , N - m ~ - ) 3.4.10 strongly computable 2.2.13, 2.2.30, realizable (B-Y-, ~- ~ ) 3.2.17 6.5.1 realizable (~,M-mr-, H , ~ - m q - ) strongly computable under substitution, 3.4.10 -" 2.2.30 recursion 1.3.4 strongly normalizable 2.2.12, 6.5.1 strongly valid 4.1. 9 recursion theorem 1.3.10, 1.9.16 recursive functionals 2.8.2 strongly valid under substitution, recursively dense basis 2.6.14 4.1.15 redex 2.2.2 strong normalization 2.2.12 red. seq. 6.4.1 strong normalization theorem 4.1.5 reduce, to - to (for terms) 2.2.2, submodel 2.4.3 substitution 1.1.2 (vii) 2.2.30 6.4°I reduction (~ & -, V-, Vr- , V1- , substitution (of deductions) 4oi.2 ~-, vE-, ~ [ i reduction (~&-, ~V-, ~ - , ]kV-, terminate, to - 2.2.2, 4.1.4 term model § 2.5 A~-) 4.1.3 reduction (~-) 4.1.3 thread 4.1.4 T- predicate 1.3.9 A reduction (VEs-, ~Es-) 4.1o3 reduction (of a deduction) 4.1.3 transfinite induction 1.9.2 true (A is - at ~) 5.1.2 reduction (V2- , X-) 4.5.2 1.5.4 reduction sequence (of a term) 2.2.2, truth definition, partial 2.2.28,6.4.1 type level 2.1.1 reduction sequence (starting from) 4.1.4 type structure 1.6.2, 1.8.2 t y p e - O - v a l u e d functionals 6.8.3
!!:i °
~.1.3
485
uniform reflection principle
1.9.2
valid (in a model) 5.1.2 variables 1oio2 (ii), (iii), (iv) wf
6.6.2
zero premiss
1.3.6.
