E. Casari ( E d.)
Aspects of Mathematical Logic Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Varenna (Como), Italy, September 9-17, 1968
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy cime@math.unifi.it
ISBN 978-3-642-11078-8 e-ISBN: 978-3-642-11080-1 DOI:10.1007/978-3-642-11080-1 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Cremonese, Roma 1969 With kind permission of C.I.M.E.
Printed on acid-free paper
Springer.com
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. 3'
Ciclo - Varenna dal 9 a1 17 Settembre 1968
t
l
OF MATHEMATICAL ~ ~
LOGIC^'~
~
~
Coordinatore : Prof. E. C a s a r i
H. HERMES
:
Basic notions and applications
of the
theory of decidability. D. KUREPA
Pag. 1
:
On s e v e r a l continuum hypotheses.
'I
55
A. MOSTOWSKI
:
Models of set theory
"
65
A. ROBINSON
:
Problems and methods of model theory
A. SOCHOR - B. BALCAR :
181
The general theory of semisets. Syntaktic models of the s e t theory.
"
267
C E N T R O INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E . )
H. H E R M E S
BASIC NOTIONS AND A P P L I C A T I O N S O F T H E THEORY O F DECIDABILITY
Corso tenuto a
Varenna dal 19 a1 17 Settembre 1968
BASIC NOTIONS AND APPLICATIONS O F THE THEORY O F DECIDABILITY by
H. Hermes P r e l i m i n a r y Remarks.
( F r e i b u r g r Germany)
The f i r s t t h r e e l e c t u r e s contain an exposition of the
fundamental concepts of some main t h e o r e m s of the theory of r e c u r s i v e functions. One of the m o r e difficult theorems of the theory of r e c u r s i v e functions i s FI-iedberg-Muxniks theorem which a s s e r t s the existence of non-trivial enumerable d e g r e e s . In L e c t u r e s
4 and 5 we prove this theorem, following the t r e a t -
ment given by Sacks, but s t r e s s i n g somewhat m o r e the combinatorial p a r t of the proof ( L e c t u r e 4). Lecture 6
deals with problems in the theory
of
primitive r e c u r s i v e functions. As a typical example of the application of the theory of recursitivy we give in L e c t u r e 7
in detail a proof for the unsolva-
bility of the domino problem in the simplest c a s e of the origin-restricted problem and ehow the
in Lecture
8
how the domino problem i s connected with
c a s e of the Entscheidungsproblem. Lecture
has
6
been given before L e c t u r e s
4 and 5. The inter-
change is due to systematical reasons. The interdependece of the l e c t u r e s may be indicated a s follows: 1
2
3
4
5 , 2
6 , 2
7
8.
Contents L e c t u r e 1: Computability, Enumerability, Decidability L e c t u r e 2:
- Recursiveness, Turing Machines, Degrees of Unsolvability
Lecture 3: Kleenesl Normal F o r m Theorem; the Jump Operator Lecture 4: Theorem of ~ r i e d b e r ~ - M u g n i kP,a r t I
v .
Lecture 5: Theorem of Friedberg- Mucnlk, P a r t I1 Lecture 6: Primitive Recursive Functions Lecture 7: The Domino P r o b l e m Lecture 8: AvA- Case of the Decision Problem of Predicate Calculus. Bibliography
H; Hermes L e c t u r e 1: C o m p u t a b i l i t y
Enuii~erability,becidability
1. Algorithmic procedures and calculi always have been a n essential part of mathematics. In the l a s t thirty o r forty y e a r s a theory has been developed in o r d e r t o study the fundamental notions which a r e connected with this part of mathematics. Everybody knows algorithmic procedures for computing the sum of two decimals. The existence of such procedllres shows that the sum-function is a computable function.
If a
mathematical theory T is given based
on a finite number of axioms and on the r u l e s of first-order logic calculus, we may generate
one by one the theorems of T. Hence the s e t of theorems of
T is' a generable set. Using lexicographical principles it is possible co get nd theorems in a sequence, s o that we may speak of the (Ith, lst, 2 ,
...
theorem of T. In this way we get an (effective) enumeration of T, and we call T an enumerable set. The notions of generability and enumerability may be identified.
F o r any natural number it is decidable whether it is a prime o r
not. Hence the set of p r i m e s is called a decidable set. The concepts of computability, enumerability and decidability a r e narrowly related (cf.no.4).
In order to be able to develop a mathematical
theory concerned with these notions it is necessary to replace intuitive concepts by p r e c i s e mathematically defined concepts.
F o r each of these concepts
different definitions have been proposed and proved to be equivalent t o each other. Practically everybody is convinced that the precise notions correspond llexactlyllt o the intuitive concepts. This fact, the so-called
Church's Thesis
(1936), may be compared with the statement that there exists no perpetuum motiile. In the following (cf. no. 5 , 6 , 8 and Lecture 2 ) we give several p r e c i s e concepts which lead to definitions of enumerability and computability. Referring to such definitions we have notions like Turing-computability, recursir veness,
/u,-recursiveness
etc. But since these concepts can be proved to be
extensionally equivalent, we l a t e r on may interchange them arbitrarily.
H. Hermes 2. In o r d e r t o compute (calculate) it is necessary t o manipulate objects, i.e.
to t r e a t objects by manual means. Not every s e t
property that every element of ch1 s e t of r e a l numbers).
S can be used in this way
S has the
(e.g. the classi-
A s e t of objects which can be used for computa-
tion may be called a s e t of manipulable objects. Typical example f o r manipulable objects a r e the words composed of l e t t e r s from a given finite alphabet A. If A h a s only one element, these words may be identified with the natural numbers. An infinite s e t
S
of manipulable objects is denumerable.
If
S1 and S a r e two (infinite) s e t s of manipulable objects t h e r e exists 2 a 1 1 mapping f from S onto S which is effective in both ways, 1 2 i.e. : if any x S is given it is possible t o compute f(x), and if any 1 1 f (y) Such a mapping is often y S2 is given it is possible to compute
-
-
.
S coincides with the s e t of natural num2 (in this c a s e f(x) is called the Gtidel number of x). In principle it
called a ~'ddelization,especially if bers
is irrelevant on which
(infinite) s e t of manipulable objects the theory is
based. Very often (following Gb'del) we
choose for this purpose the
set
of natural numbers. But many applications may be much e a s i e r if other s e t s a r e chosen. fixed s e t
- We
speak of an enumerable o r of a decidable s e t S
St of manipulable objects is given and if
only if a
=St.
3. F o r most questions concerning computability if is irreleveant whether we consider
1-place o r n-place functions ( o r similarly 1-place o r n-
place predicates). L e t us consider e.g. n=2 t h e r e exist computable functions CS 2, (1.1)
d 2 ( 6)21(~)s %(x))
=
x
(1.2)
d2(x9~))
=
x
=
Y
(1.3)
GZ2(g 2 ( % Y))
. It
may be easily shown that
d21s &22,
S.t.
for each natural number x for each p a i r x, y of natural numbers
Using these functions we may associate with e v e r y 2-place functionj f a 1- place function g,
defined by
H. Hermes
(1.4)
Now we get (1.5)
f(x,y) As
place
g( 6'2(x, Y))
=
.
f a r a s questions of computability
a r e concerned we may
re-
f by g. 4. The following statements hold intuitively:
(1.6)
A s e t i s enumerable i f f it i s void o r the range of a computable function.
(1.7)
A
1-place function is computable, iff
the
2-place relation
R is
enumerable, where R holds for y and x iff y=f(x). (1.8)
A set
(1.9)
A set
S
is decidable iff S and i t s complement a r e enumerable.
S is decidable iff i t s characteristic function f i s computable.
f (x) has the value
0 or
1 according a s
x ES
or
x 4 S.
5. Here and in no.
6 we give two definitions of the notion of enu-
merability. Here we a r e concerned with s e t s whose elements a r e words over a finite
alphabet. Let be given four mutually disjoint alphabets A, B, C, D. The ele-
ments of
A a r e called constants, the elements of
,*.
B variables, the elements
of C predicates. With each predicate i s associated a natural number a s i t s place number. the words over
AvB
a r e t e r m s , then
Ptl;,
D =
);
The words over
t e r m s . If P is an n-place predicate and t
. .;t n
by substituting a proper word for a variable, system Rule relation
and a formula p+F
in a formul:
system
@
Rule 2
p 1' 2'"" etc. a r e called
F to a formula G
the transition from an
t o the Formula
(Smulyan) is given by a finite s e t
derivable
p14p2w3
Rule 1 p e r m i t s the transition from a formula
atomic formula p
.,tn
is called an atomic formula. If p
a r e atomic formulas, then the words pl, pl+p 2, formulae.
nl 1 A called proper words,
F. A formal
4 of formulae. A formula is
, if it can be obtained by applications of
1 and/ o r Rule 2, starting with the elements of
R between words over a finite
@.
An n-place
alphabet A. is enumerabld (in the sen-
H. Hermes
s e of Smullyan) iff there i s a formal system
A, B, C, D, where A n-tuple w ble in
4
0
C
@,
belonging to the alphabets
A, and a n n-place predicate P, s. t. for each A the formula Pw o w n
w of words over 1'"" n iff R holds for w1,
.;w
.
...,
is deriva-
n
6. Another way to define enumerable relations is given by F i t c h f s
minimal logic. We s t a r t with
4 ( , ),*) . A
the 3-letter alphabet
word over
this alphabet is called a n expression if it coincides with i , o r if it may obtained, starting with words a and b
;W.
, by the rule which permits to go over from
to the word
sion. Take the s e t of all
(a, b).
(r (*+))
is an example for an expres-
expressions a s the underlying s e t of manipulable
objects. We choose certain expressions and call them = 12, 13,.
.. .
,/\,V,V,
11,
With these expressions a r e connected certain rules. We confi-
ne ourselves h e r e to indicate the r u l e s connected with (1.10)
,4
v,
=,
V, and
11:
F o r each expression a we may write down the expression = a a ( t h i s is an abbreviation for ((=a)a) (parentheses to the left, also in the
following). (1.11) F o r a l l expressions
a, b
(1.12) F o r all expressions
a, b
(1.13) F o r all expressions a b
we may go over from a t o we may go over from
sult of substituting c llabc
for
Va. the r e -
a in b, we may go over from d to
.
expression is called (lerivable (w
to
a is variable, and d
(These rules a r e s i m i l a r to r u l e s of logic, hence derivation
.
b to v a b .
we may go over from ab
(1.14) F o r a l l expressions a, b, c, d, where
Vab
if it can be obtained by the rules. E.g. the
(d), (Vjt),V\/) dhows, that
T
between expressions
r
s. t.
f o r each n-tuple
r a l . . . a n i s derivable iff
is
"minimal logicm.) An
(vV)
is derivable
.A
relation
(Fitch-) enumerable iff there i s an expression al,
R
...,a n holds
of expressions, for
a
.. , a n .
the expression
H. Hermes
7.
The l a s t example shows that the enumerable s e t s (of expressions)
a r e manipulable themselves, because they may be given by expressions, and each expression determines such an enumerable set. Unfortunately we do not have this pleasant fact for the computable functions. In o r d e r to show this l e t us assume that we have an enumerable set s . t . (a) each element of ctinn
and that
the elemets of
determines effectively a unary
of words computable fun-
(b) each such function may be given in this way (think of S
get a contradiction n
S
S
being descriptions of the computing processes). Then we a s follows : We get in a effective way for each
a prescription how t o compute a certain unary function
.
We introfn duce a new function f by postulating that f(n) = f (n)+l. According to n our assumptions t h e r e is a n m s.t. f=f This leads to a contradiction m' for the argument m (A diagonal argument of this kind is often used in
.
the theory of recursive functions). It is possible t o remedy this defect by enlarging the s e t of functions hitherto
considered. Until now we only have admitted total functions.
The domain of an
n-ary total function consists of all
n-typles of objects
in question. We now consider partial functions. The domain of an
n-ary
total function consists of an n-ary partial Tunction does not necessarily have all
n-tuples a s elements, it may even be void. Intuitively a partial
function is called computable, if there is a procedure which terminates for a given argument iff the function has a value for this argument which determines in that c a s e that value. With partial function we do not get the contradiction of no. is only possible to conclude that
f is not defined f o r the argument
If we admit a l s o partial functions, the statement
7 . It h.
(1.7) remains
true. (1.6) may be simplified : (1.15)
A s e t is enumerable iff it is the range (or the domain) of a computable partial. function.
H. Hermes
8. There a r e different important p r e c i s e
ty
definitions for computabili-
for partial functions. F o r Turing-computability and
Lecture 2.
/1L. -recursivity
cf.
Here we mention only the concept of Markovfs algorithm.
Let be given a finite alphabet A A Markovfs algorithm
and words
A 1., B.1 ( i = l ,
...,p) over A.
i s given by sequence
where "(. )I1 indicates that t h e r e niay be a dot behind the arrow o r not. (1.16) determines a unary partial function. f
. The domain
and range of f
tained in the s e t of all words over A. F o r any word mine
... of
uniquely a sequence
w=w(O), ~ ( l ) w , (~),
iff the sequence terminates, and in that case
W
over
words.
a r e con-
A we deter-
f is defined
f(W) is the l a s t element of
the sequence.. If w ( ~ + ' ) is defined we will have a uniquely determined number ( l < ~ , + ~ < p )which , describes in the sequence (1.16) the rule which responsible for the transition from We call a word
K
a
w ( ~ to )
w ( ~ + ' ).
is
part of L iff t h e r e a r e words K1, K2 S. t.
L=K KK Given K, there may be different decompositions of L of this 1 2' kind. If K has minimal length, the decomposition of L is uniquely deter1 mined and called the normal decomposition.We now procede to define w ( ~ + ' ) and p
-
n+l ' w(n+l) and pn+l a r e only defined if there i s
an i s. t.
A . is a part 1
.
n=O o r (n>O and the pth t e r m of (1.16) has no dot ) In n this case let be p the smallest i, 5.t. A. i s a part of w ( ~ ) Let be n+ 1 w ( ~ ) = KA K the normal decomposition of W(') relative to Ai. Now W(n+ll l i 2 = K1 BiKZ. of w ( ~ )and if
.
A unary partial function &whose domain and range i s contained in
the s e t of dl words over a finite slphabet) is called computable by a Markovls algorithm over an alphabet A, iff
A CA and if for each word 0
W
- 10 H. Hermes over A (a) if f (the function determined by this algorithm) is defined f o r W 0
if
f(W) is a word over A g is defined for
(b) if
0'
W
then g is is defined for W and g(W)=f(W), and then also f is defined f o r W and again f(W)=
=g(W). R E F E R E N C E S: ------------
Davis. [l], of Fitch). Kleene lyan
L e ct u r e
-------we--
2:
111 ,
117
[I,
Hermes [23
.
.
(also f o r the minimal logic
Markov [I],
+Recursiveness, Enumerability, ......................
Rogers
[I) , Smul-
Decidability.
1.
In no. 1 we use natural numbers a s manipulable objects. Let be 1 the 0-place function with value 0, S the 1-place successor-function and n U. the n-ary function whose value coincide with the i-th argument 1 O l n (i = 1, ,n). The functions Co, S , U. a r e called initial functions. The
...
initial functions a r e computable total functions. The process of substitution leads from function a function f = g(h (2.1)
1'""
f(xl,
h ),
r
....x,)
g, hl,
..., hr t o
where = g(hl(xl,
...,xr ), ..., hr(xl. ...,xn)) .
Substitution p r e s e r v e s totality and computability. The process of primitive recursion leads from functions g, h t o a function f, where (2.2)
f(xl..
..,xn ' 0) = g(xl.. . ., xn),
H. Hermes
Primitive recursion preserves totality and computability.
r-
The process of application of the tion
g
to
a function f,
( p y = the l e a s t s.t.
for all
z
y. p y
..., xn,y) = 0; in this
/IC
g(xl,.
where
g(xl,.
..,xn,y) = 0
..,xn,z)
case y =
operator leads from a
is defined iff
#
is defined and
y g(xl,.
t h e r e is a
0 and if g(x
..,xn,y)=O.
func-
y
la***
Application of the
operator p r e s e r v e s computability but in general not totality. Computabi-
lity would not generally be preserved if we would not postulate that g(xl,.
..,xn, z)
be defined for z < y.
~ h e p - o p e r a t o rmay a l s o be applied to a relation R. We define
py
Rxl.. .xny
by p y g(xl,.
..,xny ) = 0,
where
g
i s the characteristic
function of R. The functions which may be obtained starting with the initial functions and using substitution and primitive recursion a r e called primitive r e c u r s i v e functions. If we admit in addition the application of the r a t o r we get the
/L
ction is total and computable. The the computable
, h - r e c u r s i v e functions coincide with
(partial) functions. There a r e total
which a r e not primitive recursive. 2. Turing machines.
qO.
...,qp
A TM (Turing machine)
-recursive functions
M
is given by (a)
ao,...,a
of quadruples i s often called the table of M.) q a b ql, where
r
, (b) a finite (ordered) s e t of N and a ffnite (ordered) s e t of quadruples. (The s e t
a finite (ordered) alphabet states
pope-
-recursive functions. Every primitive recursive fun-
q, q1 a r e states,
a is a letter
either a l e t t e r o r one of the symbols
A quadruple is of the form (of the alphabet), and b
R ("right1' o r L (llleftll). (We assu-
H. H e r m e s
R ,L
m e , t h a t t h e s e t of l e t t e r s , t h e s e t of s t a t e s a n d t h e s e t
a r e mutually
d i s j o i n t . ) We r e q u i r e t h a t t h e r e i s a t m o s t one q u a d r u p l e i n t h e s e t of q u a d r u p l e s which b e g i n s with
qa,
where
q
is a fixed s t a t e a n d a is a f i x e d
letter. A t a p e is a two-way infinite s e q u e n c e of s q u a r e s . ( S o m e t i m e s a l s o one-way infinite t a p e s a r e c o n s i d e r e d . ) An i n s c r i p t i o n the
is a m a p p i n g of
s e t of s q u a r e s into t h e a l p h a b e t . We a s s u m e t h a t i n I
s q u a r e is b l a n k ( e m p t y ) , i . e . m a p p e d o n t h e
-c-o-n-f i g u r a t i o n
C = (I, s, q ) , w h e r e
C i s a triple
s c anned -
square
of
C, a n d
almost every
loid letter a
A (complete) 0' is a n i n s c r i p t i o n , s is a
I
s q u a r e a n d q a s t a t e . We w r i t e C = (I(C), s ( C ) , q ( C ) )
re -
I
.
s ( C ) is c a l l e d t h e
a ( C ) = I(C) ( s ( C ) ) t h e l e t t e r o n t h e
of C. C is c a l l e d t e r m i n a l , i f t h e r e is no q u a d r u p l e i n M
with
q(C), a(C)
figuration
C
. O t h e r w i s e we a s s o c i a t e with , c a l l e d t h e --s u c c e s s o r of C :
q u a d r u p l e of M which b e g i n s with C1 = ( I ' , s ,q l ) .
We put q 1 = q
q(C) a(C)
.
I' a n d s
C l e t be
. We
scanned squawhich b e g i n s
i n a unique way a c o n q ( C ) a ( C ) b q*
the
want t o define
depend o n b.
We d i s t i n g u i s h
two c a s e s : (1) If
exception
b e is a l e t t e r ,
that I ' ( s l ) = b.
( 2 ) If b = R(L), s t is t h e r i g h t
If w e s t a r t with a n a r b i t r a r y quence
C, C1, C v = (C1)I , C1I1,.
s t a r t with
C" m a y b e e x p r e s s e d
q ( C ) = q O . I'To p r o c e e d f r o m we h a v e c a s e
(I), a n d a s
h a l t s , s t a r t i n g with
C, a t
(left) n e i g h b o r of s C,
. ., w h i c h
m a y b e f i n i t e o r infinite. " T o
C t o C1 Itmay b e e x p r e s s e d a s p r i n t i n g , i f
going t o r i g h t
c*"
n-ary partial
(left), if we h a v e c a s e (2). I1M
means, that the sequence
function
C,
C:.
..
.
C"
3 . With e a c h T u r i n g m a c h i n e an
a n d It = I.
w e o b t a i n in a unique way a s e -
a s t 1 t o a p p l y M o n I(C) i n s ( C ) " , if
h a s a l a s t t e r m i n a l t e r m which is
associate
s t = s a n d I1 = I w i t h t h e G o s s i b l e )
we p u t s
M
and each natural numer
n
we
f b . (We a s s u m e in t h e following
H. H e r m e s
M
t h a t t h e a l p h a b e t of
h a s a t l e a s t t h e l e ~ ; e r - s a (blank) a n d a l . ) 0 M Let be x x natural n u m b e r s and a c o n f i g u r a t i o n with 1'"" n Cxl....,rn M M l(Cxl.. x ) a n inscription where the a r g u m e n t s q ( C x l . . . x n ) = q,, n
.
.
a r e r e p r e s e n t e d by s e q u e n c e s of (x + l ) , . . , ( x + l ) c o l ~ s c c u n 1 a and a r e s e p a r a t e d f r o m each other t i v e s q u a r e s which b e a r t h e l e t t e r 1 M ) i s the f i r s t s q u a r e on the tape by one blank s q u a r e , a n d s(Cxl. . x n which b e a r s t h e l e t t e r a F o r any configuration C l e t b e v (C) 1' ( t h e v a l u e of C) t h e n u m b e r s q u a r e s which b e a r t h e l e t t e r a 1 in I ( C ) x1,...,X
.
f,
Now we i n t r o d u c e a
s.t.
C*
case
.
x-
x ) = v(C )
n if t h e r e i s
computable
a s follows:
cM
M, s t a r t i n g with
.. .
f&(xl,
fi(xl,
. An
x ~ 9 * . *n1 x n - a r y function
s. t .
C
t h e q u e s t i o n w h e t h e r a p a r t i a l function s o m e o t h e r p a r t i a l function
i s called
f
of
g
a r e given.
t h e v a l u e s of
g
f f
that g
is c o m p u t a b l e ,
If
Turing
is c o m p u t a b l e , but a l s o
f
f
we c a l l
is c o m p u t a b l e
g-computable g-recursive) and w r i t e 2 g ,< g f o r e v e r y function
g
(where iff
may
there
under the assumption that the t h e v a l u e s of
is c o m p u t a b l e u n d e r f
,< g. A s
g
is
values
are
given
the assumption
c o m p u t a b l e r e l a t i v-e f
in
is c o m p u t a b l e if w e s u p p o s e
It i s not a s s u m e d t h a t
by a n y e f f e c t i v e p r o c e d u r e .
this
n f = f M'
b e c o m p u t a b l e o r not). T h i s q u e s t i o n h a s a p o s i t i v e a n s w e r a procedure t o get
*; i n
M a t h e m a t i c i a n s not only a r e i n t e r e s t e d
i n t h e q u e s t i o n w h e t h e r a p a r t i a l function
that
) i s defined iff t h e r e i s
, halts at
a Turingmachine
4. R e l a t i v e c o m p u t a b i l i t y .
.
. . ., x
to g
or
a n example we
have
f: .
A p r e c i s e definition f o r t h e r e l a t i o n We have
f
4g
fo'r e x a c t l y t h o s e f u n c t i o n s
f
f
4
g
r u n s a s follows :
which may b e obtained
H. H e r m e s
s t a r t i n g with the initial primitive r e c u r s i o n
and
functions (cf. no. 1)
and the application
of
g , using
the
-& is reflexive and t r a n s i t i v e
It is obvious that
fined by (f & g and g&f)
substitution,
p - operator. .
divides the c l a s s of all
Hence the relation de( p a r t i a l ) functions
in mu-
tually disjoint s u b c l a s s e s . T h e s e c l a s s e s a r e called d e g r e e s (of unsolvabi-
Ilty). L e t be -f the c l a s s t o which
<
g
set.
D
f
by
f ( g.
Hence the s e t
h a s a l e a s t element
ctions.
0
f belongs. It i s possible D
of d e g r e e s
which c o n s i s t s of
t o define
i s a partially o r d e r e d
p-)r e c u r s i v e fun-
all (
Not in e v e r y d e g r e e we leave a total function (Medvedev
cf. R o g e r s [I]
.
)
1955,
D e g r e e s which have total functions a s e l e m e n t s a r e cal-
l e d total d e g r e e s .
A total d e g r e e may be identified with t h e s e t of a l l to-
t a l functions belonging t o it. The s e t
T
of total d e g r e e s i s not
only
partially o r d e r e d but in addition a semi-upper-lattice:
Let
unary functions and
h(x, y) = d 2 ( f ( x ) , g(y)). Then
i s the l e a s t upper
bound of
with
f and g 1.f
5.
g
r e s p e c t to
quadruples
q a
---
C is not t e r m i n a l guration
b qt
where
(no. 2) we
total
< .
i s a unary total function the relation
fined using a n extended concept
be f, g
of T u r i n g machines.
b is a state a s s o c i a t e with
(cf. no. 2)
f < g may be de-
H e r e we admit a l s c .
. If
a configuration
C a s i t s s u c c e s s o r a confi-
C1 g begins with
depending on g. C r i s determined by the quadruple which g q(C) , a (C) (cf. no. 2). L e t be q a b q' this quaduple.
If b
a s t a t e we define
is
not
C1 = (I(C), s(C), qf ) w h e r e q9 = q n ask
otherwise.
C' = C1 (no.2). If b = q" we define g qa = q t if g( G2l ( v ( ~ ) ) ) )= G' 2Z(v(C),, and
Hence in o r d e r t o get
Cf one h a s in g e n e r a l t o g a n "oracle" (Turing) about the value of g f o r a c e r t a i n argument. With
g and
each
Turing machine
e a c h natural n u m b e r
n
(of the extendet kind), e a c h function
we a s s o c i a t e an n - a r y function f'" M, g' n The definition r u n s l i k e the definition of f M in no. 2, 3, but with
in place of
C'
. Turing computable relative t o n (off the extended kind) s.t. f=f M, g
An n-ary (partial) function i s called iff there
6.
is
It
a Turing machine
.
is often convenient to identify a predicate (set, relation) with
its characteristic function
(which i s a total function). Hence we may speak
of the degree of a relation and extend
REFERENCES : -------------
See
Lecture
<
to relations.
1.
Lecture 3: Kleene's Normal F o r m Theorem; the Jump ............................................... 1, degree is not
dl
With each total degree (the j u m p of d)
an upper neighbor
troduce Kleene's
2.
d
it
Operator,
i s possible to associate a
which has the property that
d<
of d\. To prepare the definition
normal form for
d'
of
(but d'
d' we in-
g-recursive total functions.
In the following we a s s u m e that we have mutually disjoint infi-
nite seouences
ao, a l , a 2 , .
..
Bnd
qO, q l , q 2 , .
. .,
and that
the alphabets
and s t a t e s of
any Turing machine a r e initial segments of these sequences
(which i s not
a s e r i o u s restriction). A Turing machine
M (of the extended
H. Hermes
kind, cf. Lecture,
no. 5) may be described by its Godel number
We a s s u m e in addition that a l s o finite figurations
a r e described by
sequences
Giddel numbers
Co, C1,
. . .,Cm
.. . ,Cm ).
G(CO,
G(M). of con-
The follo-
wing constructions depend on the (fixed) ~ b ' d e l i z a t i o n G. We want to introduce a unary total function
and for each n an
U
T~ (which depends on a total function g) in o r d e r to g F (cf. also Lecture 2, nos. 2, 3 , 5 ) Let be M, g
(ni-2)-place predicate
.
describe the function
the value (3.1)
v(C ) i f t h e r e is a sequence of configurations m cO, 'm s.t . g = G ( c ~ , , c,),
...,
u(g) = 0
(3.2)
...
otherwise.
n T zx . x y iff there a r e a Turing machine M and configurations g l M , Cm), C = C CO, , Cm S. t. z = G(M), y $ G(CO, 0 Xl...X ' n C. = (C.)' depending on M) and Cm is terminal. 3+1 J g
..
...
.. .
Now we repeat the definition of
fn (Lecture 2, no. 5) by writing M, 2
This i s Kleenets Normal F o r m T h e o r - It shows that each n-ary sive function may be represented
.
by U ( ) ~ ~ T ~ .xny) X ~ .with
It is e a s y (but somewhat tedious) to show that
g-recur-
suitable z.
U is a recursive function
T~ a g - r e c u r s i v e predicate (i. e. it-s characteristic function is g n g-recursive (g-computable)). F r o m this we infer that U ( p yTg zxl..
and
is an n-ary g-recursive (partial) function for every z. Hence,
the number
z = 0,1,2,
.. .
of
varying
we get e v e r y n-place g-recursive function.
In addition it can be shown that in o r d e r t o obtain r a c t e r i s t i c function
.xny)
U and the cha-
Tn starting with the initial functions Bn'd t h e fung
H. Hermes ctions g, (cfr. Lecture 2, no. 4)
it is not necessary to apply the
.
F
-Ope-
r a t o r ( U and T~ areItg-primitive r e c u r s i v e f f ) Hence (3.3) shows that g we get every g-recursive function by applying the -operator at most
P
once.
F o r l a t e r application it i s convenient to notice that M, starting with
Finally we r e m a r k G(Co,
. . ., C m )
, halts
C:
iff
that for the usual choice of
g
and of
21
we have
1
This shows that in o r d e r to check whether for
1 VYT G(M)xy g
T zxy holds o r not, the oracle g is asked' only for arguments which a r e l e s s than y
.
3.
F o r each total unary function
g
we define
1 g' = characteristic function of the unary predicate VyT xxy. g We want to show that (3.5)
(3.6)
g < gf
(3.7)
-g # -g1 (hence
(3.8)
f
Using (3: 6), (3.7), (3.8)
with (3.6) g < g1),
,< g 4 f t < g ' . we may extend the operator '(jump) to elements
of T(tota1 degrees) by defining ():I
4. We obtain
>
=
(a. It follows that
(3.7) by proving (Church) :
d < df
.
H. H e r m e s
Otherwise let be g t
5
We introduce a total function h by
g.
{
h(x) =
(3.10)
U ( p T g x x y ) + l , if t h i s i s defined f o r x , 0
Under o u r a s s u m p t i o n
gt
otherwise. we find
.
h
i s g-computable Henee a c c o r d i n g 1 g f o r e v e r y x. We t o no. 2 t h e r e is a n u m b e r s. t . h(x) = u ( r y ~ zxy) 1 h ( z ) = U(/U y T zpy), which c o n t r a d i c t s ( 3 . 10). get a s a special case g 5. In o r d e r t o obtain (3.6) and ( 3 . 8 ) we p r o v e t h e following T h e o r e m (Kleene). L e t be g
a u n a r y and t o t a l function, R a 2-place r e l a t i o n a n d
R < g. Then t h e r e i s a computable total u n a r y function (3.11)
1
VYTg
V y ~ x iff y
Proof:
C, M(x)
s t a r t i n g with
L
n a t u r a l n u m b e r ) we get
s.t.
.
r(x) r(x) Y
x i t i s p o s s i b l e t o c o n s t r u c t effectively a n e x -
F o r each number
tended T M M(x) s . t .
r
a sequence
co
=
(where M(x)
ct
t
is an
c,
,
arbitrary
=(Co);:,
C2=
=(C )I of configurations s. t. t h e following s t a t e m e n t s hold : T h e r e 1g"" CM(x) i s a k s . t . I(C ) is void. F o r a n u m b e r k > k we have k 0 x, 0'
-
Using t h e a s s u m p t i o n not, t h e r e will be a whether halt.
Rxl
R k
M now
> k o s.t.
o r not, e t c .
But if t h e r e a y
"checkstt whether Ckl
be
s. t.
r ( x ) = G(Nl(x))
-
M(x) - cx, 1
.
Now
If t h e r e e x i s t s no y s. t. Rxy, M(x) will halt.
M(x) , s t a r t i n g with
(3.12) Now l e t
1
< g,
0
.
RxO
o r not. If
M ttcheckstf
R x y , M(x) d o e s not
Hence we have
c ~ ( ,~ h a) l t s iff t
Vy Rxy.
r i s a computable total function. F r o m (3.4)
we infer: (3.13)
~ ( x ) s, t a r t i n g with c ~ ' ~ ) h, a l t s iff t
VY T
1
G(M(x))) t y
.
H. Hermes Comparing (3.12) and (3.13) we get (introducing r(x)) (3. 14)
1 V y T r(x)ty , g
V y 'Rxy ; iff
which gives (3. 11) for
.
t = r(x)
6. We now apply Kleenels Theorem in o r d e r t o prove (3.6) and (3.8). Proof of (3.6):
We introduce R by postulating RxY
iff
. Hence
It is obvious that
R< g
ble total function
r
(3.15)
vy R x y
according to Kleene we have a computa-
s. t.
The left side i s equivalent t o obtain
g ( G z l ( x ) ) = Gz2(x) A Y = Y.
g < g l ( r ) , and trivially
V ~ T ' r(x)
iff
g
g(
r(x) y
cg 2 1(x)) = 6/ 22 (x) .
gl(r)
which shows that
7.
f1 = g f ( r ) <
The upper-semi-lattice
g1
Hence from (3.15) we
6 g1 .
Proof of (3.8) : L e t be f 6 g. We define Rxy by 1 Tf \< f < g. Using Kleenels theorem we have
. ~ ~y: x x y
.
1
Tf xxy
1 iff V ~ Tr ( x ) r ( x ) y g
.
R \< g, since
.
. T ' (lecture
2, no. 4) with the additional
jump-operator i s a very cornplrex s t r u c t u r e which h a s been intensively studied.
I want to mention only two results:
(1) Every countable partially ordered s e t i s imbeddable in
T.
H. Hermes
(2)
The complete degrees (e.g. de with the degrees
> -
the degrees of the form dl) coinci-
0'.
8. Of special interest a r e the degrees of enumerable sets. These de-
g r e e s a r e called enumerable degrees. About the enumerable degrees we --have the following elementary facts: (a) 0 i s an enumerable degree, since every decidable s e t belongs to 0 and every decidable s e t (b)
i s enumerable.
Going back to the intruitive notion of enumerability it is easy
t o s e e that every enumerable s e t may be expressed in the form Vy Rxy, where
R
is decidable. Conversely each s e t of this form with decidable
1 is enumerable. If is a computable total function (e.g. f = S ) then 1 1 Tf xxy i s decidable. Hence YyT xxy is enumerable. This shows that Y O 1 is an enumerable degree (cf. (3.5))
R
.
(c)
As we have s e e n
expressed in the f o r m
in
(b) , each enumerable s e t
S may be
V y Rxy with decidable R. Using Kleenets Theorem
(3.11) for a computable total unary function g, we obtain the result that degree of We have shown that
S = g t ( r ) < g1 = 01.
---
-
0 and 0' a r e enumerable degrees and that for
every enumerable degree we have 0 < d < O t . Post (1944) has asked whether
- -
t h e r e a r e enumerable degrees other swered (positively) not before
R E F E R E N CES: ------------
then O , O 1 . This question has been an-
196617. Cf. Lecture 4 and 5.
See Lecture
1.
- Kleene
and P o s t
[I]
, Post
ll]
H. H e r m e s
L e c t u r e 4: T h e o r e m of F r i e d b e r g - ~ u & n i k , P a r t ....................................
I
1. T h e F r i e d b e r g - M u c n i k T h e o r e m a n s w e r s question
in t h e a f f i r m a t i v e t h e
whether t h e r e a r e enumerable degrees besides
.
L e c t u r e 3, no. 8)
0 and
0 ' (cf.
We follow t h e t r e a t m e n t of S a c k s who t r i e s t o s e p n r a -
t e a combinatorial
part
of t h e proof
(which h e c a l l s " p r o r i t y method")
f r o m t h e r e s t which u s e s r e c u r s i v e concepts. T h i s l e c t u r e is devoted t o the combinatorial part.
T h e p r o o f is finished i n t h e next l e c t u r e
. For
o t h e r p r o o f s cf. t h e r e f e r e n c e s . 2. E, F, F'
T h e individuals c o n s i d e r e d h e r e a r e n a t u r a l n u m b e r s . u n a r y a n d H, D
binary predicates, and g
Let be
a total unary
ction (whose a r g u m e n t s and v a l u e s a r e n a t u r a l n u m b e r s ).
We i n t r o d u c e
t h e following a b b r e v i a t i o n s :
-
(4.1)
Lrs
for
O
(4.2)
Ps
for
Vr(gr)
(4.3)
Qs
for
V r ( g ( r ) = g(s) A L r s ) ,
(4.4)
$) s k
for
(4.5)
sk
for
(4.6)
@ (k)
for
[s:
(4 7)
p(k)
for
p:Y/sk]
T h e derivations
in
O<s A
s A Fs
TFI
Vr(r<s A
4rkA ~
4 rk)
A2: A3:
g(s)
-
.
t h i s l e c t u r e a r e b a s e d on t h e following
S
A
FsC)~ES
~ H r . s- 1 A H r s --, D r s A y E s Hss +Es
1,
= k,
H r s 1 4 Hrs),
,
Axioms: Al:
n
fun-
H. Hermes
A7:
-
Hrs
1
Hrs
4
We f i r s t prove s e v e r a l lemmata. The most important a r e Lemma Lemma 5 which relate the predicates used to derive sets
Lemmata 8
(k) and
Axioms
6
y(k)
and 9
C$
v . These lemmata
and
which show that for each
k
4 and are the
a r e finite. This immediately leads to Lemma 10.
and 7, not used hitherto (and no
in o r d e r to derive Lemma
other axiom) will be used
11. In the next lecture we apply only Lemmata
10 and 11. 3. Lemmata
Lemma 1.
Proof: Lemma 3.
1.
Axiom
ysk--) 7 E s . Axiom
4 sk-
2.
+ss.
Lemma 1, Axiom
Proof: 7-
r <s
Lemma 4: Proof:
( p s k we have
have
0 < r < s, 7F1r
@ r k gives
Lemma -
5:
4 rk
@ s k ->Vu(r
A
g ( r ) = g(s). Hence and F r (from u s.t.
yuk.
,Qs
gives
4 rk), we get
r < u < s,
4
/\ Y u k )
and
1Hru-1
i(i
-
.
1,Axiom 5). Using O r k
(Lemma 3 ) . Comparing -->Hrr
that t h e r e i s a number with
A
3.
F r o m @ s k we get -Qs(Lemma
and
w e obtain ,Hrr
5.
4s k 47Es.
Proof: Lemma 2:
1 to
.
L r s . Since we
Hrs-1. F r o m
rk
Hrs-1 we fina Hru.
This together
H. H e r m e s P r o o f : U s i n g t h e definition of qrk,
r s. t .
we h a v e a n u m b e r
'?+/$K
r<s,
H r s - 1 , H r s . L e t b e i = g ( s ) . R e m e m b e r t h a t k = g ( r ) ( f r o m @-k).
7
T h e n we h a v e g ( r ) = g ( s ) a n d
If we a s s u m e k~ we get a c o n t r a d i c t i o n :
L r s ( u s i n g 4 r k a n d 7 H r s - 1 ) . H e n c e we get Q s a n d , with Axiom 5, E s . But w e a l s o h a v e i E s by L e m m a 2. If we a s s u m e k
--
-
< g(s),
L r s ( a s a b o v e ) a n d D r s ( u s i n g A x i o m 2). Hence w e g e t Ps and (with A x i o m 4) E s . But h a v e i E s
b y L e m m a 2.
1
Hence w e h a v e i
< k. F r o m y s k we g e t 7 E s ( L e m m a 2), t h e n -,Ft s a n d Fs (Axiom 1). S i n c e 0 < s a n d g ( s ) = i we h a v e $ s i 7 4. L e m m a t a
6 t o 9.
L e m m a 6. Proof:
Lemma card
L e m m a 7.
r < s,
5.
($ (k)
Proof;
According to L e m m a
($r k
and @sk,
is infinite t h e n @(k) =
1
also
So,...,S
'j
s
with
0
< s <.
y(0)< 2'
1
n
c a r d y ( k ) < 2k by induction:
. (b)
Let
(a)
Lemma
s. t . p u k .
If $ ( k )
. .< S
n
t
.
:-
< < -
we h a v e n u m b e r s
Yulk
,..., y'unk.
(0) is void (cf. L e m m a 5 )
iCk ick
2 ick
be
i
. Hence
( c a r d 2fl/(i)
card
< k. T h e n we h a v e
c a r d @(d)(~emma 6)
2i
Hen-
.
8 be true for all
card fik) <
@ (k)
H e n c e if
is f i n i t e l e t
. Then
s
.
t w o n u m b e r s r, s with
between
v ( k ) is infinite.
a. Proof
4
card y ( k ) + l
t h e r e is a n u m b e r u
s.t. s c u <S n + l = c a r d $(k). ce card Lemma
5
+
1) (Lemma 7)
(induction h y p o t h e s i s )
H. H e r m e s
zk .
<
c a r d @ ( k ) & Zk
L e m m a 9. Proof:
L e m m a t a 7 , 8.
A r /I
i (u
L e m m a 10:
Ak v u
o f :
T h e g i s t of L e m m a t a 8.9
finite f o r e a c h
is t h e fact that $(k)
k. F r o m t h i s we i n f e r i m m e d i a t e l y
L e m m a 11:
and y ( k ) a r e
L e m m a 10.
L e t be
(1)
g(s) = k ,
(2)
-F'
s, 7 H s s - l ,
(3)
u < s,
(4)
/\ r /\ i(u < r A i < k 4 l @ r i A
(5)
Ar(r
,<
7y
ri),
.
u+qDrs)
Then we have
V r ( q r k 4 ,dm Proof: F r o m a t t h e definition
(4) we get f o r
-I
Hrm)
r = s and i = k t h a t - j @ s k . Looking
(4.4) we find that TFS. Hence
Ps V Q s
V
H s s - 1 (by
Axiom 6). H s s - 1 is excluded by (2). Ps
c a n b e excluded a s follows: A s s u m e Ps. T h e n t h e r e is a num-
b e r r , s. t . i = g(r).
r
\< u.
0 < r < s,
T h e n we h a v e Hence -,Dm
F r
4ri. Now
0 < r < s,
have
@ rk.
H i
-
1
= g ( r ) < g ( s ) = k.
Using (4) we g e t
Q s . By definition of Q w e have a n u m b e r r,
F
F
lHrs-l,
g(r) = g(s) = k
. For
this
r we
W e want t o show that ~ H r m f o r e v e r y rn.
We have
-
~ H r s 1. Hence by Axiom
If now H r m f o r s o m e m , we have a n u m b e r and H r m .
g(r) < g(s), Drs. Let b e
by (5), c o n t r a d i c t i n g D r s .
T h e r e f o r e w e have s.t.
F
7 we h a v e - t H r m f o r m < s - 1.
m - 2 ~ 9 .s.t ~ H r m - 1
T h e r e f o r e w e would have q m k (by definition (4.5)). Now (4) s h o w s
t h a t 7 H r m which c o n t r a d i c t s H r m . R e f e r e n c e s : F r i e d b e r g r17, Mucnik
rll , Lachlan Ll]
, S a c k s [I],
Shoenfield
1
H. H e r m e s
Lecture 5: T h e o r e m of F r i e d b e r g ------------ ~ u F n i k , P a r t I1 ----------------------1. We want t o s n o w t h a t t h e r e a r e e n u m e r a b l e d e g r e e s
do$' d l a n d dl&do. 0
<
d
Since for e v e r y enumerable d e g r e e
d
do, d l , s . t .
we h a v e
,< O f (cf. L e c t u r e 3, no. 8) n e i t h e r d n o r d
can be 0 o r Of. 1 In t h i s l e c t u r e e v e r y s e t ( r e l a t i o n , function) h a s n a t u r a l n u m b e r s a s 0
elements (arguments, values). Small l e t t e r s r e f e r t o natural numbers. F o r 0 1 0 e a c h s we w i l l d e f i n e s e t s A.s, A s t . t h e b i n a r y r e l a t i o n s x E . A and s S 1 a r e r e c u r s i v e ( i . e . d e c i d a b l e ) . Now we i n t r o d u c e t h e s e t s AO, A x&A1 S
by
x PA'
F r o m the equivalence b l e (cf
. Lecture
3, no. 8)
define s e t s
Ts
s.t.
(5.2)
a function
S
x eTs
n 2. T h e s e t s
. The A'
T h e d e f i n i t i o n s of
iff v s x e ~ : w e i n f e r s a m e holds f o r
a n d A'
A0 is e n u m e r a -
.
w i l l b e i n t e r r e l a t e d . We l a t e r o n
is a b i n a r y r e c u r s i v e r e l a t i o n a n d i n t r o d u c e
1 n € A n iff 2 n t l e T s-1'
s iff 2n E T s- I '
CAO
Ts
A
that 1
will b e d e f i n e d t o g e t h e r with s e t s
FS, HS a n d
g ( s ) by s i m u l t a n e o u s r e c u r s i o n . We i n t r o d u c e t h e following a b b r e -
viations:
FIS
for
c
F'
Hrs
for
B' n
Es
for
T
S
F f o r F' c T
T ~ -,
=
s
T~ j
O.
Ts-l
-
D r s f o r H~
S
n F'
,
j 0,
H.. H e r m s s
T h e d e f i n i t i o n s of
s
F' , H',
= 0. F o r t h e c a s e
n i t i o n s (5.9), (5. 1 2 ) f o r (5. 14)
T
S
a n d g ( s ) a r e g i v e n by (5.6) f o r t h e c a s e
s > O we d i s t i n g u i s h C a s e 1 w h e r e we h a v e t h e defi-
(5. l o ) , (5.11) a n d C a s e 2 w h e r e we h a v e t h e definition
FS, H',
and g(s)
. In
both c a s e s
T
. 0
F
(5.6)
= H
0
s is g i v e n by ( 5 . 1 3 ) a n d
= T o = 0 (void s e t ) ,
g(0) = 0.
s- 1 In o r d e r t o define FS , H' and g ( s ) f o r s > O we s u p p o s e t h a t F , 1 A : a n d A' a r e g i v e n by a n d g(s-1) a r e defined. H e n c e a l s o 1 (5.2). Let b e f , f t h e c h a r a c t e r i s t i c f u n c t i o n s of A,: A:. L e t b e
-
~ ( s )= 0
(5.7)
if
s is e v e n ,
e ( s ) = 1 if s i s odd,
e ( s ) = t h e n u m b e r of p r i m e f a c t o r s
(5.8)
3 occuring in the pri-
m e n u m b e r r e p r e s e n t a t i o n of s. Consider
t h e following condition ( w h e r e p is t h e
kth p r i m e n u m b e r ) :
We h a v e
C a s e 1 if (*)
C a s e 2.
be
r(s)
the greatest
is s a t i s f i e d , o t h e r w i s e
m
f o r which
V y (...)
,
In C a s e 1 l e t
T h e n we define :
H. H e r m e s
In C a s e 2 w e put
s S F = H =0,
(5.12)
g(s)=O.
In both c a s e s l e t b e (5.13)
T s = Ts- l , of Ps V Q s V H s s - 1 ,
(5.14)
Ts = Ts,l
Here
P
and Q
a r e introduced
F~ o t h e r w i s e .
U
b y (4. I), ( 4 . 2 ) , ( 4 . 3 ) , w h e r e F', F, H, D
a n d E a r e defined i n (5. 3), ( 5 . 4 ) , ( 5 . 5 ) . Using (5.13), (5.14) a n d (5.2) we g e t i m m e t i a t e l y
In t h e definition quantifiers i n (*))
.
( e . g. V n ( n
of
5
s
F',
HS, T
s
a n d g ( s ) w e h a v e only r e s t r i c t e d
~. .) i n (5.10) and V m V y ( m 5 s A y
In t h e t h e o r y of r e c u r s i v e functions i t is p r o v e d
5 s ...
)
that this fact
g u a r a n t e e s t h a t e v e r y t h i n g is r e c u r s i v e .
By i n s p e c t i n g (5.9), ( 5 . 1 0 ) a n d (5.12), (5. 13), (5.14) a n d (5. 2) we s e e that (5. 16)
S
S
0
F h a s a t m o s t o n e e l e m e n t . H , T s, A S a r e finite. 3. Connection with L e c t u r e 4. F i r s t we find t h a t
S
T h i s is t r i v i a l e x c e p t f o r C a s e 1. H e r e a c o m m o n e l e m e n t of F a n d m u s t b e both of t h e f o r m
HS
2 p + l - & ( s ) and 2n+ & ( s ) which i s i m p o s s i b l e .
-
H. H e r m e s
Using
(5.4)
H'
a n d t h e d e f i n ~ t i o nof
it is e a s y t o v e r i f y t h a t
-
(5.18)
~ H S S 1.
Now w e m a y c h e c k ( u s i n g 5. 17) t h a t H e n c e a l s o l e m m a t a 10 a n d
Axioms
1 to
7 ( L e c t u r e 4) a r e t r u e .
1 1 of L e c t u r e 4 hold.
~'1
.
4. We want t o s h o w t h a t A' (In a s i m i l a r way it c a n b e shown 1 1 ' t h a t A' A ) -We a s s u m e t h a t A < A a n d look f o r a c o n t r a d i c t i o n . 1 1 L e t b e fO, f t h e c h a r a c t e r i s t i c f u n c t i o n s of A' r e s p . A According to
j .
.
K l e e n e ( L e c t u r e 3 ) t h e r e is ( u n d e r o u r a s s u m p t i o n ) a n u m b e r e , s . t . 1
(5: 19)
r
f ( x ) = U(
yT
1
exy)
P
.
We now f o r m u l a t e a condition (5.20). In no. 5 we s h o w t h a t we g e t a c o n t r a d i c t i o n a s s u m i n g t h i s condition, a n d i n no. 6 the negation is a s
(5.20). This
finishes t h e proof
we show t h e s a m e a s s u m i n g . 1 0 for A < A . T h e condition
follows:
(5.20)
F o r each finite s e t
5 . We n e r e a s s u m e tc L e m m a
L t h e r e is a n u m b e r s, s . t .
(5.19) a n d ( 5 . 2 0 )
10 ( L e c t u r e 4) t h e r e is a n u m b e r
. Let
b e k = 2e+l. According
u s.t.
L e t be
L is f i n i t e
. By
(5.20) we h a v e a n u m b e r
s
s. t .
(*+j
.
We now c h e c k t h e
- 29
H. H e r m e s
a s s u m p t i o n s of L e m m a 1 1 ( L e c t u r e 4). ( 1 ) g ( s ) = 2 e + l = k. ( 2 ) Hss
-1 /
H e n c e F' re
u
by (5. 18). (3) S i n c e g ( s ) = 0 we h a v e s S
If s < u then
0.
L fI F 3 F
< s. (4) is i d e n t i c a l w i t h
D r s , i. e .
H
r
fI F
s
(++). H e n c e 7 D r s . q r k and that
#
0, we would h a v e
- NOW L e m m a
s)
> 0 and --C a s e 1.
= 0 corltradicting
(5.21). ( 5 )
Let
L fI F
S
be
#
w). T h e r e f o -
r 5 u. 1:' we a s s u m e
0 in c o n t r a d i c t i o n t o
11 shows that t h e r e i s a number
r s. t.
~ H r m for each m.
F o r e a s i e r reference we rename former
S
F's by (4X),
r by
s (and f o r g e t about
the
.
H e n c e w e have : (5. 23)
a n d ~ H s mfor each
@sk
B y (5.23) we h a v e c e by (5. 11)
g(s) = k
g(s) = 2e(s)
= 2e+l
+ &
rn.
{ 0. We t h e r e f o r e h a v e
(s)+l. That gives
E ( s ) = 0,
C a s e 1. Hene ( s ) = e and
by ( 5 . 9 )
where ber y
r ( s ) i s determin2d a s
in
( 5 . 9 ) . A c c o r d i n g t o (*)
we h a v e a num-
s. t.
(5.25)
r(s)
5
1 s , T f0
s , y L
e per(S)y, U(y) = 1
S
H e r e we m a y r e p l a c e -------
f0
s
by -
that in o r d e r to check wh?ther
fO: In L e c t u r e 3 , no. 2 we h a v e m e n t i o n e d
T
1
zxy holds o r n ~ t ,we h a v e l o a s k t h e g o r a c l e f o r g only f o r a r g u m e n t s which a r e l e s s t h a n y . In o u r c a s e we
y < s . H e n c e i t i s sufficient t o show t h a t m e values for each x <s. ------have
(a)
0
If f ( x ) = 0 t h e n we h a v e x
e AUs'
f0
S
and
-
f0
x e A O , cO(x) = 0
h a-v e -
.
the sa-
H. H e r m e s (b)
If
fO(x) = 0
n, 2x B T n wise
let
by be
we would h a v e
0
then
xe A , x
A0 n ( 5 . 2 ) . We want t o show t h a t
0
fs(x) = 1. H e n c e
H' nTn-
4 0,
X ~ A ; . Then
i. e . Hsn-
for a certain number 0
fs(x) 2x
H
= 0, if x S
-< s.
Other-
b y ( 5 . 1 0 ) . But t h e n
c o n t r a d i c t i n g (5. 23).
-
Hence we
have: (5.26)
r(s)
5
s.
Now i n (a) w e d e r i v e fl(p:(s))
y
5 s,
T
1
f0
e p:")
y, ~ ( y =) 1
= 0, a n d i n (b) fl(pr(s))
( a ) We h a v e Fs ( f r o m $sk.
(5.23)), i. r . F'
.
= 1.
C Ts,
hence
2 ~ ' ( ~ ) +1C T s by (5.24). pe e '(') A's+l (5.21, p:(S) A', (b) F r o m ( 5 . 2 6 ) we get u(J.LyT1 e p;(s) y ) = 1 , h e n c e f0 f ( p S ) = 1 b y (5. 19)
fl(~~~(~))=0.
.
6. We no!-assume
L
a finite s e t (*it+)
Let be
m
( 5 . 1 9 andetle_-n_etaJ&n_ of ( 5 . 2 0 ) . s. t . s(s
> o ~ g ( s )= 2 e + l 4 ~ 1 s+ L ~ F ' # 0)
.
m 2p e + l
is g r e a t e r t h e n e v e r y e l e m e n t of L. m ( c ) t h a t t h e a s s u m p t i o n p e~ A' l e a d s t o a c o n t r a d i c t i o n a n d
a number s . t .
We d e r i v e i n
In t h i s c a s e we h a v e
m pe E A 1 a l s o l e a d s t o a c o n t r a d i c t i o n . m ( c ) We a s s u m e t h a t p e e n 1 . H e n c e t h e r e i s a n u m b e r s 1 s . t .
i n (d) t h a t t h e a s s u m p t i o n
H. Hermes
p
m
1
.
6 As
s > 0, s . t .
number
Since T = 0. t h e r e is a 0 1 S m ~ E T ~ - T ~ - THence ~ - ~ . F ={zpe + l ? This
F r o m this we get
1
2p
m
shows that we have Case Now we have
s
>
+
2pe +i,sTs
.
l(no. 2),
and that
.€
( s ) = 0, e ( s ) = 0, g(s) = 2e+l. s 0, g(s) = 2e+l, 7 F ' s (because-F C Ts,,). Applying
w)
m
2 p . +1, the only element we get L Il FS f 0. But this cannot be since e s of F , i s not an element of L according. to the choice of m. d)
We a s s u m e that
pyl.
Using (5.19) we get
e
s. t.
1
e pmy and U(y) = 1. There i s a sefO quence of configurations Co, , Cn) Going through , Cn, s.f. y = G ( Co, this sequence, the T M with Gbdel number e has only a finite number Hence we have a number y
T
.. .
of question
about the value of
a r e also given by
0
.
f
0
. For
.
sufficiently l a r g e
Hence in (5.27) we may replace f s ficiently large s. We may a s s u m e in addition that (5.28)
f
. ..
6(s)=0, e(s)=e,
Now we get from
m
5
0
-s these values 0
by fs
with suf-
s , y L s.
(5.27) :
This shows together with (5.28) that we nave Case 1 (no.2). Therefore thes + 1 = {2pr's)+1j, g(s)= r e exists a number r ( s ) 2 m, s. t . F e = 12pe(s) = 2e+l. We will show in
a moment that - I F ' S . Now applying (ff*Y) we get
2pr(s)+l, only element of F', cannot be an element of e F~ , cannot be a n element of L according to the choice of m , since
L~F'
0.
But
r ( s ) 1 m;
It remains to show that F ' s . Otherwise F
S
C
T
s- 1'
hence
H. H e r m e s
.
'('I + I , g T s - l Hence t h e r e is a n u m b e r q s. t. Z ~ ' ( ~ ) + ~ E - TT '"e e 9 q-1' T h i s Leads t o .rq= )2p:(s)+l] Helice w e h a v e C a s e 1. w h ~ r e F q =
.
-
.
( q ) + ~ &(q)] B y c o m p a r i s o n we g e t 8 (q) = 1. r ( q ) = r ( s ) w h e n c e i 2 p e (q) (i. e . - - t P q c T g(q) = 2 e + l . Now we h a v e q .> 0 , g(q) = 2 e + l:lF1q ). q- 1 Applying (*k4 we get L n F~ # 0. But 2 ~ : ( ~ ) + 1 , f h e o n l y e l e m e n t of F~ , i s not arl
e l e m e n t of
L.
R E F E R -----E N C E-S : S e e L e c t u r e 4. ------
n c-t i-on? -L -e c-t u- r-e- -\'i;--.-P-r i-m-i-t i-v-e -R-e-c-u-r s-i -v e- -F-u1. In L e c t u r e 2 ,
no. I
the c l a s s
-P
of p v i m i t i v e r e c u r s i \ : r t'crn-
c t i o n s h a s been i n t r o d u c e d a s t h e s m a l l e s t c l a s s of ( t o t a l ) functions t o which 1 belong S a n d e v e r y vn and w h i c h is c l o s e d u n d e r s u b s t i t u t i o n a n d C: p r i m i t i v e r e c u r s i o n . T h i s c l a s s i s a p r o p e r s u b c l a s s of t h e c l a s s o f a l l
.
r e c u r s i v e t o t a l f u n c t i o n s , a s h a s b e e n shown by A c k e r m a n n : t h e function d e fined by
H. Hermes
i s total and recursive, but not primitive recursive. The c l a s s of recursive functions i s not an a r b i t r a r y mathematical construction but has an llabsolutenmeaning, because it coincides with the c l a s s of computable functions. It i s not known whether the class of primitive recursive functions i s also *absolute11 in a s i m i l a r sense. The definition of P
depends
on the c l a s s of natural numbers a s the underlying do-
D of manipulable
main
ning of primitive
objects.
We may a s k ourselves: what i s the mea-
recursivity for other c l a s s e s
-P
2. We may consider
C
of manipulable objects?
a s being generated by two different kinds
of operations: (1)
independent of the pecularities of N: To these open n rations belong the substitution, and every U. (a function U . may be considered a s a
Operations
0 - a r y operation )
.
(2)
Operations depending on the pecularities of N: Here we have thf 0 1 0-ary operations C and S and the primitive recursion, which i s a bi0
nary
operation leading from functions
function
f
(cf. Lecture 2, no. 1)
g, h to a a functions g, h
to a
.
N may be considered a s arl absolute f r e e algebra of type < 0, 1 >, 1 i s the 0-ary operation and S the unary operation. This inwhere C: dicates immediately that
0
Co and S'
a r e dependent on
N; the s a m e holds
for the
primitive recursion where in the f i r s t equation occurs 1 in the second equation S
.
C:
. and
3 . How can we transfer: the notion of primitive recursivity of another
domain D
D of manipulable objects ? Let be g a
onto N.
longsn
to
~ s d e l i z a t i o nwhich maps
Then we can associate with each (total) function N a s its
g-transform-
a function
fg
which
f which Itbenbelongstt to D,
H. Hermes
by the following definition :
.
We define
P a s the c l a s s of the g-transforms of the elements of P_ If -g we choose different GBdelizations g l , g2, the c l a s s e s P_gl and P may -g2 differ from each other.
On the other hand we may use the processes (1) and imitate the p r o c e s s e s (2) (cf. no. 2) in a natural way, in order to get a class
P-D
of
functions which henceforth may be considered a s genuine primitive recursive functions. We want to s a y that we have a satisfactor cept
of primitive recursivity for
y
D if it i s possible to introduce
a natural way and if there is a Godelization g s . t .
4.
class
As a f i r s t example for a satisfactory
of words over
solution for the con-
a finite alphabet
ED in
P = P -g -D'
solution we mention the
{ ao, . . . ,a N l
(including the empty
. This < 0,1,. . .,
word E)
c l a s s may be considered a s an absolute free algebra of 1 1 type 1>, where E i s the 0-ary operation, and So ' ' . . ' s N 1 a r e unary operations, defined by S .(w) = wa. (concatenation). In o r d e r to J E, So , , S; and define priimitate (0) we now select the functions
1'
...
mitive recursion by the following N+2 equations:
A s s e r has proved that. we have a satisfactory solution by indicating an appropriate GBdelization.
H. H e r m e s 5. T h e e x p r e s s i o n s of
m a y be considered a s an have t h e
expression
defined by
Fichts
m i n i m a l l o g i c (cf. L e c t u r e 1, no. 6)
absolute f r e e algebra as
of type <0,2>, w h e r e we
t h e 0 - a r y o p e r a t i o n a n d a b i n a r y o p e r a t i o n J,
J(x, y ) = (xy). H e r e w e h a v e t h e following p r i m i t i v e r e c u r s i o n :
In a s i m i l a r way w e m a y p r o c e e d i n the c a s e of a n a r b i t r a r y a b s o l u t e f r e e Mahn h a s 6.
.
n >, w h e r e n a r e n a t u r a l n u m b e r s ( j = 1,. . ,r ) . 1'"" r j shown t h a t we have a s a t i s f a c t o r y solution in a l l t h e s e c a s e s .
a l g e b r a of type
T h e r e a r e d o m a i n s of manipulable o b j e c t s whose e l e m e n t s i n
a
n a t u r a l way m a y b e c o n s i d e r e d a s finitely g e n e r a t e d but g e n e r a l l y not in a unique way.
We c o n s i d e r two e x a m p l e s :
(a) T h e d o m a i n operation
0
of
(empty s e t )
the binary operation
U
0
and t h e u n a r y o p e r a t i o n
{
(unit s e t )
and
of non-emptv w o r d s o v e r a finite alphabet
is g e n e r a t e d by t h e 0 - a r y o p e r a t i o n s
m e n t of t h e alphabet)
5
0-ary
(union).
(b) T h e d o m a i n a ,...,aN
"totally finite s e t s n i s g e n e r a t e d b y t h e
and t h e b i n a r y operation
0
a , 0
. .. , a N
(the e l e -
(concatenation).
In t h e s e e x a m p l e s we have a dificulty with t h e p r i m i t i v e r e c u r s i o n . T h e n a t u r a l way t o d e f i n e t h i s o p e r a t i o n e . g . following t h r e e equations:
in c a s e (a) would b e b y t h e
H. H e r m e s
T h i s o p e r a t i o n p r o d u c e s a function f out of given f u n c t i o n s if --
g, h, k, k u t only
(6.9), (6.10). (6.11) a r e not c o n t r a d i c t o r y . Hence i n g e n e r a l t h e p r o c e s s
of p r i m i t i v e r e c u r s i o n is only defined i n t h e c a s e w h e r e t h e " n a t u r a l e q u a t i o n s t f d o not involve
a contradiction.
Rodding h a s s h o w n that we h a v e a s a t i s f a c t o r y s o l u t i o n in c a s e ( a ) . It is unknown to t h e a u t h o r w h e t h e r we h a v e a s a t i s f a c t o r y s o l u t i o n i n c a s e (b). 7.
Tile next p a r a g r a p h s include t h e d i s c u s s i o n of s o m e h i e r a r c h i e s .
A ------h i c . ~ . a r c h yf o r a c l a s s j the c l a s s
13.. -1
1I.
-J
(Sometimes
H is a s e q u e n c e
i s a proper subclass J
runs over
a s e g m e n t of
of t h e b e s t known h i e r a r c h i e s
;.e f u n c t i o n s o v e r t h e d o m a i n This
F. J
.. . ,
S.
t. f o r e a c h
is t h e union of a l l
l l c o m p u t a b l e l to r d i n a l s ) .
One
E f o r t h e c l a s s P_ of p r i m i t i v e r e c u r s l -n of n a t u r a l n u m b e r s is d u e t o G r z e g o r c z y k .
h i e r a r c h y according to Ritchie
we introdlice f u n c t i o n s
of
El, _HZ, gj+la n d H _Ho,
m a y b e i n t r o d u c e d a s follows:
First
by postulating:
E of t h e G r z e g o r c z y k h i e r a r c h y is t h e s m a l l e s t c l a s s -n a n d which is c l o s e d u n d e r which c o n t a i n s t h e f u n c t i o n s c:. s', u:, F n Now t h e n ' t h
class
substitution and r e s t r i c t e d primitive r e c u r s i o n .
T h e o p e r a t i o n of r e s t r i c t e d
H. H e r m e s
p r i m i t i v e r e c u r s i o n l e a d s f r o m t h r e e functions g, h, j
t o a function
f iff
t h e following t h r e e equations hold:
E c o i n c i d e s with t h e c l a s s of e l e m e n t a r y functions which h a s b e e n -3 introduced by KalmBr-Czillag a s t h e s m a l l e s t c l a s s which c o n t a i n s t h e o n functions U . s u m , difference, product, x (greatest natural Co' 1 number < x ) y ( g r e a t e s t n a t u r a l n u m b e r 5 x y if y { 0, 0 o t h e r w i s e ) ,
1 1 yI
?
se),
and is c l o s e d u n d e r
s u b s t i t u t i o n and t h e o p e r a t i o n s
and
n.
0
8. T h e
Grzgorczyk hierarchy for P -
0
may s e e m somewhat artificial
b e c a u s e i t depends o n t h e p r o c e s s of r e s t r i c t e d p r i m i t i v e r e c u r s i o n . But it is p r a c t i c a l l y i d e n t i c a l with a n o t h e r h i e r a r c h y
introduced by Heinermann and 0 1 which contains Co , S, e v e r y longing t o
R-n- I substitution. If n
A x t. U:
En
R f o r P which h a s b e e n -n is t h e s m a l l e s t c l a s s of functions,
-
e v e r y f obtainable f r o m functions be-
b y one - p r i m i t i v e r e c u r s i o n , a n d which is c l o s e d u n d e r is g r e a t enough we h a v e R
ved f o r
n > 6 by Meyer,
Ritchie
and
for
for
= This has been pro-n %+ 1. n > 5 by K. ROdding, f o r n > 4 b y D. M.
-
n -> 3 by Scwichtenberg.
9. R. W. R i t c h i e h a s p r e s e n t e d a h i e r a r c h y--F f o r the c l a s s of e l e m e n n 0 t a r y functions ( s e e no. 8) _F c a n b e t a k e n a s t h e c l a s s of a l l l i n e a r func-
.
tions
( o r e l s e a s t h e c l a s s of 611 t o t a l functions which a r e I1cornputable by n+ 1 finite a u t o m a t a n ) . A function b e l o n g s t o F iff it i s total and r e c u r s i v e n s. t t h e a m o u n t h of t a p e which is u s e d f o r the a n d if t h e r e i s a function g E F computation of f(xl,
. . . ,xn)
is bounded by
g(xl,
. . . , Xn) .
H. Hermes
Re fe r e nce s: ----------
[I]
ill.
Asser
(primitive r e c u r s i v e functions over different
Grzegorczyk [I) R. W.
Ritchie
, Heinermann
il] , [2]
Lz] , RBdding domains). Axt fi] ,
Hermes 127 , Mahn r l ] , P e t e r
[ly
, Meyer I11 , D.M.
, Schwichtenberg
111
(hierarchies)
Ritchie [l]
,
.
L e c t u r e V I I ; The Domino Problem ........................... 1. The proof of the Ifundecidability of the domino problem" i s a ty-
pical example f o r the application of the theory of r e c u r s i v e functions. The problem (cf. no. 3) h a s the advantage that it does not presuppose the knowledge of a mathematical theory. The fact that a special f o r m of the domino problem i s unsolvable has been related to the Decision P r o b l e m
(~fEntscheidungsproblem") of the f i r s t o r d e r logic calculus. In this way it has been shown that ble (Lecture 8) problem for
.
the
AvA-
c a s e of this Decision P r o b l e m i s
unsolva-
In no. 2 we show a s a preliminary s t e p that the halting
T Ms
i s unsolvable.
2. The halting problem for Turing machines. compare Lecture
2, no. 2.)
( F o r the terminology
With r e g a r d to the intended application it
is convenient to use not extended
T M s with
a one-way infinite tape going
to the right. Hence the tape has an initial square. If we have a configuration
C=(I, s , q) where s is the initial s q u a r e , and if the quadruple q s l q f is an ele ment of
M
we have to a l t e r the general prescription of Lecture 2, no. 2
by postulating that
C
i s terminal.
H. Hermes
M
We say that
Kt!
if M, applied on the void inscription in the
initial square, stops after a finite number of steps. We say that M, applied on the number z, h_aKs_ if Pvl,
starting with
C:
, stops after a finite
number of steps. We want t o show briefly that the halting problem for cidable (i.e. that the s e t of TMs where M
TMs i s unde-
halts, is not decidable).
ad absurdum and assume that the halting problem i s
proceed by reductio
decidable. Then it a l s o would be decidable whether
M, applied on G(M)
(GSdel number of M, under a given ~Edelizatinn), does not halt t h e r e is (7.1)
a TM
We.
.
Hence
Mo, s . t .
F o r every
M: M
0 '
applied on G(M), halts iff
M, applied on G(M),
does not halt. This leads immediately to a contradiction for
M = Mo
.
3. Domino problems. A domino is an oriented square plate with colors
on each edge.
A
d ~ m i n omust not be rotated o r reflected. Two
dominoes a r e of the s a m e --type
if the corresponding edges have the s a m e
colors. (We sometimes confound dominoes with their types). Let be given a finite s e t D
D i s called -good if it i s possible
of domino types.
fill the plane with dominoes who (i.e. whose types) belong to
D , , in such
a way that adjacent edges have the s a m e colors (matching condition)
Fig. 1)
.
i
Fig. 1
to (see
H. Hermes In the origin
- restricted
domino problem
domino types. The p a i r
D,D
0
is given a finite s e t
DO
of
is called good if it is possible to fill
the (first) quadrant with dominoes belonging to c o r n e r domino belongs to
DO
D in such a way that the
and the macthing condition i s satisfied.
If in this definition we replace the c o r n e r by the
(main) diagonal we get
the diagonal-restricted domino problem. In each of the domino problems and the diagonal-restricted) for
(the general, the origin-restricted
it may be questioned whether the property
0
D (resp. D, D ) of being good is decidable. In each c a s e we have a
negative answer, which was given by Wang
(1961) for the origin-restricted
case, by Kahr, Moore and Wang (1962) f o r the diagonal-restricted c a s e by B e r g e r (1966) f o r the general c a s e
. We
and
want to give the proof f o r unde-
cidability in the origin-restricted c a s e in this lecture. In Lecture 8 s o m e r e m a r k s a r e included about the diagonal-restricted case, which
is not s o
easy. Even m o r e complicated i s the general case. 4. We want t o associate with each (one-way) TM M a finite s e t DM of
domino types and a (one-element) subset D
0
M
of DM s . t .
the fol-
lowing holds : 0
(7.2) M does not halt iff DM, DM is good (for the origin-restricted c a s e ) . F r o m (7.2) we get immediately that the property of being good is undecidable. Let be given a o r d e r t o describe sidered 0.1,2
TM M (by i t s l e t t e r s , s t a t e s and its table). In
DM,
DL
we may suppose that the s t a t e s of the con-
TMs a r e given by numbers
O,1,2,.
.. and the
l e t t e r s by numbers
,... . We have the following colors: a, qa, qR, qL, qaR,
qaL,
W, I, V and H ,
H. H e r m e s
w h e r e a r u n s through the l e t t e r s ,
q through t h e s t a t e s and R ( " r i g h t n ) ,
L("leftn), W(nwestn), S(llsouthn), I(tlinitial",
V(nvoidtt) and H(llhorizontaln)
a r e s p e c i a l s y m b o l s . Using t h e t a b l e of M we introduce a function
C
=
qa = C(q, a ) (ltcolor functionl1) whose v a l u e s f o r given a r g u m e n t s q, a a r e c o l o r s
defined a s follows:
V ,
T o the s e t
0
L7
M
if
n o quadruple beginning with q a is i n M
belongs only one type, given by
Fig. -
( H e r e W, I, S. W
a r e t h e c o l o r s and
(K) ( t t K o r n e r t t )i s the n a m e of t h e do-
mino. S i m i l a r l y f o r the o t h e r t y p e s t o b e defined T o the s e t
2
subsequently).
DM Selong ( b e s i d e s (K)) t h e following types ( w h e r e again
a r u n s through t h e l e t t e r s and q
through
the s t a t e s :
H. H e r m e s
Fig. 3 5. Hence
re C .
We a s s u m e t h a t
we h a v e a n infinite s e q u e n c e of c o n f i g u r a t i o n s I
3+1 = 1, 2, 3,
C
P r o o f of (7.2), F i r s t P a r t .
= C
J
. . .)
and
C
C1, C2,.. ., whe-
Co' = (void i n s c r i p t i o n , f i r s t s q u a r e , 0). L e t b e B. ( j =
J
0
the behavior
( P ( " p r i n t ") o r R o r L ) which l e a d s f r o m C.
J- 1
. We s h o w
j with
M d o e s not h a l t .
to
that D is good by defining a filing of t h e q u a d r a n t n, D L (K) in t h e c o r n e r a n d m a t c h i n g c o n d i t i o n s s a t i s f i e d by defining h o r i z o ~ 1 -
tal s t r i p s
of d o m i n o e s
So) SC,Bl~.sC ' ' ' . 2 2
which a r e l a i d down o n e
H. H e r m e s
a b o v e t h e o t h e r a c c o r d i n g t o -Fig. 4 .
Fig. 4 -
Let be
... .
a . .; q a n a r b i t r a r y configuration (where a a is o n o 1 "' t h e i n s c r i p t i o n , n . t h e n u m b e r of t h e s c a n n e d s q u a r e a n d q t h e s t a t e ) . Let be
C = a
Now we define
( ( 7 . 6 ) is d e f i n e d only if
n
> 0.) Now a l l t h e c o n d i t i o n s a r e s a t i -
s f i e d : In t h e c o r n e r we h a v e d o m i n o (K). E v e r y
s t r i p satisfies the hori-
z o n t a l m a t c h i n g condition. T h a t a l s o t h e t h e v e r t i c a l m a t c h i n g condition i s s a t i s f i e d m u s t b e shown f o r S
.
c j + ~Bj+l tion, t h a t S the
S
and S
and for CIBl 'C.B F o r t h e induction s t e p it i s c o n v e n i e n t t o u s e t h i 0
and observa-
and S h a v e t h e s a m e u p p e r c o l o r s . We l e a v e C P ' 'CR CL details to. the r e a d e r .
H. H e r m e s
6.
Proof -
good. H e n c e to
of (7.2Q~cond
--P a r t .
we h a v e a c o r r e c t filling
0
D M' DM
We a s s u m e t h a t
F
is
of t h e q u a d r a n t . We now want
s h o w by induction o n t h e n a t u r a l n u m b e r j : (1) M,
a p p l i e d o n t h e void
at least j (2) T h e
inscription
in the initial square, p e r f o r m s
steps.
j t h ( h o r i z o n t a l ) s t r i p of
F
i s uniquely d e t e r m i n e d
.
C . = a ... a q i s t h e c o n f i g u r a t i o n which w e h a v e a t e r J o n"' t h e j - th s t e p , t h e u p p e r c o l o r s of t h e j- th s t r i p of F a r e
(3) If
We f i r s t ---------(K).
a s s u m e t h a t j = 0. (1) is t r i v i a l . (2): We h a v e t o s t a r t with
T h e r i g h t c o l o r of (K) is I. T h e only d o m i n o which h a s
c o l o r is (I). Hence t h i s way of
(K) m u s t
that the initial s t r i p
h a v e (I) a s i t s r i g h t of
F
I a s i t s left
n e i g h b o r . We find i n
i s So. We v e r i f y
( 3 ) by i n s p e c t i o n
So. We now assume -
t h a t ( I ) , (2), ( 3 L h o l d f o r j
--
. Let be
d e t e r m i n e s t h e next b e h a v i o r of M. In o r d e r t o s h o w n 1, w e h a v e t o exlude t w o c a s e s , n a m e l y t h a t
Now t h e p a i r q, a (1) f o r
j
+
(i) t h e is no
q u a d r u p l e in
M
beginning with
( i i ) n = 0 a n d t h e r e is a q u a d r u p l e in
hn
qa,,
and that
of t h e f o r m
qa Lq' n
(then C . would b e t e r m i n a l a c c o r d i n g t o no. 2). Exclusionof of t h e
jt th
J
(i) : A c c o r d i n g t o t h e induction h y p o t h e s i s (3) t h e r e i s a d o m i n o s t r i p of
t o (7. 3) w e h a v e
C
qan s t r i p with lower color
F
which
= V.
V.
In F
has
a s its upper color. According qan t h e r e m u s t be a d o m i n o i n t h e ( j + l ) s t C
But t h e r e i s not d o m i n o a t a l l with
lower
H. Hermes
color V. Exclusio*
(ii);
Assume (ii). According to the induction hypothesis (3) the
.
second domino 3f the jp!h s t r i p has C
a s i t s upper color According ¶a Hence t f e second domino of the (j+l)st
= q f a L. 0 qao L a s its lower color. Inspection of Fig. 2 and Fig. 3
to (7.3) we have C strip
has q'a
0
shows that this domino i s c e qL of F.
must be the right color of
lqa'
LJ i s
of the jth s t r i p must
be
a* a*
qL
as
of the f i r s t domino in
. Hence
. But
its left color. Henthe (j+l) th s t r i p
>a* L; (where a* i s open). The
We find that this domino must be
lower color is
(qaoL). (qaoL) has
the upper color of the f i r s t domino
by induction hypothesis (3) this color
W. This finishes the proof of (1) for
j+l s t e p s
.
performs a t
least
In o r d e r to show (2) and (3) for h+l we have t o distinguish three
c a s e s whether the quadruple of M printing
j+1. Hence M
which
o r a going t o the right o r a
begins with qa describes a
going t o the left. In each c a s e we
find , starting with the f i r s t domino of the
(j+l; s t s t r i p and going to the
F
i s uniquely determined by the
right, that each domina of this s t r i p of matching conditions. In this way we show to
(2) and (3). The details a r e left
the reader.
R e f e r e n c e--s: --------
Wang
ilf .
111 Buchi
, Kahr-Moore-Wang 117 , Rerger
H. Hermes
Lecture
AVA - C a s e o f t h e D e c i s i o n P r o b l e m of P r e d i -
8:
c a t e -- C a l c u l u s . 1. A formula
y
of predicate calculus i s called satisfiable if it has
a model. It has been proved by Church (1936) that there i s no algorithm to check the
satisfiability for a r b i t r a r y formulae (undecidability of the Deci-
sion Problem).
One may be interested in the s a m e question, where
tJ)
i s restricted to a subclass A of the c l a s s of all formulae. Among these subclasses those a r e of special interest which may be described by a prefix.
E.
2.
t b r e belong to the subclass AAv a l l I\x
ten in the form x, y , z
/\ y /\
I+ , where
z
i s quantor-free
may be writ-
and has a t most
a s f r e e variables. F o r some of these subclasses satisfiability is de-
.
cidable, e. g. for V . .VA/\V. e.g.
I.+
formulae which
for AAV.!.
.. .
F o r others satisfiability is undecidable,
F o r a long time the c a s e AV/7 was
before the y e a r 1962 that Kahr, Moore and
Wang have proved that satisfia-
bility for/\VA is undecidable. They connected this 0
D, D
ty whether
unsettled. It was not
fact with the undecidabili-
is good for the diagonal -restricted domino problem.
E a r l i e r (1961) ~ i i c h iwith a method which he related t o the origin-restricted domino problem has shown that satisfiability for the
qfi A \/A
-cas.e is
undecidable. In this lecture we want to show by reduction to the resul of Lecture 7
that satisfiability f o r the class
t e how the
I\VA P,
which
i s undecidable and to indica-
s a formula
(8. 1)
s: P s
iff
_O of manipulable objects and an undecida-
is defined for those objects. Let us assume that
to each object F o r all
VAv
-case may be settled.
2. Let be given a class ble property
J'A
is associated in an effective way,
(Qs i s satisfiable.
s. t.
H. Hermes
theorem that satisfiability i s unde-
This would be a proof of Church's
cidable. Furthermore it could be that for every s a
subclass
A
of the c l a s s of a l l formulae. Then (8.1)
the theorem that a l s o for A In the following we domino s e t s D, and P
0
D, D
choose 0
where D
O_
has exactly one element and i s included in
a s the property of beidg good for the origin-restricted c a s e , In shown that
is undecidable.
P 0
D, D a formula
YD, DO we a r e interested i n ) , s. t.:
yet the definitive formula 0
D, D is good for the origin-restricted c a s e iff
(8.2)
would lead to
a s the c l a s s of all p a i r s of (finite)
We want to associate with e v e r y not
ys belongs
satisfiability i s undecidable.
the preceding lecture we have
is
the formula
' yD, Do
(which
i s sati-
sfiable D,
DO
is defined a s the conjunction of formulae (8.3),
We a s s u m e that
D = idl,.
the dominoes a r e numerated given by
9.5:
..,dN],
cl,..
., c M
DO
= [dl]
. . .,( 8.9).
, that the r u l o r s of
and that the colors of
dn
are
Fig. 5
The formciIae (8.3),
. . . ,(8.9)
a r e built up using the binary predicate sym-
H. Hermes bols
Dl,.
bols
Z
. .,DN,
C
1
1 . .., 54 ,..., CM ,...,
4 CM , the unary predicate sym-
and a unary function symbol f. These formulae b e a r names whicl:
a r e explained in no. 3. Unicity condition f o r dominoes: .a\xA y v'!n Dn x y (V!nDn x y
is a n abbreviation f o r (D 1 xy y..
.
y DNxy)./\~(Dlxy~DZxy)
A t ( D 1 x y A D 3 ~ y ) . A . ..n - \ ( D N - l ~ y A DNxy). Hence V!nD nxy is quntor-free. vln..
may be read: there is exactly one n, s . t . .
..)
Unicity condition for colors: AX/\ y
1
~ mCm ! x
y
... ~
4 -~,4xAy~!m~~xy.
Domino color relations (cf. Fig. 5) : 1
A X A ~ ( D ~ X Y . xy + CA ~ ...
nc;
xy)
(n=l..
1 4 (Here we have N formulae. )
(8. 6)
Horizontal matching condition:
(8.7)
Vertical matching condition : 1 Ax Ay(Cm
(8.8)
XY
3 . 3 Cm xf(y))
Origin Condition A x Ay(Zx A Z y
(8.9)
Existence of Z e r o
--t D1xy)
.
. ..M
(m=l,
...,M)
H. Hermes 3. Proof
want t o show that (8.9))
D, Do is good.
of (8.2). F i r s t P a r t . We assume that
YD,Do
'
(i.e. the conjunction of the formulae (8. 3),
i s satisfiable. We get a model for
Cnoose a filling F
We
of the quadrant
.. .,
in the following way : D, DO which meets all conditinns for the
'Y
origin-restricted case. Numerate the squares according to
2.6:
Fig. -6
t
Take the s e t of natural numbers a s the individual domain w . Related to w 1 4 N), C m , . Cm (m = 1, M) we now give to the symbols f, D (n = 1 , . n a meaning by the following stipulations:
. .,
. .,
...,
f(x) = s u c c e s s o r of x.
zx -
iff
-Dnxy iff
(8.12)
cP xy
(8. 13)
-m
x i s the zero.
in
F on square (x, y) we have domino
(p=l,.
. ., 4 )
the edges s e e
.. . , (8.13)
.
iff in F on s q u a r e (x, y) on edge p we have color
Under (8. lo),
dn
all formulae (8.3),
(for the numerating oi F A . 5).
. . .,(8.9) a r e
exactly what i s indicated by their names. Hence
7'
f D, Do
valid: they express is satisfiable.
i s satisfia+D. DO D, Do is good. Let be given a (non-empty, pos-
4. Proof of18.2). Second P a r t . We a s s u m e that
ble and want to show that
and a model of YD,DO over w. r etc. the meaning associated with the predicate symbol D n'
sible finite) individual domain w Let be
g,;. . .
' '
by the model. (8.9) guarantees that there i s an element
z r 0
w s. t .
_Z z 0' Take one
H. H e r m e s
of t h o s e a n d c a l l it z
0
o b e -f ( z0 ) = z 0
. Let
' -
f
;+I '
i ( z0 )=f -(f ( z )) (i=O, 1, 2,
We now i n d i c a t e a filling F of t h e q u a d r a n t b y d o m i n o e s d
. . . ). AN
(which d e p e n d s o n t h e c h o s e n m o d e l a n d o n z ): L e t b e (i, j) a n a r i b t r a r y 0
. . ). S i n c e o u r m o d e l s a t i s f i e s (8. 3) t h e (n = 1, . . ., N) s. t . D fi(z ) f J ( z ). We now defi-n-- 0 - 0
s q u a r e of t h e q u a d r a n t (i, j=O, 1 , 2 , . r e is e x a c t l y o n e n u m b e r ne
F
n
b y t h e s t i p u l a t i o n t o put d o m i n o d
It is now e a s y t o (8.8) g u a r a n t e e s
checfi t h a t
(with t h i s n) n F m e e t s e v e r y condition:
o n s q u a r e (i, j). T h e v a l i d i t y of
. From dl w e i n f e r t h a t f o r e a c h i, j, p w e h a v e e x a c t l y o n e m
t h a t w e h a v e on (0, o ) t h e c o r n e r d o m i n o
t h e v a l i d i t y of (8.4) p i . s. t . C f (z )fJ(z ). T h e validity of ( 8 . 5 ) g u a r a n t e e s t h a t c is t h e c o l o r -m00 m of t h e p r t h e d g e of t h e d o m i n o d which i n F is o n s q u a r e (i, j).
n
F i n a l l y t h e validity of
(8. 6)
and ( 8 . 7 ) show t h a t t h e m a t c h i n g condition
is s a t i s f i e d .
5. -.. The formula we s e e
If i n ( 8 . 8 ) we i n t e r c h a n g e Y D , DO i m m e d i a t e l y t h a t t h e r e is a q u a n t o r - f r e e f o r m u l a
exactly three f r e e variables
(where
'Ix) U
x, u, y,
s. t .
is a s u b s t i t u t i o n o p e r a t o r )
TD,DO
. Now
x
and
y
with D, DO is equivalent t o
it is a well-known f a c t
("Skolem f u n c t i o n s f f )t h a t (8. 14) is s a t i s f i a b l e if a n d o n l y if
is s a t i s f i a b l e
. (8. 15)
c u l u s . We identify
(8. 16)
D, Do
(but not (8. 14) is a f o r m u l a of p u r e p r e d i c a t e c a l -
yD,
DO
with (8. 15). Now f r o m ( 8 . 2 )
i s good f o r t h e o r i g i n - r e s t r i c t e d c a s e
T h i s s h o w s t h a t s a t i s f i a b i l i t y f o r \Iv AV
/\
we get
iff s a t i s f i a b l e .
is undecidable.
H. H e r m e s
6. F i n a l r e m a r k s .
VA
i n (8. 15) (e. g.
would
l e a v e out
T h e r e i s no straightforward possibility t o omit V x Z x in A x V u 11 y
by including
3(D,Do).
If we
( 8 . 9 ) we would not b e a b l e t o c a r r y out t h e p r o o f
4, s i n c e we would not b e s u r e t h a t t h e c o r n e r - c o n d i t i o n is m e t .
i n no.
(Wecould
o m i t (8.8) and ( 8 . 9 ) a n d u s e t j r V y ~ l x y i n s t e a d . But t h e n a g a i n we would have existential quantifiers).
We c a n o v e r c o m e t h i s difficulty by r e p l a c i n g i n ( 8 . 2 ) t h e o r i g i n - r e s t r i c t e d c a s e by t h e diagonal r e s t r i c t e d c a s e s a r y condition t h a t f o r r e s t r i c t e d c a s e ) is elements, e.g.
D
0
.
In o r d e r . t o h a v e t h e n e c e s -
D, Do t h e p r o p e r t y of b e i n g good (in t h e d i a g o n a l -
undecidable, w e h a v e t o a d m i t , t h a t =
) dl,.
. ., d p ) . Then
D
0
has several
in p l a c e of ( 8 . 8 )
and (8.9)
.
w e c a n u s e t h e f o r m u l a A x ( D xxV. . VD xx) which d e s c r i b e s t h e d i a g o n a l 1 P condition. T h i s f o r m u l a c a n b e m e r g e d into A x A u A y CXDlD0 ( c f . ( 8 . 1 5 ) ) , w h i c h s h o w s t h a t s a t i s f i a b i l i t y f o r / l v / j i s undecidable. Of c o u r s e t h i s p r o o f p r e s u p p o s e s t h a t b e f o r e h a n d t h e d i a g o n a l - r e s t r i c t e d c a s e is t r e a t e d , w h i c h is, a s i n d i c a t e d i n no. restricted case. T o
conclude
I
1 , not a s e a s y a s t h e o r i g i n -
want t o m e n t i o n t h e e s s e n ! i a l i d e a s
( 2 ) which h a v e b e e n u s e d by K a h r , M o o r e and Wang (K-M-W) ( 1 ) In L e c t u r e 7, no.
(I),
in this case.
the jfth configuration C j C . b y the jlth diagonal
5 we have represented
b y t h e j t t h h o r i z o n t a l s t r i p . K-M-W
represent
( c o n s i s t i n g of t h e s q u a r e s (i, i+j], i = 0, 1, 2 , .
..) .
J
C. J i n t h e j t t h s t r i p . But t h e t o t a l i n f o r m a t i o n of t h i s i n s c r i p t i o n i s included e . g . i n t h e f i r s t 2j s q u a r e s of t h e [ape. K-M-W r e p r e s e n t only t h i s p a r t of t h e (2) In L e c t u r e
7
we h a v e r e p r e s e n t e d t h e t-o t a l i n s c r i p t i o n of
i n s c r i p t i o n in t h e jxth diagonal, References: See L e c t u r e
7.
but t h e y r e p e a t t h i s p a r t p e r i o d i c a l l y .
H. H e r m e s BIBLIOGRAPHY -------------A s s e r , G.
R e k u r s i v e Wortfunktionen . Z e i t s c h r . f. m a t h . L o g i k und G r u n d l a g e n d e r Math. 6 (1960, 358-278.
Axt, P.
I t e r a t i o n of P r i m i t i v e R e c u r s i o n . Z e i t s c h r . f. m a t h . L o g i k und G r u n d l a g e n d e r Math. 11 (1965). 253-255.
B e r g e r , R.
T h e Undecidability of t h e D o m i n o P r o b l e m . M e m o i r s of t h e A m e r . Math. Soc. 66 (1966). 7 2 . p p .
Buchi, J. R.
T u r i n g - M a c h i n e s and t h e E n t s c h e i d u n g s p r o b l e m . Math. Ann. -1 4 8 (1962), 201-213.
D a v i s , M.
Computability a n d Unsolvability. McGraw-Mill, New Y o r k (1958). XXV + 210 pp.
F r i e d b e r g , R. M.
Two R e c u r s i v e l y E n u m e r a b l e S e t s of I n c o m p a r a b l e D e g r e e s of Unsolvability. P r o c . Nat. Acad. S c i . USA 4 3 (1957), 236-238.
-
G r z e g o r c z y k , A.
S o m e C l a s s e s of R e c u r s i v e F u n c t i o n s , R o z p r a w y M a t e m a t y c z n e 4 (1953), 1 - 4 5
H e i n e r m a n n , W.
Untersuchungen bber die Rekursionszahlen r e k u r s i v e r Funktionen. D i s s e r t a t i o n M u n s t e r 1961. (Unpublished).
.
H e r m e s , H.
1 E n u m e r a b i l i t y , Decidability, Computability. S p r i n g e r ~ e i d e l b e r g t ~ eYwo r k (1965). IX+245 pp.
H e r m e s , H.
2
T h e Concept of C o n s t r u c t i v i t y . In: C o n s t r u c t i v e A s p e c t s of t h e F u n d a m e n t a l T h e o r e m of A l g e b r a . S y m p o s i u m IBM, Z u r i c h . Wiley a n d Sons . ( T o appear).
K a h r , A. S . , M o o r e E . F. 1 a n d H. Wang
Entscheidungsproblem reduced to the V w ~ a s e . P r o c . Nat. A c a d . S c i USA 4 8 (1962), 365-377.
Kleene, S. C.
1
I n t r o d u c t i o n t o M e t a m a t h e m a t i c s . North-Holland P u b l i s h i n g C o . , A m s t e r d a m (41964). X+550 p p .
Kleene, S. C.
2
M a t h e m a t i c a l L o g i c . John Wiley a n d Sons, New Y o r k (1967). XI11398 pp.
K l e e n e , S. C. a n d E . L . P o s t
1 T h e U p p e r S e m i - L a t t i c e of D e g r e e s of R e c u r s i v e Unsolvability. A n n a l s of Math. -52 (1954), 379-407.
L a c h l a n , A. H.
T h e p r i o r i t y Method I. Z e i t s c h r f. m a t h . L o g i k und G r u n d l a g e n d e r Math. 1 3 (1967), 1 - I ? .
1
.
H. H e r m e s M a h n , F. K.
P r i m i t i v - r e k u r s i v e Funktionen auf T e r m m e n g e n T o a p p e a r i n A r c h i v f. m a t h . Logik und Grundlagenforschung 1 2 (1969)
Markov, A.A.
T h e o r y of A l g o r i t h m s . I s r a e l P r o g r a m f o r Scientific T r a n s l a t i o n s , J e r u s a l e m (1962). 444 pp.
M e y e r , A. R .
Depth of N e s t i n g and t h e G r z e g o r c z y k H i e r a r c h y . Notices of t h e A m e r Math. Soc. 12 (1965), 342.
~ u g n i k ,A. A.
Solution of P o s t ' s Reduction P r o b l e m and S o m e O t h e r P r o b l e m s of t h e ' T h e o r y of A l g o r i t h m s I. A m e r . M a t l ~ . Soc. T r a n s l a t i o n s (2) 2 (1963), 197-215.
P e t e r , R.
R e k u r s i v e Funktionen. A k a d e m i e - V e r l a g , B e r l i n (21957). 278 pp.
P e t e r , R.
Uber die Verallgemeinerung d e r Theorie d e r redurs i v e n Funktionen f u r a b s t r a k t e Mengen g e e i g n e t e r S t r u k t u r a l s Definitionsbereiche. Acta Math. Hung. 12. - (1961), 271-314.
Post, E.L.
R e c u r s i v e l y E n u m e r a b l e S e t s of P o s i t i v e I n t e g e r s and t h e i r D e c i s i o n P r o b l e m s . Bull A m e r Math. Soc. 5 0 (1944), 284-316.
.
.
Ritchie, D. M.
Complexity Classification of P r i m i t i v e R e c u r s i v e F u n c t i o n s b y t h e i r Machine P r o g r a m s . Notices of t h e A m e r . Math. Soc. 2 (1965), 343.
Ritchie, R. W.
C l a s s e s of p r e d i c t a b l y C o m p u t a b l e F u n c t i o n s . T r a n s a c t i o n s A m e r . Math. Soc. 1 2 (InrjQ), 139- 173.
Ritchie, R. W.
C l a s s e s of R e c u r s i v e F u n c t i o n s B a s e d o n A c k e r m a n n ' s Function. P a c i f i c J o u r n a l of Math. 1 5 (1965), 1027-1041.
Rodding, D.
P r i m i t i v - r e k u r s i v e Funktionen Gber e i n e m B e r e i c h e n d l i ~ h e rMengen. A r c h i v f. math. Log. und Grundl a g e n f o r s c h u n g 10 (1967), 13-29.
R o g e r s Jr. H.
T h e o r y of R e c u r s i v e F u n c t i o n s a n d Effective Computability. McGraw-Hill, New Y o r k (1967). XIV+482 pp.
Sacks, G.E.
D e g r e e s of Unsolvability. P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , N. J. (1963). IX+174 pp.
H. H e r m e s
Schwichtenberg, H.
1
Rekursionszahlen und die Grzegorczyk-Hierarchie. To a p p e a r in Archiv f. math. Logik und GrundLagenforschung 1 2 (1969).
Shoenfield, J. R.
1
Mathematical Logic. Addison-Wesley Publ. Co., Reading M a s s . (1967). VIII + 344 pp.
Smullyan, R. M.
1
Theory of F o r m a l S y s t e m s . P r i n c e t o r n University P r e s s , P r i n c e t o n (21961), XI+147 pp.
Wang, F
1
Proving T h e o r e m s by P a t t e r n Recognition 11. Bell S y s t e m Technical J o u r n a l 40 (1961), 1-42.
-
-
C E N T R O INTERNAZIONALE MATEMATICO ESTIVO (C. I. M . E . )
Djurio
Kurepa
ON SEVERAL CONTINUUM HYPOTHESES
C o r s o t e n u t o a V a r e n n a ( C o m o ) dal 9 a1 1 7 s e t i e m b r e .
ON SEVERAL CONTINUUM HYPOTHESES by Djuro Kurepa (Beograd) 1. Continuum hypothesis. The classical Cantor's continuum hypothesis states that for infinite s e t
S
the cardinality of the s e t
PS
is the immediate follower of the cardinality
of a l l kS
every
the subsets of
of
S
S. i. e.
1.1. Continuum problem. The general continuum problem consists t o evaluate the number k P s for any transfinite s e t S
.
2. Conditionned c o n t i ~ u u mproblem. The conditioned continuum a s k s 2.1 F o r any s e t S and any set property
the following:
T, let
P,S
all the subsets of S , each possessing the property 2.2.
Determine the cardinality
2.3.
F o r what properties
kP$
of
Q
be the s e t of
n;
.
TC is
f o r some cardinal numer ( s ) x. F o r each S and each property yb it makes s e n s e to put t o preceding problem. 2.4
In particular , S might be a structured se, e . g. ordered, a graph, a
topblogical space, etc. and the property connected to the structured s e t
S
7C might b e a r also a character
.
2.5. Examples. 2.5.1. ( ) 1 $ks)'
If
5%
of all
means
"to be of cardinality 211, then P S denotes the system X
the 2-point-subsets of S; the cardinality of
and the question whether
(2) S equals
- 58 -
D': Kurepa for every
is linked t o the choice axiom, the identical relation x = x2
sfinite cardinality
.
2.5.2
If
x
family
then for any topological space
P%S: = F(S) of
connexion t o kF(S) one h a s
e.g.
t h e r this number is necessarily with
.
being equivalent to the choice axiom (A. T a r s k i )
m e a n s n to be closed",
one has the
tran-
all
F - s e t s of the space S
the problem of De Groot, a s
a 2-power, i. e. of the form
S
; in
to whe-
2X ;
help of the continuum hypothesis the problem is settled by affirmati-
ve for every regular accessible aleph. 3. Cantor's continuum hypothesis. 3.1.
In
1883 Cantor stated ([I]
( I ) 2kw0 = k i l ;
so
Hilbert published of H i l b d t
I s .
far
no t r a c e of
a uproof"
3.2.A system
C
provided in 3.3.
of (1) (Luzin)
of axioms the adjunction
C
itself
of
. K. ~ E d e lel, 2) proved that in s o m e
(1) will provoque no contradiction
succeded in
which the negation
of
(1) holds and another one
1963 t o exhibit a model
(1)
Consequently, the independence of (1)
a s e t theory in which
1925
t h e r e is no contradiction.
Finally, P. Cohen l l - 3 1 s e t s in
proof was found. In
(Luzin in particular). One started to
great s u c c e s s was when
of a theory of 3.4.
Cantor's
of (1) ; but, the doubts on the c o r r e c t n e s s
'proof w e r e published
doubt on the provability
, p. 192, 244) to have a proof for
is is
in
holding. proved and
which
one has
(1) does not
hold. 3.5. Function
N(q)
.
Now, if for e v e r y ordinal
and if we do not a s s u m e
O(
we write
N(o) = 1 , what is the value
of
N( q ) ?
By
D. Kurepa
a Zermelo-Fraenkel theorem the number
.
3.6.
peut w
0
-
en
Kurepa [52]
pour
N(o)
particulier,.
premier
on
w
to
p. 657 hint
N(o)
poser
N(o)
.
N(o)
is
any
the following:
nous croyons qulon
nlimporte quel ordinal > o peut
wq
.
0
(8) we read
z k W O = kw.
to
not cofinal
non cofinal avec
= 1, wl
ou
N(o) = l e
ordinal inaccessiblen.
. This
3.6.2 T. 1202 ,
where an
formulated under be increasing
analogous statement for
of
Easton
strictly:
,
N(q)
the assumption that the function
98,p. 17 )3.6.3
assumption was proved by Easton
(not necessarily
by the statement end of
and non cofinal
analogue en posant
prendre
'
>o In
l1D1une facon
is
Therefore one might a s s u m e that in particular
ordinal number 3.6.1.
N(a)
'5
and any D<' E
increasing )'
the following statement
(cf. Mathias [I] ord-+N
.
In
(5)
is ord
particular,
(2ublished in [89] ,
is confirmed not to led to a contradiction:
. Hypothesis:
F o r every infinite cardinal
n
the
number
an
is the f i r s t inaccessible cardinal > n (I1 P e r ogni numero cardinale infin nito n, 2 6 il primo numero' inaccessibile >nn)
.
The function (3.5.1. ) was introduced in 1937 (cf. Kurepa C157 , formula (2))
.
The existence of the function (3.5.1.)
is implied by the axiom of choi-
ce; probabljr ,the converse holds too. 3.6.4.
Here is a s e r i e s of nice continuum hypotheses.
Let n
be a
natural number; '.then f o r every ordinal q
(the case n = l yields the Cantor's 3'. 6.5.
A
general continuum
hypothesis)
one has
.
,great variety of continuum hypotheses is obtained in the
D. Kurepa
following way:
F o r every ordinal number
3.6.6.
a
l e t m(q)
be a positive integer; then
Hypothesis.
F o r el erg-
diadic
gp(G{O,.i'Jone h a s
Ord -+
ordinal mapping
of the preceding continuum hypotheses in acceptable.
Each
3 . 7 . On factorials.
--
Fo" any ordinal (cardinal) number c>--'inal (cardinal) numbers m the
,ar-dinal number of
dinallty
11
all
such
n
that
let m
< n.
.. . =
2n
.
F o r natursal Inirlgers w e have
a s well a s ( 3 . 7 . 3 . ') where
such
P ( n ) denoiea the s e t of a l l uniform mappings
that
be ? h e s e t of a l l the If
the permutations of
, then f o r every transfinite cardinal n
( 3 . 7 - 1.)
In
.
n
is defined a s
any s e t
S 3f
car-
D. Kurepa
(3.7.5)
o<S
nite cardinals
n,
if
w e r e holding, t h e r e would be by
an
What would yield
(3.7.6.)
.
15.5 €j
holds f o r e v e r y o r d i n a l n u m b e r . A s t o t r a n s f i -
(3.7.3)
(3.7.6.)
,$
for
= kP(n)
(3.7.1)
.
n=kw ? It would b e 1
:
c o n t r a c t i n g t h ~:,ontinuurn
hypothesis. Ana?ogously, one p r o v e s t h a t t h e
identity
imply
(3.7.6. )
would
and
i n p a r t i c u l a r that
nal
number
4.
<
s e t (2) (12)
the f i r s t inaccessible
t h e s e t of which
all
PO
the
d2generated
subsets
of
(O,<), i. e .
t h e c o m p a r a b i l i t y r e l a t i o n is t r a n s i t i v e ; the e m p t y
(I)), e.g.
equals
linearly and
. Of by
course,
if
(1)
i s degenerated
o r d e r e d o r a n antichain , t h e n
the question concerning t h e c a r d i n a l i t y of
is exactly t h e continuum pyoblem. 4.2.
cardi-
let
is c o n s i d e r e d a s d e g e n e r a t e d ( d - s u b s e t of
is
wo,
If
) be
s u c h orles in
i
ordinalO(>
.
is a n y o r d e r e d s e t
PD ( 0 ,
c ,>i, w h e r e
for every
.
Ordered sets 4.1.
2k% = 2k
We - f o r m u l a t e d t h e following.
Q-hypothesis. E v e r y infinite
t r e e (T, ,<) s a t i s f i e s
the
set
D. Kurepa
(Kurepa 161, p;133).
Remark. Every finite t r e e and the empty t r e e satisfy the
4.2.1.
preceding inequality, As a matter chain that
C
fact, if (T,,<) is any finite t r e e , then t h e r e is a
of
in (T,
kC(resp. kA) be
chain
A* in
well
as
an
> , k c * (resp. kA*)
. Certainly,
(T, <)
. Since
6)a s
kc
antichain
for
kA >, kT;
if
kPA
< k PDT.
Therefore, the requested inequality
Q(T)
T
that
Now, Q(T) By case
such
PA C P D T , one
the
positive
integer
is satisfied a l s o a similar
kA
kA 4 2 , 3 , 4 3
argument on permuting
Number
1s the number
b.
4.3.1.
any
kC< kA,
< kPDT
then kA i. e. 2
<
is
. . ... .On
.
the other
hand,
A, C
one
proves
Q(T) for the
In
(1) ?
particular,
is the number
t r e e (T,,<) ?
Ramification hypothesis
The affirmative answer
is
[R
HI.
called ramification
hypothesis and is
i6)p.
130, proposition
. The ramification
4.3.2.
principle f o r
hypothesis
is equivalent to the reduction
t r e e s : E v e r y infinite t r e e has the s a m e cardinality a s one of it
degenerated subtrees (D. Kurepa on p. 138)
C1 (anti-
a s one verifies directly.
connected t o the general Suslin problem (v. Kurepa
1
suck
Let
b(O,<) reached in
b(T, <) reached in
p1 : r73
,
.
kA < k c 4.3.
for
(T,
holding for e v e r y kA satisfies ( k ~ )< ~ 2
kA=1, 5, 6 , .
this relation is satisfied for
in
every chain
has
( k ~ ))I kT ~
A
. It
fi] p.
130 (Prof. P2); equivalence
is interesting that one h a s the following
PPP2
D. Kurepa Theorem
4.4.
. The
kPD (T,Q = ( k ~ ) +for every infinite t r e e
(4.1) conjuction
of
hypothesis
(Kurepa [92] )
the general continuum
In particular, infinite
tree;
on
incompatible with of
reals
preceds b
in
the
and
well
5.
2kW0 =
Without
if
akwl
continuum
(R;
if
+
2
contradicting
the
some ordinal o( satisfying
every infinite t r e e
As a matter
of
for every
and
well
Q(X)
ordering of
a c t b ea s b
of
and
the relation Q(W) would
.
(T,<)
every
W one has
of cardinality
fact, the numbers
for
kwd
infinite t r e e .
N(Ql)>q+l , then for
we
have
b T, kT a r e either equal o r consecu-
(T,,< ) s u c h that
kX
>
; consequently,
kw
d-subset
PX C P D ( O , 6 ) and
therefore kP
D
(0, ,<) >kPX >,
prove the relation
(6.1. )
Notations.
kw,-
> kw
'
= kw,,
means means
O(
or
X
= kwd
, we a r e not
.
.
the imediate predecessor
the immediate follower
of
O(
of O(
.
kT =
6+1
f
kX means cardinality of
do(+
2
. If a = q , kT
and this proves the relation (6.1. ) able t o
a
then
tive (D. Kurepa [6]p. 105, theor. 1) ; in either case t h e r e ,is some
K
is
hypothesis we a r e not able to prove
+
If for
any
R ,
C H
kPD(T, <) ,< (kT) 6.
is
of
KG
Q(T)
t r e e s the equality means
W
(kr)
imply
W
6' )
ordering and
RH
f o r non
G C H ; e.g.
k PD(R,
hypothesis and of the yamification
GCH and the
the contrary, the
is equivalent to the
.
the
R
the set
identity
.
D. Kurepa
BIBLIOGRAPHY
Cohen, P.
The independence of the continuum hypothesis. Proc. Nat. Acad. Sci. Usa, 50(1963) ,f143-1148;
[I].
I1 P a r t , Ibidem 51(1964) 105-110.
[2]
H .Set theory and the continuum hypothesis,
New York-Amster-
dam , (1966) 4+154.
~ b ' d e l , ~ . [I].
The consistency of the axioms of choice and of the generalized continuum hypothesis (Proc. Nat. Ac. Sci. USA 24(1938)556-557). The consistency of the axiom of choice !and of the generalized continuum hypothesis with the axioms of s e t theory Princeton, N. J., (1940)4+69.
123.
"Uber das Unendliche, Math. Annalen, 95(1925)161- 190.
Hilbert, D. Kurepa,D.
[6] [7]
Ensembles ordonnes e t ramifies. These, P a r i s , 1935. Publ. math. Belgrad, 4(1935)1- 138.
.
L1hypothese de ramification, comptes rendus 202,1936. 185-187.
&5]. L1hypoth&sedu continu et l e s ensembles partiellement ofdonnes, Comptes rendus, P a r i s , 205 (1937)1196-1198. [51].
Sur une hypothese d e l a theorie des ensembles, Comptes rendus, P a r i s , 256(1953) 56+-565 (seance du 09.02.1953.
l52].
Sur un principe de l a theorie des espaces abstraits, Ibidem 655-657 (seance du 16.02.1953).
[89]
.
Sulltipotesi del continuo, Rendiconti del Seminario Matematico, Torino, 18, 1958-59, 11-20.
[92]
.
Sur une proposition de l a theorie d e s ensembles, C. r. Sci. P a r i s 249 (1959) 2689-99.
Mathias A.R.D. Sierpifiski,W.
A survey of recent r e s u l t s in s e t theory university, 1968).. Hypothese du continu, Warszawa-L*ow
II+76 (Stanford
1934, 6+192.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.
A.
I. M. E. )
MOSTOWSKI
MODELS O F S E T THEORY
Corso tenuto a Varenna
dal
1 9 a1 1 7 S e t t e m b r e
1968
MODELS OF SET THEORY by A. Mostowski
(University of Warszawa)
Lecture
I. Aim of the lectures: t o outline various methods used recently
in construction of models for axioms of s e t theory.
No completeness
in pursuing this aim is attempted. In the introductory lecture I we describe three s y s t e m s of axioms for abstract s e t theory.
In all these systems t h e r e a r e
two primitive notions: "class" and
"membership".
We define s e t s
a s c l a s s e s which a r e capable of being members of other classes: x
is a set if and only if there is a c l a s s
y
such
that
x
& y.
We also define atoms a s objects which have no elements. The distinction between s e t s and c l a s s e s was noted already who distinguished between the "consistent sets" and
by Cantor
"inconsistent s e t s " i.e. terminology.
s e t s and (proper) c l a s s e s in the modern
As we shall s e e t h e r e a r e axiomatic systems in which
the existence of proper c l a s s e s is assumed and other systems in which their existerice i s excluded. Set
-
theoretic formulae:
different l e t t e r s (the variables).
Let
xl, x2,.
. be
We shall often replace
a sequence of "x" by any
other letter and omit subscripts. elkxi), xi t x . , x i = X. a r e formulae; the J J v a r i ~ b l e xi is the unique f r e e variable m the f i r s t of them and the (i)Expressions
A. Mostowski x ,x. a r e the unique free variables of the remaining two. i J (ii) If F and G a r e formulae, then s o is the expres-
variables sion
(F)
] (G);
i f it is free in
the variable xi F
(iii) If x.
1
i s free in this formula if and only
o r in G . F
i s a formula, then so i s
is free in this formula if and only if
I
(The symbol general quantifier. from as
1
(x.jF; the variable
J
if j and
F.
xi is free in
i s the Shefferls stroke and
(xi) i s the
We define the propositional connectives different
in the usual way; also the existential quantifier i s defined
- I ( x ~ ) IF.).
We denote by i s a free variable of
i
F r ( F ) the set of those
for which
xi
F.
Predicative formulae a r e those formulae in which all quantifiers a r e limited to sets or atomes. We obtain a precise definition of this class of formulae by replacing the rule where
k
(iiil) If
I?
i s not in
F r ( F ) and
(iii) by the following:
i s a formula, then s o i s k
(xj)(xk) :(xj
xk)
-+ F]
+ j.
The above expression can also be written a s (xj)(xd [(xj t xk)
I (FI F$
Definitions:
x
i s a set
(in symbols
S(x)): C ~ ( X ) & ( E Y )Y); (X€
x is a proper class (Pcl(x)):Cl(x)&7 S ( x ) ; x i s an atom; (At(x)): (y)(y
g x)
We now list axioms of the different systems which we shall consider
.
The first
9 axioms a r e common to them all:
Axiom for classes: Ext.
r Cl(x) & Cl(y)& (z) [ (z E x )
Cll.
Pcl(x)
4
(Ey)(y e x);
(z E
yg + (xZy);
A. Mostowski
C12.
(y g x) +Cl(x);
Fund.
( y c x)
-+
(Ez)
{
( z E x ) & ( t )[ ( t E z)--t(t
# x)]] .
Remarks. Ext is the familiar axiom of extensionality. C l l says that proper c l a s s e s have elements, i. e. , a r e not atoms. C12 says that whatever has elements is a class.
Fund is the axiom
of foundation and says that each c l a s s which has elements contains a t least one minimal element, i. e . ,
one whose elements a r e not elements
of the given class; of course the minimal element may be an atom. Axiom f o r sets.
fib) v ~ t ( x ) ] & [ ~ ( y ) V ~ t ( y ) ]-+ (Ez) ( ( ~ ( z )& (t) / ( t G Z )
Pair.
Lit = x, "(t =
sum. S(X) Pot.
(EU
~ ( u &l
(2)
S(x) -+ (Ep) [s(P)&(z)
Emp.
(Ex) [ ~ ( x )&(y) ( y e x u
Inf.
(EX) ( ~ ( x & )
Remarks.
Y)]!)
;
( Z E u) g ( ~ t [(z ) E t ) & ( t EX^ {) { (215p ) z ( Cl(z) & (t) [ ( t ~z ) 4 t E xd))l
(EU)(U EX)&(V)
[(vex)
+ ( ~ w ((w )
EX)
di
These axioms state the existence of the unordered
pair of any two objects (which may be s e t s o r atoms), of a union of s e t s which belong t o a given set, of the power s e t of a given set, of t h empty s e t and of at least one infinite set. will be the usual one: the union,
3 x, y )
The notation for these s e t s
for the unordered pair,
P(x) for the power s e t and
0
i r x for
for the empty set.
The
uniqueness of these s e t s follows f r o m Ext. F u r t h e r axioms will fulfill a twofold role: f i r s t they will determine the number of atoms; secondly they will express the idea that an image of a
s e t under an a r b i t r a r y mapping is again a set.
The f i r s t question does not s e e m t o be v e r y important for
A. Mostowski
the abstract set theory and it is nowadays customary t o dismiss it by simply assuming that with the exception of 0 t h e r e a r e no atoms We shall follow this custom and adopt the following axiom:
altogether.
Noatoms:
(x E y)&(xf 0)4 S(x).
The second aim is much m o r e important. It is achieved in many different ways according t o the system of axioms which one adopts. The system of Zermelo-Fraenkel.
In this system we f i r s t
of all assume that t h e r e a r e no proper classes: Nopcl:
Cl(x)+S(x).
This axiom together with Noatoms allows u s t o simplify previously given by omitting everywhere the expressions
the axioms "S(x)".
The idea that an image of a s e t is again a s e t is expressed in ZF
by the following axiom schema: SubstZF.
in which
F
(x)(E!y ) F
-+
(a)(Eb)(yl { ( y e b) 5 (Ex) [(x E a ) & ~ ] ]
is an a r b i t r a r y formula such that b is not i t s free variable.
System of ~ 8 d e l - B e r n a y s . In this system we assume that every predicative formula determtines . a , c l a s s and e x p r e s s the idea that the image of a set is again a set not by means of a schema but by a single axiom involving the notion of. a class. F i r s t we define the ordered pairs:
1 Fi,xj {] .
<xi, xj> = xi,
A c l a s s whose elements a r e ordered p a i r s is called a relation; it is called a function
(in symbols
Fn(x)) if it satisfies
(w)(u)(v){[ ( < p , v > ~XI&( < u , w > E x u We denote by (Ez)
x"a the c l a s s t
[ (z e a)&(d z,, y> 6 )])
the condition:
4
such that
(v = w)3
(y)
.
(y E t ) . ~
provided that such a c l a s s exists.
A. Mostom ski
The
axioms which we admit in the system GB a r e now a s
follows: Clex
(Ex)(Cl(x) & (u) S(u)+ k u E x ) P F]{)
GB' SubsbB.
Fn(x) & S(a) 3 S(xUa).
Remarks.
In Clex
( c l a s s existence scheme) F is a GB predicative formula in which the variable x is not free. The c l a s s
1
whose existence is stated in Clex
{u: u =
U]
i s denoted by u:F GB is the universal c l a s s consisting of a l l sets.
x''a
e. g.
is called the axiom of substitution; note that the
SubstGB class
);
exists by virtue of the class existence scheme. System of Morse. In this system we leave the axiom of
substitution unchanged and extend the c l a s s existence scheme by allowing a r b i t r a r y formulae
We denote these axioms by Clex
F.
Comparison of the s y s t e m s
ZF, GB and
M
and Subst
M'
M.
The following theorems a r e easily established: I.
ZF
Proof.
is interpretable in
We interpret c l a s s e s of
11. GB is a subtheory of 111.
GB. ZF
a s s e t s of GB.
M (obvious).
GB is an inessential extension of
ZF.
By this we
mean that each preaicative formula provable in GB is also provable in
ZF. Proof.
by GB.
If
F
is predicative and provable in
I (and i t s proof),
F
is provable in GB.
In order t o establish i t s provability
ZF, then
Let F be provable in
in Z F it will be sufficient
t o show that every model of Z F can be extended t o a model of GB in such a way that s e t s of the new model be identical with s e t s of the old model.
Let therefore
M
be a model of
ZF
and let
M' be
A. Mostowski
the family of sets here
F
i t t M: (t, a l ,
...,an)
i s a predicative formula with
satisfies
n+l
M
in
F
)
;
free variables and
al,...,a
a r e elements of M. Interpret classes a s elements n M' and the membership relation a s The resulting model satisfies
.
of
all the axioms of GB and contains a submodel isomorpkic with
it G M:
and consisting of sets
t
x
1.
All the sets of
M
M' have
this form which proves the theorem. We shall show l a t e r that
M
is an essential extension of
GB. Relativisation.
Let
A(x) be formula in which
variable; A may contain free variables other than formula (xi)
we may obtain a
F
by
F
and called the formula F
and
A
ieFr(F). relation
F r o m every
formula by replacing each quantifier The new formula i s denoted by F (A)
relativised t o
We assume that formulae
A.
do not have any variable in common. It is
( X ~ ) F ( ~ ) if
.
i s a free
new
...] .
(xi) [A(xi) -2
x
x
i
6 Fr(F)
The
1
and
symbol
-
( F ( G ) ( ~q ) ( F ( ~ )G ( ~ ) ) (, ( x ~ ) F ) ( ~ ' =
obvious that
((x~)F)*
.= denotes
F
x i A
if
here the logical equivalence
(not the connective). Functions whose values a r e classes.
We insert here a note
about functions whose arguments a r e s e t s and values a r e classes. Of course this notion is important only for systems
GB and
M.
A function all of whose values a r e non vojd classes and
whose arguments a r e s e t s can be defined a s a relation. of such a function fx =
{ y: < x, y>Ef]
f
.
for the argument
of a function, then we define
a relation
ad
x
If we want t o admit f
a s a pair
The value
is simply 0
a s a possible value
a, r where r i s
A. Mostowski a relation and a is a class which contains Dom(r) a s a subclass. The value of x
f
for the argument
is an element of We write
is
x
r
or
X
Dom(r) o r an element of "f:a
-;L~"
instead of
domain a whose values a r e subclasses of
0
according a s
a-Dom(r).
"f i s a function with
y".
The elementary operations
on functions such a s superposition o r restriction of the domain t o a subclass can easily be defined for such functions.
i
domain a whose values for by
Fii.
f+(i/u)
f..
If
f:a-rz,
in a
ijka
A function with
a r e explicity given will be denoted
and
u s z,
then we denote by
the uniquely determined function with domain
a
uli) whose
restriction to a i s f and whose value for the argument i i s u. Semantical notions defined in set theory. The basic semantical notion i s that of satisfaction: a sequence
x
satisfy a formula
F
in
x.
f
of subclasses of a class
If we want t o discuss this relation in
s e t theory, we have t o replace formulae by certain sets. The following theorems a r e provable in
IV.
1
integers
There is a smallest set
,< 2
i j
i, j and is
then s o do <3,
ZF: Frm
a r e elements of
7
such that whenever
< a, b 2)
and
( 4,
such that F r m for
a, b
<' i, a>>
,0 , i >,
arbitrary
belong t o
Frm,
for arbitrary i
in cd
(the set of integers). We write 1 write We call
for 0
i
e
rx. and
1
x?
J
Ei =
for < 3 , c a , b , > a n d Frm the s e t of
By F r m
for
< 1,
< i, j >-.> and similarly
for ( 2, c i, j%>
r(xi)al
for
<
.
We also
4,..
" ~ 8 d e lsets" of formulae.
we denote the subset of Pr ~ b l d e ls e t s of predicative formulae.
Fm consisting of
A. Mostowski V. F
There is a function
( x i ) ) = i
= Fr(a)
y
5.ex?
Fr(
,
F r defined on ) = Fr(
J F r ( b ), F r ( ' T x i ) a = F r ( a ) 1
-
F r m such that
5. ='z = l i , jjand 1 J
1 it .
We call
Fr(l>lb7) =
F r ( a ) the
set of f r e e variables of a. We call a formula
S
with
3
f r e e variables a predicative
satisfaction formula for one of our t h r e e systems
ZF, GB, M
if
the following formulae a r e provable in these systems: Fr(x)); (i) S(x, y, z ) 4 ( x E F r m )&Cl(z)&(yE z Pr (ii) S( % l ( x i P , 24 i, y>], z ) (YE z);
Gig xJ. 1 S( ' xi x?
S(
,j < i , y l > ,
=
(iii) S( ralb7, y, z ) z b s ( a , y
(iv) S(
r(xi)a7
(v) S( 'Txi)2 A formula
.
j, y">] ,z )
, { 4 i, yt.> ,
c j, Y')]
= (ytE
, z) =_ ( y t
Y");
= Y");
[ F r ( a ) , z ) V f S(b, y \ F r ( b ) ,z)]
;
, y, z ) 3 S(a, y, z ) if i f C . ~ r ( a ) ; otherwise:
(dl(u c z ) + S ( a , Y u[
y. 2) 5
with t h r e e f r e e variables will be called
S(x, y, z)
ZF, GB o r M
a satisfaction formula f o r
if the following formulae a r e
provable in the corresponding system: (it)
S(x, y, z)
(ii')
S( r ~ l ( x i j l ,y, z )
yi
S( T-,i€x?
.Y~Z)=
Y ~ GYj;
S(
Gi
,y,z)=
yi = y.,
( i t ) and i
t
(vt)
4
= X?
as
(x E F r m ) & C1 (z) & (y: F r ( x ) .+z),
(iii)
S( r(xi)a7 , y,
In case of the system
Z)
z,
J
and (iv),
z
ZF
(u)
[ (u C
a ) --t S(a, y + (ilu). zj]
t h e r e is no need t o distinguish
between the satisfaction and predicative satisfaction formulae because sequences of c l a s s e s can be identified with sequences
of sets.
.
A. Mostowski The following theorems exhibit essential differences between systems
Z F , GB and
M:
VI.
T h e r e is a satisfaction formula f o r
VII.
There is a predicative satisfaction formula for M
but - provided that VIII.
M
If
-
is consistent
GB
ZF.
no satisfaction formula f o r M.
is consistent, then t h e r e is no predicative
satisfaction formula for
GB, and a l s o no satisfaction formula for GB.
We shall sketch the proofs of these theorems. In o r d e r t o prove
VI
we define
s e t s of finite sequences: Str(z, kl ,k2, dl, d2) =
;) YC
Qu(z, k, i, d) = k if =
z
ie
dl" d2 d;
(in
Z F ) two operations on
:( Y
I d l # kl) V ( ~ I d 2 4k2)
otherwise : (u) [(u
25 y E zd
We obtain a predicative satisfaction formula for expressing in the language relations:
z
is a set,
sequence
f
such
1 p, q
or or
o r finally there is a last t e r m of
f
is
j
x;
such that for each
y g z Fr(x);
for each
i
if
fi
is
if
f
is
f.I =
rx
and
i
in Dom(f)
either
a p
such that
fi =
there is a finite sequence in
fi
k
is
f o r some f. =
Cp)f:: 1
with
Tf.J ) f
z
the
Dom(k) = Dom(f)
Dom(f):
r ~ l ( x p F , then
5P ex1q
, then r
ki ={{]
q
:
UEZ];
ki = f{
,]: ( ~ € 2 )&
if
by
t h e r e is a finite
E o r it is = x l P P q j, k both s m a l l e r than i such that
there exist
ZF
the conjuction of the following
x g Frm,
that
it is
of Z F
yki, u>) E k]{
z)+ y
E
, then ki=({xp,u,
.)
(VEZ)
(u e v ) & ;
:uez},
A. Mostowski
if if
f
= f
fi =
then
,
r
(xp) f;
ki = S t r ( z , k., ky, F r ( f j ) ,P r ( f 4 ) ) .
J
, then
k.= Qu(z, k . , p, i 3 r ( f j ) ) ;
J
1
y is an element of the l a s t t e r m of
.
k
We proceed s i m i l a r l y in o r d e r t o prove the positive p a r t of VII. The negative p a r t of theorem
VII is proved by using the
well known technique of ~ 8 d e l . We a s s u m e that t h e r e e x i s t s a satisfaction formula n
in
S
for M
F r m with
s
and denote by Fr(n) =
{
a function such that for every s ( n ) is the
0jthe s e t
resulting f r o m the formula with the ~ 8 d e ls e t f o r i t s unique f r e e variable. i S ( s ( x o ) , 0, V) where
V
Let
n
induction on the length of
the G
...,xP
s
that
and
.. ,xP
if
We prove by
~ 8 d e sl e t
p
g, then the
M:
S(g, F i p ~ i V). ,
the r e s u l t that the equivalence
M
s(n,).
M.
is a formula with
G
with the
Using this equivalence for
and hence
c a n be defined in
Gbldel s e t
following equivalence is provable in G(xl,.
n
i s the universal class; we r e l y h e r e of
Then 7 S ( s ( n o ) , 0 , V ) has xl,
by substituting
be the ~ 8 d e sl e t of the formula
0
course on the fact that the function
f r e e variables
n
~ 8 d e ls e t of the formula
T
G= lS(s(no),O,V)
we obtain
G G ~ would be provable in
M
would be incbnsistent.
The detailed proof of theorem
VIII will be given later.
The main idea i s this: we assume the existence of a predicative satisfaction formula for set which is a model f o r
GB ZF.
and prove i n
GB
the existence of a
F r o m t h i s we derive (always in GB)
the existence of a s e t which is a model f o r
GB and
thus infer that
A. Mostowski
GB is consistent.
Thus the consistency of
would be provable in
GB
GB which a s is well known entails the inconsistency of GB.
We conclude with a theorem which we shall need l a t e r and which can be proved by using the s a m e technique a s the one used in the proof of theorem VII.
M.
F o r each predicative formula
following formulae is refutable in
T
the conjuction of the
ZF :
(x)(y) [T(x, y) A (x E F r m p r ) & (y
(is')
is a sequence of
s e t s ) & (Dom(y) = F r ( x ) ) ] , (iil')
[ T(
(i)W (u)
'?l(xi)
I
,
1< i, u > l )
similarly a s above with
(iid
(a)(b)(y)( ( a E F r m
(iv")
(a)(i)w (y)
Proof. Hence the (i")
-
(v")
Let
f
"
we
infer by H
Hence the conjuction of
"
.
=
"
;
Pr
Txi)n3,
rk ~
IT(y x i ) ~ ,y) % (u) T ( a , y u { ~ i , u ) r ) ) ] ] .
( aY]],
) & ( ei F r ( a ) )+
be the ~ 8 d e set l of the formula l T ( s ( x o ) , 0).
induction
that if
h
Fr(H), then
H = CIT(to, 0), we obtain (i")
-
(v")
s ( t ) = to.
is the
whose f r e e variables a r e
is a sequence with domain
If we take in particular
replaced by
1 B E Frrnpr)b-(i$?!Fr(a))-+I~(
~ 8 d e sl e t of the formula 7 T ( s ( t ) , 0 ) is
a predicative formula and if
t
E
)& (b E F r m ) 3 ) TE ( ?;l y ) ~ pr pr L-,T(a, y IF r ( a ) ) v-T(b, Y 1 ~ r ( b ) j ) ) )
(a)(i)w (y) {(a E F r m
(v")
"
C U U ~,
Using
~ 8 d e ls e t of Xij,
(j = O , l ,
T(h, f) H E YH.
leads t o a contradiction.
...,k-1)
H(fio,.
..,fik-l).
A. Mostowski Terminological remarks.
If
S
i s a satisfaction formula
(or a predicative satisfaction formula), then we shall write instead of
S(x, y, z).
we shall replace
F ' x
x
,,...,
xn
F
and
let
x
by
and not simply Let
If F
i s the
z
~ 8 d e set l of a formula
x [y j F , then
(although it would be more exact t o write
in these formulae).
F
be a predicative formula with the free variables be its relativisation to the formula
F(')
x 6 z.
Then the formula (~,E.z)L
... &
( y n E z)-i
z t ~ D i l , ~,.+., ~ >
I = F ( ~ ) ( Y ~ , ,yn),$
-
...
i s provable in each of the considered systems of set theory. Let (y E
Zyr(X))
zl,z2
be classes,
& (z1C X
is an elementary subclass of
(z2 z2
. If (x)(y) ( ( x ~ ~ Prr )m& x ly])], then we say that zl z
and write
L.
zl < p r ~ 2 .
A similar
definition can be given for the notion of extension in c a s e when ranges over arbitrary formulae not just the predicative ones. shall not use this notion however.
x We
LECTURE
I1
The backbone of the whole set theory is the stratification In discussing
of the universe into levels(simp1e theory of types).
this phenomenon we Shall a s s u m e a s known the notion of ordinals and the t t e o r e m on definitions by transfinite induction.
We denote by On
the c l a s s of all ordinals; in the c a s e of the s y s t e m
ZF
a r e no c l a s s e s we think ,of On a s of a formula s o that
On(x) means
Sometimes we shall use the expression
"x is an ordinal".
the s a m e a s
where t h e r e
On even in Z F treating it a s equivalent with On(x).
x
We define by transfinite induction s e t s Ro = 0,
RL
= P(R
Rg :
R,=U~R,,:~
1,
'
r6)
is a
(f
limit number). Note: What we define is in GB (or any stronger system) a function which c o r r e l a t e s with each ordinal f c a s e of
Z F we have a formula
that
[odd
(4
Rt
.
In the
R(x, y) such that it i s provable in Z F
.
+(E ! Y)R(X.y)]
a set
This unique
y is denoted by
R
.
xB
it satisfies the equations given above.
I. F o r every s e t
x
t h e r e is an ordinal a such that
xSa.
This r leorem is provable in each of our t h r e e systems. The proof is obtained easily from the axiom Fund. The l e a s t ordinal a such that
x l R a is called the rank of x.
We want t o discuss the problem whether some of the R a t s a r e models of and prove in
In o r d e r t o answer this question we shall formulate
ZF. M
the following theorem:
I1 (Scott-Scarpellini). L e t each ordinal
&-,
AT
for each limit ordinal
is a set,
T
.
f
A= A*
u (A C_
A
:f e o nf
and
Under theses assumptions:
where f o r Af
=U{A+,,:fkf]
A. Mostowski
(a).
F o r each
continuous mapping then
A,)= F
Frmpr
+ On
fF: On
t h e r e is an increasing and
such that if
fF(a) = a and
in
There is a function
Frm
the mapping
Pr (a).
formulated in
f: F r m
X On A O n suah that for Pr a + f ( F , a ) satisfies the conditions
(c) There is an increasing and continuous mapping such that
if
t(a) = a, then
proof.- (a) which occur in
we take
f
G
prA.
If
fH.
I H?
?G
is
F
of connectives
is
F
f~
then we take a s
T(xi)~'l
and
fF the compo-
i$Fr(G),
then
The only c a s e which r e q u i r e s a m o r e elaborate
fF = fG.
proof is one in which
F
h a s the form
I-
(xi)H1
and
i 6 Fr(H).
a : ~ ~ ~ ( On, ~ ) and +
We f i r s t define auxiliary functions b:On
t:On+On
has no connectives, then we take a s
F
If
and
Aa<
. We use induction on the number
F. If
the identity map; sition of
Fr(F) xEA a '
F[X]SA/=F~X~. (b).
each
F in
.-+ On a s follows: a(x) = min b ( r ) = sup
{
r E On:(Eu) [(u .s Ar) & ( A k 7 H [xu { ~ ui $, a {
a(.):
xEA
r
~
~J (
~
)
It follows from these definitions that if
1.,
[(A
b H [ u 14i. u~j]))
x 6
(u) (AkH Ab(r)
r
, the11 (1)
l x*
(*)The subscripts after quantifiers denote t h e i r relativisation t o the formula x d a.
Thus
(x).
means the s a m e a s
(x)
1 (x €
a) 9
...J .
A. Mostowski NOWwe define u p
c(0) = 0, c(r+l) = max(c(r),b ( r + l ) ) and c ( r ) =
for limit numbers r.
) r
= f o c. F H This function i s obviously increasing and continuous. If f (x) = r, F Fr(F) we have c(r)> b(r). and for every x in then fH(r) = c ( r ) = r
(
Finally we put
f
Ar
It follows
+
u,@s(u)~
(ulA ( A ~ H [x+i, r
A ~ + F 1x1
r
( A ~ H [xV(ci. u ] ] )
and similarly
A
~
(b).
[ FX J ~(u),
(ACH [ x u {
We define
f(F,x) = x
(without logical operators), f ( r F I = f(H, x)
if
i
4 Fr(H).
= f(H, c(x)) where
c
If
)F ,:
if
-+(u),
r
(AI=H[
xu{ci.
F i s an atomic formula
x) = f(F1, f(F2, x))
i d F r ( H ) , then we put
and
f( q x i ) ~ ?x)=
f ( ? x i ) ~ l ,x) =
is defined similarly as in part (a).
The proof
i s identical a s in the case (a). (c).
We define
F E ~ r r n ~ ~ ,] t(r) ) = sup If
t ( r ) = r, then f(F, r ) = r
hence, in view of
t(0) = 0, t ( r + l ) = max ( t ( r ) , sup t ( r '):r' 4 r
$
if
f
f(F,r+l):
r is a limit number.
for each predicative formula
F
and
(b),
Ar< prA. Remarks on the Scott-Scerpellini theorem.
We want to
discuss the question whether theorem I can be so reformulated a s t o become provable in ZF o r in GB.
A. Mostowski
The case of t h e system ZF.
Since there a r e no proper
c l a s s e s in ZF, we must replace the c l a s s
A
and the relation
x GA
r
If we do this, then we cannot use the satisfaction formula
by formulae.
since t h e r e is no f ~ r m u l adescribing the satisfaction of an a r b i t r a r y x
in
F r m in the domain of all s e t s satisfying a given formula A.
We come around this difficulty by remarking that the satisfiability of an explicitly given formula
in the indicated domair. can be expres-
F
(A)
sed by the relativised formula
F,
Thus we shall have not a single
theorem starting with the general quantifier "for each F in ~ r m "but a theorem schema which can be proved separately for each explicitely given formula F.
A final change in the wording of the theorem con-
c e r n s the function fF: since there a r e no mappings of On into On in Z F we must replace the mapping fF by a formula which
the s y s t e m describes it
The Scott-Scarpellini iheorem takes thus 'in Z F the following form: Let x and. x, y
A(x)
and
B(x, y) be two formulae with the f r e e variables
respectively; let
C
be the conjunction of the following
F
with two f r e e variables
formulae:
-
F o r any formula
DF be the conjunction of the formulae: (XI
[~n(x)
(E
x, y
let
A. Mostowski
With these notations the following holds: formulae
A(x), B(x, y)
H(xl,.
and
..,xn) there
F o r arbitrary
is a formula F(x, y)
such that the implications
a r e provable in ZF. Notice that the formula x's satisfying over
C
A is the union of s e t s
says that the domain of a l l
{ x:B(x, r)]
with
r
ranging
ordinals and that these s e t s form an increasing and continuous
family while
DF
F
says that
defines an increasing and continuous
mapping of ordinals into ordinals. A similar reformulation of the Scott Is also possible in the case of the system GB.
-
Scarpellini theorem
Again we must express
the theorem a s a scheme because there is no satisfaction formula for
GB. However the assumptions of the theorem and the statement
concerning the existence of a mapping can be expressed as in system
M.
Finally we notice that a theorem similar to the Scott-Scarpellini theorem can be proved of the form
u { A,
:r
(in any of our three systems) for unions s
3
where
s
is an inaccessible cardinal.
Applications of the Scott-Scarpellini theorem. The following theorem is proved in
M; it c&n also be proved in any system obtained
from G'S by adding to it new axioms which secure the existence of a
a predicative satisfaction formula: There is an increasing and continuous function f: On +On
111.
such that whenever a model of
f ( r ) = r , then
Rr < p r ~ ; in particular, R
is
ZF. Proof.
The decomposition
V =
U{ Rr
:r
on]
satisfies
the assumptions of the Scott - Scarpellini theorem. Since all axiom of Z F a r e equivalent t o predicative sentences and a r e valid in V, we infer that if R is an elementary subset of V, then R
r is a model of ZF. We can now supply proofs of two theorems which we announced
in Lecture I: IV.
There is no predicative satisfaction formula for GB
provided that GB is consistent. Otherwise we could repeat in GB the proof of theorem I11 and infer that there is a set a which is a model of ZF.
Hence by
adding t o a i t s definable subsets we would obtain a model f o r GB. Since this proof would be formalizable in GB we would have a proof in GB that GB is consistent which would entail
the inconsistency of
GB. V.
M is essentially stronger than GB,
provided that GB is
consistent. Proof.
T h e r e is a predicative satisfaction formula for
M but none for GB. VI.
Z F is not finitely axiomatisable, provided that it is
consistent. Proof. We denote by of axioms of
K
the conjunction of any finite number
ZF and show using the version of the Scott
theorem which is provable in
ZF
-
Scarpellini
that t h e r e is an ordinal r such that
A. Mostowski
R
4'
K. Hence the consistency of K is provable in Pr consistency of the whole Z F is not s o provable. VII.
Z F while the
GB is finitely axiomatisable.
We merely sketch the proof of this theorem. Call particular instance of the axiom (scheme) t o the f o r m ~ l a F.
SubstGB
which corresponds
It can then be shown that all the particular instances
-
GB formulae: (Eul)(Eu2) l ( u = < u1.
UZ>
)&(ul E
...F*9 where F 1 . . .F g a r e the
F;,
can be derived from the axioms
of Subst
F V the
U 2 o
, (Eul)(u= < ul. ul> 1,
1~ < ?
? ( u ~y),
(UE
(ul E Y)
DU
fill =
=
1
< u , t > C y), (Eul)(Eu2)
y) & (u G z ) ,
.
< u1
(EU)(U=
< u2,
u3>>
< .I>
4U2,
U3>>
u
>
2 & (
=
U1.
U2>
) &
) & ( C u Z . ul> E Y). ( E U ~ ) ( E U ~ ) ( E U ~ )
ul>> C- Y)
) & ( < u 2 >< u l 9
U~)>EY)]
.
(Eul)(E~2)(E~3)
.
These axioms state the existence of the following classes:
E = ~ < u , v > :u ~v
E ~ .3I = ~ < u . u >: u e v ) .
V
Dom (Y), Y x V,
Y,
Cnvl(Y), Cnv2(Y)
E
t h e r e is a c l a s s
u
f o r which
H(u).
Details of this proof
in G8de11s monograph.
Natural models. of
H
and I by means of the operations enumerated
above and consisting of all can be found
Y, Y A Z .
and our theorem reduces t o
the statement that f o r every predicative formula Y constructed f r o m
v -
Z F , GB, o r
If
r
is an ordinal and R
is a model
M, then we call it a natural model of the corresponding system
In theorem I11 we established (in M) the existence of a "tower" of natural models f o r
Z F which is ordered by the relation
-
We now show
that the existence even of a single pair Rr' Rs such that ~~dR Pr s We do cannot be established in ZF provided that ZF is consistent. t h i s by proving in Z F the following:
A. Mostowski VIII. (Montague a model of
-
Vaught).
+0
r there is an (Exl)(xo
x xl)
r # 0.
and in
r
then
R
Rs
1.f
is
t t r , then
and
R eR and hence t r Thus the formula
s
x
r' t +l#r.
Using the fact that ZF
0.
R x. t Rt in R and hence it must also
Thus there is an
check that all axioms of
je:
Rs, it i s t r u e in R
such that
is satisfied by
be satisfied in R and therefore
is a limit number
(Ex)(xZx) is true in
Since the formula R
R,(~,R,;
ZF. Proof. We f i r s t show that
hence
If
r
in R
such that
R E x
t
is a positive limit number we easily
a r e valid in
.
The verification is
Rr
evident in all caseswith the exception of the axiom scheme Subst
ZF
which requires a separate treatement. Let. F be a formula with
k+2 free variable and let
5 E R:~'~)-
'i~ 'f be such that
Rr
+ xi)^! x j
in R
let
f(x) be the unique
y
in
4,y
.
Let
x and x P and l e t a be an element of and satisfies the condition
R
such that
be variables
Rr' R s
' F[ J
The s e t
. R~
For
-x
!= F[~U,&,
which do not occur in Im(f, a) = b
~ [ ~~ vpa>, ,i
( q,
b>]]
belongs t o where
H
is the formula
It follows that t h e r e is a s e t R r k H [ ~ u ) < p a> , , < q, b>i] Thus axiom
Subst ZF
be also proved in
M.
is valid
b'
in
Rr
such that
and we easily prove that in
Rr'
Theorem
b1 = Im(f, a).
IX can obviously
x,, F
Rs
A. Mostowski
<'
In o r d e r to obtain m o r e information about the relation in the c l a s s
R 's
of a l l
We c a l l an ordinal t h e r e is a sequence and that
R
f(t) t
f
r
extendable if f o r any ordinal
of o r d e r type
s
such
for
prRf(o
( c ) of the Scott-Scarpellini
that t h e r e a r e extendable ordinals.
a r e elementary extensions of
R
On
so
and
Rr
<pr~s
so
then R s
that because of the extendability of such that
s
Rr<pr Rs
from a certain
s
0
>r
>r
r
R m l f ) and R r(prRm(f). follows :
-(
s
0
such
such
which
<
pr'
hold.
r
R Pr s
( l e a s t ) extendable
r
with the properties : whenever We notice
t h e r e a r e a r b i t r a r i l y high
and our assumption s a y s that each such
on can further
>
Rr
< Pr R t f o r s o m e t > s. be extended t o an
F o r a r b i t r a r y finite sequence m ( f ) the l e a s t ordinal
such that
us a s s u m e that t h e r e is a
such that t h e r e e x i s t s an ordinal
X
r
M.
R (pr~t
does the relation Let
R 's
F o r each extendable ordinal
a r b i t r a r i l y high ordinals
Proof.
theorem it follows
and well o r d e r e d by the relation
(Ryll Nardzewski).
but f o r no t > s
>
such
consisting of
The next theorem is provable in
s
t, t '
Moreover t h e r e a r e ordinals
that t h e r e is a sequence of o r d e r type
there are
f(0) = r
that
each p a i r
s
.
F r o m part
IX.
P
we introduce the following definition:
f
Rt
.
of s e t s we denote
that all t e r m s of
f
by
belong t o
Consider the formula T(x, f) defined a s
s s
A. Mostowski
we
shall show that this formula has the charateristic pro-
perties (if')-(v")of the truth predicate (cf. Lecture I, theorem IX). Since we know that we can refute the conjunction of (i") - (v") we shall have the proof that our assumption leads to a contradiction a s soon a s we veri= fy that T has the properties ( i f ' )- (v").' Of these, ( i f ' ) , (ii") and (iv") a r e obvious In o r d e r to verify the remaining two we prove a lemma:
R R , thenlS< Pr t pr t '
I f so < s < t a n d n r 5 R R r S,
Proof of the lemma. F o r each s >s such that R 0
mentary extension of R
S
is an ele-
we denote by s 1the least ordinal )s
such that
The existence of s 1follows from s' our a s ~ u m p t i o n s.Now we s t a r t from given ordinals S, t and construct two Rs, is an elementary extension of R
sequence f and g saticfving the inductive equations f(O)=s, f(n+l)' = (f(n)I1,f ( u ) =sup {f(n): n c u 3
sup {g(n): n < u ) ,
R
=
g(n) <,,R~(,)
U{
Since R
We obviously have R
<pr
: n
G On
-
X
I- Y; =-
]
.Let x t F r m
V we infer that R s +
we prove similary that Rti=
< R and f(n) p r f ( m )
f o r n.4 m; moreover V =
Rg(n)
S
, g(0) = t , g(n+l) = (g(n)I1, g(u)=
Rt
v k x [y]
XprV
U Pr
x [y]
;
on)
R ~ ( ~ n,
and y
(F
V
=
RF r ( x ) S
b x
. lyj
;
and therefore
whence ( R . ~ = x
[Y])
L (RtkX
The lemma is thus proved. Verification of condition (iii"). Let us assume that
[Y3)
.
A. Mostowski
This assumption is equivalent t o the statement that either R
I-
m(f)
+..,x1
I-
the lemma R
m(f)
pr Rm(f ) F r ( x . )
'fi
Fr(x2)]
. Eince by
for i= 1, 2 we infer that the
1
previous statement i s equivalent t o the disjunction of the statements Rm(f
I Fr(xi)) *?x:
[f
I
Fr(xi)]
for
i = l,2
and this disjunction
is precisely the right hand side of (iii"). Verification of condition (v"). We a s s u m e that x E F r m and i
F r ( x ) . Let us consider the
statement T ( ' (x.) 2
P
'
, f);
this statement says that f is a sequence with domain F r ( x ) and that x [f 0 ( < i ,
j= Rm(f)
u>j]
for every u in R
m(f)'
We hLve t o
rf u)-=i,
prove that this statement is equivalent t o the following: for every u, R m ( f ~ { < i ,u s ]
x
u1 .1
.
Obviously the first statement is implied by the second because m(f w ((
>2
NOW we m(f) ' assume the f i r s t statement and choose an a r b i t r a r y u. By the l e m m a
<
Rm(f)
p r R m ( f ~<{i , u)
) = m(f) whenever u is in R
f
) and hence the f i r s t statement implies
@(xi). [gwhence we infer that the second statement is Rm(fu{< i. u > ) ) valid. Theorem IX i s thus proved A theorem similar t o M can also be proved for other transfinite sequences of sets, for instance for s e t s L shall discuss later.
which we
A.
Mostowski
In connection with theorem M we discuss briefly the ordinals which a r e not extendable. Let us call a
function f whose do=
main is an ordinal r and which satisfies the condition
<'pr Rf(,, f o r a r b i t r a r y m E n E r a chain of length r s t a r = f(n) ting i n f(0). An ordinal s i s not extendable if and only if t h e r e is R
an ordinal r such that t h e r e is no chain of length r s t a r t i n g at s. The l e a s t such r is called the height of s. The height of an extendable ordinal could additionally be defined a s On. We don't have an exact
charscterisation of ordinals
which a r e heights of non extendable o r d i n a l s . However we can exhibit a r a t h e r l a r g e number of examples of such ordinals.
->0
We call an ordinal r
is a predicative formula F with
s E On and r Rs
C
F
e
[ (
R- definable ordinal i f t h e r e
an
Fr(F) =
$
0
f
such that whenever
Rs, then r is the unique element of R x,f]
S
while 0 is a unique such x if r
such that
4
R
.
We c a l l F a definition of r. We now prove in M:
X. ( W i l m e r s ) . F o r every R-definable ordinal r t h e r e is
.
an ordinal whose height i s r+l. Proof. Using p a r t (c) of the Scott - Scarpellini theorem we easily prove that t h e r e a r e ordinals x such that R element a chain of length r
+
+ 1 which belongs t o Ra
b = f(0) and claim that the height of b is r + 1 Since f i s a chain of length r
is a chain
g
of length r
+
contains a s
1. Let a be a smallest such ordinal
and l e f be a chain of length r
obvious that the height of b i s
X
3
r
+
.
. We put
+ 1 starting at b, it is
1. Now we a s s u m e that t h e r e
2 starting at b and derive f r o m t h i s a s s u m =
ption a contradiction. There e x i s t s a predicative formula H with
A. Mostowski
f
Fr(H) =
p and a r b i t r a r y u, v in R
H P is a chain of length (1
P
< 0,
R
such that for an a r b i t r a r y ordinal
0, 1 1
u
>
<
,
I, v
(V
E On)&(u
7).
We shall indicate below how to construct such a formula. Assuming that we c a r r i e d out the construction we proceed a s fol= lows.
-
We easily show that n G Rn+l) c
n in r + 1; since g(n) G R R
g ( r + I).
Now
g
1
g(r+l)
R
g(n+l)
we s e e that g
( r + 1 ) is a chahof length
for a r b i t r a r y
1
(r+l)E
r
+
1 and hence
R g ( r + 1) I = H [ { < o , g ( r + l ) > , T] Denoting by F a definition of r we obtain from the l a s t formula
Since R
g(0)
EX^) EX^) EX^)
(
R g ( r + 1)
is an elementary subset of
can replace in this formula g ( r
+
1) by
infer that there a r e elements c, d of R I
F [{<2,
r>jj
[ F ( x ~ ) & ( x 2 # 0)
and R b
b
H [{do,
R
g(O), i. e. b
g ( r + 1)
by b. We thus
such that k f 0, R
d>,
<
1, c
we
b 'P
+ I>?]-
.
A. The first
of these formulae provesthat
Mostowski
c = r
and the
second that d is a chain of length r + l . Since d 62 Rb and b
<
a ,
we obtain a contradiction with the definition of It remains t o construct the formula
H
. Let
R' ( x , y ) be a formula which defines the relation =
(cf. p
R
C Frm
(f): (
)
79
P='
and let
) & (f
.
a
x =
G ( z, t ) be the formula
c zFr(X))
!=
6, (
xCfJ)--.
The required formula is On(x
1
& i 6
) & F n ( x ) & ( D o m ( x o ) = x l ) & ( i ) ( j ) ( z ) ( t ) 0 j
)
&
+G(
(
j x ~o ) & (
o
j
,
t>fx0)+
)I .
t
z,
>
i
Natural models of G B and of M . These models a r e si= tuated much l e s s densely than the natural models of ZF: we shall prove the following result in the system
ZF
ZF
+
AC
resulting from
b y adjunction of the axiom of choice:
XI. (Shepherdson) anly if
r
is of the form
.
Rr
s + 1
is a natural model of
and
s
is a
.
M
Proof. If r
is a successor ,
R
is a
model of
GB
if and
strongly inaccessible
cardinal; this condition is also necessary and sufficient f o r be a model of
GB
o r of
M
R
r
to
, then
r = s + 1 because there exists a universal
of the model. F r o m the satisfiability of the axiom Subst it follows that if
t
is an ordinal <. s
and
f
a mapping
of
t
A. Mostowski Im ( f, t ) & R s and hence t h e r e is an ordinal
into s
, then
t1< s
such that
Im ( f, t )
t
t'
.
This proves that
s
is
weakly inaccessible. In o r d e r t o show that s is strongly inaccessible we have
l e s s than
s
<
t
t o show that if
.
s , then the cardinal nwmher of
By our assumption
Since we assume that
t
P ( t )
is
and hence p( t )
Rs
Rs.
R
is a model of GB and since t h e r e s + l is a well ordered set which belongs to R and has the same power S
as
P ( t ) , we infer that t h e r e exists an ordinal number in
which has the s a m e power a s is s m a l l e r than
If all
s
P ( t )
. Thus
Rs
this cardinal number
.
s is strongly inaccessible, then we easily check that
the axiom of
M a r e t r u e in
axiom Subst is satisfied because if
Rs + f
.
In particular the
is an9 mapping of
into R s and a belogs to Rs, then Im(f, a) belogs t o R A comparison with theorem
problem: how t o characterize ordinals sfying the relation
Rr
+
VIII r ,
XprR s +
1 ?
Rs
s'
suggests the following s ( r
(
s )
sati=
It i s easy t o s e e
that this problem is wrongly expressed because the relation
in
but not of R is the l a r g e s t element of R r r+l s+J. Therefore we modify the problem and discuss not the relation of question never holds (R
elementary extbnsion but a closely connected relation of elementary embeddability. Definition. t h e r e is a function
f
Rr
is elementarily embeddable in
whith maps
Rr
isomorphically
Rs (with
if
A.
E
respect t o the relation
>
r
Rr,
t r u e of
f
. Since
+
Rr +
strongly inaccessible. Since and
Rs
Rr
onto
is a model of
where
p
s
=
+
r + l
GB, the same is
s
, where
1
is
is the largest element of
Rr
the largest element of
it follow that
Rs + 1
( i. e. f( Rr ) = R s ). s Similary f( r ) = s since
Rr+, maps
f
R
and s R r + l hand the range of f Of
I
such that
i
p
( r
Rg ( f
because an ordinal in r
+ 1
r
is
the largest ordinal of
for reason of cardinality.
ment in
+
be a function which embeds R
+ p and hence r ' + p
R r'
Rr'
is measurable.
Proof. Let into
Rs is a strongly inaccessible ordinal
r
i s elementarily embeddable into
R r + l r, then
.
onto an elementary subset of
)
XII. (Reinhardt). If and r'
Mostowski
1(
+
1 )
the l a r g e s t ordinal
.
Rs +
On the other
s
is strictly contained in
Hence t h e r e is an ordinal Obviously
r + 1 ) ).
Rg ( f )
+
1
s
p
P f R g ( f )
must be the value of
f
for an
argu=
.
F r o m the properties of
f
it easily follows that
f ( x n y ) = f ( x ) nf ( y ) , f ( x V . y ) - f ( x ) V f ( y ) and Rr
f( x - y ) = f( x )
.
-
f ( y )
for arbitrary
x,
y
in
F o r instance the f i r s t equation is established by no=
+
ticing that
x 0 y
is characterised a s the unique element
togcther with
x and y satisfies in
(t)
I[(t E xl
(t 6 x 0 )
together with R s + l '
Rr
& ( t
f ( x ) and f ( y )
c
+
x2)
z
which
the formula
.
Hence
f ( z )
satisfies the s a m e formula in
A. Mostowski Now
F
claim that
we put
t-multiplicative f o r each If and If
x
C
x,
y C r
and hence
= s
y e F,
r G F
then
r - x
-
.
of
F
.
f ( g )
g
F
The intersection unique element of
+
1
x )
is
and
Rr +
.
r
; both these formulae
-
x ) = 0
, then t
g
, it follows that
.
F
is t-multipli=
consisting of elements
satisfies in
Rr
is a function with domain
Rs
f ( t )
n Rg
+
.
( g )
Hence either
prime.
t & r
satisfies in
it is a function with domain
Rr
-
s
p
f ( x ) fl f ( r and
p CE s = f ( r )
and we infer f ( r ) =
Since
g e R r + l
formula which s a y s that infer that
x )
be a sequence of type
Since
From
is not void.
F
f ( r
p
belongs t o F
g
F.
y
Now we show that if cative. Let
f ( y ) = f ( x A y )
then f ( x ) = f ( x ) n f ( ~2) f ( ~ )
F,
f ( x ) or
n
.
r = x d ( r
p 6 f ( x )
and
which is prime and
p € f ( x )
and hence
cannot be true because x or
r
is a filter of subsets of
F
= f ( r - x ) y
either
r : p 6 f ( x )]
.
r
then
x
and
Hence x 5 r,
5
x
p E f ( y ) , i.e.,
it follows that If
<
t
x n y 6 F
hence
{
F =
is a filter of subsets of
a
+
t
, we
the formula saying that
1
can be characterised a s the
which together with
g
satisfies in
the formula ( t E x O ) = ( w ) ( v ) [(
V>E X1)+(t
Hence the same forfnula is satisfies in f ( n R g ( g ) and
f A g ) )
.
R s + l
E by
vd
A. Now we notice
that the s a m e formula
irj
Rg(f(g)) = s + l by (1Rg(f(g)) and f(g) and hence Since Rg(g)S F, we s e e that pE f(x) f o r e v e r y x in Rg(g). tice that x 6 Rg(g) is equivalent t o t=(Ey)( ( y, xo) d x l ) [ { Rr+l R s + l I= ( ~ y ) ( E x l ) [{< 0. f ( x ) >
.
.c
also sati=
f(nRg(g)).
n
sf'ied i n R
Mostowski
< 1, g>]l
Now no=
and hence t o
1, f ( g ) > ~ ] ; i. e..
to
f(x) E R (f(g)). Thus p is an element of e v e r y m e m b e r of Rg(f(g)), g Rg(g) E F. i. e. , p& flRg(f(g)) = f ( 0 Rg(g)). Hence
0
This proves theorem XIII. Notice that this proof like the proof of theorem XI was c a r r i e d out in the s y s t e m Z F
+ AC.
A. Mostowski Lecture
111 In this lecture we shall apply the Scott
-
Scarpellini
theorem to obtain various families of sets which form models of We call a set in f
A , every in aFr (
e
F
Frm
) the set
a
,
belongs to A.
The
of a determined by
with
P'
set
Fr ( F ) =
SF ( a, f )
and
b F [ f
$
and for every
x>j]j;J2
u +O.
will be called the section
f.
We shall show that all sectionsof
elements
a
of
A
can be obtained by iterating finitely many operations Let us con= sider the following operations:
.
predicatively closed if for every a
Fx e a : a
= F
A
ZF
A.
I. If A
rationa
-
1
A
0
e
Mostowski
and A is closed with respect t o the ope=
1
1 2 , then
is predicatively closed.
A
We shall only indicate the essential steps of the proof. F i r s t we notice that an a r b i t r a r y finite sequence whose t e r m s a r e integers belongsto
. Hence
A
c e s with a common finite domain does the set of all sequences
f
s
if a set
of finite sequen=
d SCd ~ e l o n g st o
o x where
x
A , then s o
s
ranges over
and
i s a fixed one one mapping of a s e t d' onto d. Fr( F Hence if the s e t
f
.
.
{ x ,ka F r ( F'
belongs t o A , then s o does the set
where F ' a r i s e s from F by a permutation of variables.
A
By i t c r a t i n g suitably the operations prove that if a
d
& A.
1
- A
F r o m this we
s
easily infer that if
e
A
a s e t of finite sequences with a common finite domain ranges contained te s e t
df
in
a
where
such that
d
5
a
is in
we
a G A , then
is a finite s e t of integers and
d
12
and d
s
is
and with
A , then for e v e r y fini=
d j ~ Wthe s e t
)
f (. ad' : f
d E s t
belongs t o A. Finally we show that i f integers and f s
r
$
n
ad : ( Eu )
d
,s
ad
[ f U{ < n,
a
" ln3, u>]
cs A ,
d
is
a finite s e t of
then the s e t s]{ belongs to
A along with
.We show this by noticing that in o r d e r to obtain
this s e t f r o m
s
it is sufficient to subtract from every member
A. Mostowski
s
of
the s e t of all s e t s of the form \
{f
=
consider the set
E aFr@) : a )= F
f
I] .
rxo t
one of the formulae r ~ l ( x 0 D ( a ) btlongs t o
A1O -
Using the operation
case when A
3
F
e
A
for
we use
F =
a.
L
=
If
a E A
x:
,
and
F is
rxo = x z
, then
A
we extend t h i s result t o the
7
A
2,
established above we show that if
i = 1, 2, then the same result is t r u e for the
Fi
c a s e s when
DF ( a )
t
has a r b i t r a r y variables. Using the operations
a s well a s the r e m a r k s
D ( a )
where
a s we easily see inspecting the operations
A
F
t>]
r F1
I
and
F :
A I 2 in o r d e r to construct
r(
F =
.
xi ) F :
SF( a, f )
from
Finally
DF( a )
.
The main result of the present lecture is a s follows: 11. let
A
be a family of s e t s (indexed by ordinals)
which satisfies the assumptions of the Scott If in addition
A
a =
u { As
and predicatively closed for every
-
s E 0nj
:
r
Scarpellini theorem. and
in On, then A
A
is transitive
is a model of
ZF. Proof. The verification of most of the axiom is immediate We discuss only the axioms Inf, Pot and Subst which a r e slightly mo= r e difficult to verify,. Axiom of infinity r
n
of ordinals: r 0 = 0,
.
We define by induction a sequence
r n+l = min
s
:
Ar
E n
A
and put
A. Mostowski
r = sup r
then A
n'
satisfies the conditions stated in the axiom Inf.
Axiom of power set. F o r let
s ( x ) = min
: x g
P ( a )
& ( x
At ).
f
.
Then
the formula ( v )
[. ( v
tively closed ,
+
xo )
2 f ) -
Fr( F ) in
AFr(
A k ( x O )( E ! x l ) F f ( x ) = m i n
< 1.
y
$
"
>
PI)]?
At E A
of
A
1
>
(xCa) &
u
and F
and since
A u
is predicaz
ip].
a
A
and let
'1
F
be a
p
be
such that
For
x
in
A
let
[{
( y e A S ) & ( A * ' F
{
t = max( r , sup
f(
40, x>
x ) : ( x C-- Ar u
Using the Scott - Scarpellini theorem we determine an ordinal such
that
u
>
t
and
is
.
3 . F u r t h e r m o r e let
C
and put
s ( x ) :
where
U
7
(v E x l )
. Let
0,
s : ( E y ) [
f
P(a) n A
and hence t o A .
U
Axiom of substitution formula with
a>))
p( a ) .n A
it belongs t o A
in
CP(a ) =
u such that
< 1,
SF( At,
This section is equal to
a sequence
x C A -3(x
Now take an ordinal
determine the section
and x
max (r , sup
s: x g A s $ , t =
A]
r\
a CS Ar
for arbitrary
x, y in
AU the following
equivalence holds :
-
Finally we choose an ordinal
and determine the sectiom. an integer such that (xl)
{
( x1
E
x j
xO)
SG( Au.
I
< j, F
isjnot f r e e in ( Exo
[
v such that
(
XO
a
>y and
G
X.J
AU E Av
p ) where G
j
is
is the formula &F]Z
.
,
)I)-
A. Mostowski This section which we denote by
b
belongs t o
A
v
and satisfies the
condition
Remark. Theorem I1 was proved above on the basis of the system
M
established in
.
There is a version of this theorem which can be
ZF. Similary a s on p. 82 we consider a formula
with two free variables and define a formula (Er)
[ On( r
.
) Ri B( x, r
4 formulae listed on p. 8 2
the s e t
1x
Let
C*
B
A by be the conjunction of the
and of the following formulae:
(The f i r s t of the above formulae expresses the fact that : B(x, r ) ]
is transitive and the second that this
is closed with respect t o the operations
set
.
A
- A obviously,the 1 12 ' second formula should be expressed in t h e language of Z F which
can easily be done by writing down the definitions of
Ai(x, y )
as
set theoretic formulae). Imitating the proof of theorem I1 we can derive from C in Z F all formulae obtained f r o m the axiom of Z F by relativising all quantifiers to the formula A. In other words the formula A define an interpretation of
ZF
in
ZF
+
C
%
.
Examples of c l a s s e s which determine models of Example
1. V =
Example 2:
UI Rr
.
constructible sets. We define f o r an a r b i t r a r y t
set a a' =
: r E On]
ZF.
f
SF(a, f) : (F E F r m
Pr
) & (0 b F r ( F ) ) & (f 6 a F ~ ( F -)
103)j
a ' is the family of all sections of a determined by an a r b i t r a r y formula
F
in
Frm
in a. We now put Lo = 0;
+
Lr
The union
= L',;
1
L =
and an a r b i t r a r y sequence
Pr
=
Ls
u{ L
U{ L
f
with t e r m s
3
: r t s (s is a l i m i t number)
is
: r E On)
called the c l a s s of con=
structible sets. It i s e a s y t o construct a relation which well o r d e r s the c l a s s L. We define it a s the union Xo = 0,
+
X
of F r m
X
u
in
Lr
+
s
: r
is obtained f r o m
1
every
{
=
X
- Lr
: r
E
where
on]
s is a limit number and where
if
by the following construction. we denote by
such that f o r s o m e
Pr
Vi Xr
f
F
U
Fr(F) -
L
in
the e a r l i e s t
{
i'or element
03
The t e r m "earliest" r e f e r s t o a fixed well ordering of the denumerable s e t
Frm
Pr
which we think
If t h e r e a r e many sequences equation is t r u e
then we denote by
f
U
of a s fixed in advance. f for which the above the e a r l i e s t of them in the
lexicographical ordering 44 of finite sequences induced by the r e = lation
r
. Now we define
pairs
< u,
v>
where
Xr
u
+
and
v
a s the union of
Xr,
both bdong to
Lr
of the s e t
+
1 - Lr