Constructive Order Types

STUDIES IN LOGIC AND

T H E F O U N D A T I O N S O F MATHEMATICS

Editors

A. HEY T I N G, Amsterdam A. M 0 STOWSKI, Warszawa A. ROBINSON, New Haven P. SUPPES, Stanford

Advisory Editorial Board

Y . BAR-HILLEL, Jerusalem K. L. D E B O U V k R E , Santa Clara H. HERMES, Freiburg i/Breisgau J. HINTIKKA, Helsinki J. C. SHEPHERDSON, Bristol E. P. SPECKER, Ziirich

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM

-

LONDON

CONSTRUCTIVE ORDER TYPES

J O H N N. CROSSLEY Professor of Mathematics Monash University, Melbourne, Australia

1969

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM

*

LONDON

0 NORTH-HOLLAND PUBLISHING COMPANY

- AMSTERDAM

- 1969

No part of this book may be reproduced in any form by print, photoprint, microfilm or any other means without written permission from the publisher

Library of Congress Catalog Card Number 75-78330

PUBLlSHERS :

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY LTD - LONDON

P R I N T E D IN T H E N E T H E R L A N D S

PREFACE

At the suggestion of J. C. Shepherdson the author first attempted to construct an analogue for order types of DEKKER and MYHILL’S monograph (1960) where they studied a recursive analogue of the theory of cardinal numbers. (We give a brief survey of their work, insofar as we parallel it, in chapter 13.) Their work to some extent corresponds to TARSKI(1949) and the reader will find that the present work is related in a similar way to TARSKI(1956). This latter monograph and Dekker and Myhill’s were the chief influences in the development of the present theory; for constructive order types satisfy all the finitary postulates for an ordinal algebra. (Incidentally, we note that if everywhere below we replace “recursive” by “arithmetic” then the results are even closer to those of TARSKI (1956).) However, the problem of constructive order types was also suggested to the author by G. Kreisel and his interest arose because it was felt that the theory might throw light on the question of what is a “natural” wellordering. We give an answer to this01* for ordinals less than em, the o t h solution of W”=CI. But although it is possible to extend the methods used here to larger ordinals (cf. S C ~ T T E 1965; , CROSSLEY and PARIKH, 1963; ACZEL,1966; FEFERMAN, 1968) nevertheless we have no general method or answer for arbitrarily large recursive ordinals and this despite the fact that the first significant attack on the problem was essentially made by V E B L E N ( ~(We ~ ~ ~note ) . that it is certainly not true that the wellorderings given by Kleene’s 0 (see KLEENE, 1955) are “natural” for any

*

See the notes on p. 217.

6

PREFACE

ordinal 2 w 2 under any generally accepted necessary conditions of 1966a; CROSSLEY and S C ~ T E , naturalness (see, for example, ACZEL, 1966). Moreover the present author has felt for some time that constructing actual well-orderings is an ultimately fruitless way of achieving a definition of “natural” or of showing that there is a bound strictly less than wl,the first non-recursive ordinal, to all ordinals which have a natural well-ordering of their type. The reasons for this view may be expressed briefly as follows: if one can construct well-orderings for ordinals
PREFACE

7

We do not restrict ourselves, in our methods of proof, to constructive means but employ whatever classical means are available. Nevertheless we do feel that a great deal of coherence has been achieved through the thoroughgoing use of (partial) recursive functions. Of course, as Kreisel pointed out to the author in 1962, the theory should work for other sets of functions satisfying certain axioms. Now that we have amassed a quantity of results in the recursive case, it is clear that any countable set of functions closed under recursive operations will give the same results as part two. But we are also interested in developing analogues of Nerode and Myhill’s work (see NERODE,1961). We have recently found that this is possible but the generalizationsgo beyond sets and linearly ordered sets to embrace a multitude of algebraic systems and work on these is in progress in collaboration with Anil Nerode. Amongst the many who have encouraged and aided me I should like to single out for special mention Alan Hamilton who read the book at the proof stages and suggested corrections and many improvements and Anil Nerode with whom I have had many hours of pleasant and fruitful discussion, especially during my visit to Cornell University in 1966.

All Souls College, Oxford. November 2 1968

JOHNN. CROSSLEY

INTRODUCTION

Ordinal number theory may be approached in two distinct ways. First we may consider equivalence classes of well-ordered sets under one-one, order preserving, onto maps. On the other hand we can define ordinals set theoretically by 0 =0 (the empty set), successor of a =a’ =a u ( a } and if Q is any set of ordinals then W Q is an ordinal. This second approach was the first to be considered when it came to studying ordinal theory in recursion theory. Thus CHURCH and KLEENE(1936) set up systems of notations for an initial segment of the ordinals (the so-called recursive ordinals). However, though this approach has had many fruitful results it has not yet led us to a deeper insight into the structure of the ordinals. In utilizing the other approach SIERPINSKI (1958), for example, sometimes explicitly noted when the methods employed were effective in some sense: the present work is offered as a further contribution in this direction. We shall often talk about “classical results” and by this we simply mean the results of ordinal theory as in BACHMANN (1955) or SIERPINSKI (1958) and since we often deal with analogies this has turned out to be a useful figure of speech. One classical method, namely definition by recursion, is not available to us since we have not been able to give a constructive version of the sup (or lim) operation. Moreover we believe that the best that can be achieved in this direction will be obtained by following the lines initiated in chapter 12. In order, therefore, to overcome this lack of definition by recursion we define all our operations by set-theoretic means as in SIERPINSKI (1958). In chapter 1we define recursive isomorphisms02of linearly ordered sets.

12

CONSTRUCTIVE ORDER TYPES

Recursive isomorphisms are one-one partial recursive maps which are order preserving. We believe it is necessary, in general, that the maps should be one-one on the whole of their domains and not just on the (possibly non-recursively enumerable) linear orderings. Perhaps it was because this was not realized at the time that RICE(1956) did not develop the theory of constructive order types for ordinal o.With this definition of one-one for recursive isomorphisms we obtain an equivalence relation of recursive isomorphism and the equivalence classes we get are what we call constructive order types (C.0.T.s). We also insist, though we did not do so in our earlier work (CROSSLEY, 1965, 1966; ACZELand CROSSLEY, 1966), that the recursive isomorphisms be order preserving wherever they are defined. The difference this makes is generally small with the exception of the results due to M. Morley in section 2.4. Since there are only countably many recursive functions and uncountably many one-one functions we see that C.0.T.s give a finer classification than (classical) order types. In section 1.2 we define an ordering R of almost the same type as the rational numbers (namely type 1 7, where q is the order type of the rationals) which is recursive. Then we give (what is essentially) Cantor’s proof that any two such orderings are recursively isomorphic. From this point to the end of the monograph we restrict our attention to C.0.T.s of linear orderings which can be embedded in our standard ordering R by a recursive isomorphism. This is a restriction we did not impose in CROSSLEY (1965,1966) and ACZELand CROSSLEY (1966) but we have found it has two advantages. First of all it removes certain features of the theory which the author considers pathological (see example IV.5.1 of CROSSLEY, 1965) and secondly it facilitates the development of a theory of losols paralleling DEKKER and MYHILL’S (1960) theory of isols. 03 Addition of C.0.T.s is defined in chapter 2 in a straightforward imitation of the classical set-theoretic definition. However we now have to ensure not merely disjointness but recursive separability. We include a useful criterion for this due to Dekker and Myhill. Once this is done it is easy to show that not only is the definition of A B satisfactory but even that there is a partial recursive functional which acts appropriately on recursive isomorphisms between representatives. Addition is associative, but not, of course, commutative as it is not commutative in the classical case. In the penultimate section of chapter 2 we establish two basic results. The first is the separation lemma, namely if A =B+ C and A is a linear ordering in the C.O.T. A , then we can choose BEB and

+

+

INTRODUCTION

13

C E C such that the sum of B and C is actually equal to A and not merely recursively isomorphic to A. This lemma easily yields our directed refinement theorem (2.3.2). This theorem is used repeatedly in the sequel. The final section of chapter 2 contains M. Morley’s proof of the analogue of the classical theorem of Lindenbaum and Tarski that if an order type a is similar to an initial segment of the order type B and fl is similar to a final segment of a then a=fl. A fairly easy consequence of this is the result that A * n = B . n if, and only if A = B (where n >0). Quords, which are C.0.T.s of quasi-well-orderings, that is, linear orderings with no recursive descending chains, are introduced in chapter 3. We show that the condition that there be no recursive descending chains is equivalent to every non-empty recursive (or, equivalently, recursively enumerable04) subset having a least element. It is easy to show using SPECTOR (1955) that there are (even recursive) quasi-well-orderings which are not well-orderings though this also follows directly from Parikh’s counter-example below in chapter 7. Quords are closed under addition and, conversely, if A = B + C is a quord then so too are B and C. But quords have other nice properties. For example, we can cancel on the left. But they do not have perhaps as many properties analogous to those of ordinals as one might expect (in particular see chapter 7). One question which we barely touch on and to which we would like an answer is: What order types can recursive quords have? For it seems that if one considers linear orderings of a certain class (say, recursive ones) with no descending chains of another (not necessarily different) class then there are well-ordered initial segments whose length only depends essentially on the class of functions considered (see, e.g. GANDY(1960)). In the present situation we merely show that recursively enumerable infinite quasi-well-orderings have types of the form w + z where z is an undetermined order type. In chapter4 we study our orderings of C.0.T.s and since these are most importantly employed with respect to co-ordinals we also define the latter. These are the C.0.T.s of well-orderings. Co-ordinals are closed under addition just like quords and the same sort of converse holds (see above). We define A c B if there are representatives which are settheoretically included in each other (with the ordering on the sets correct)

14

CONSTRUCTIVE ORDER TYPES

and A S B if there are representatives such that one is an initial segment of the other. I* is defined similarly with “final” instead of “initial”. Classically, with c ,< interpreted in the obvious way, if a, /3 are ordinals, then a c implies a 5 /Ibut the proof of this fact requires non-recursive methods and the corresponding implication does not hold for co-ordinals. Iis the more important ordering and we show that it is a partial ordering of quords and a partial well-ordering of co-ordinals; but it also has the extremely useful tree property namely, if A, B I C then A, B are comparable i.e. A S B or B S A. Finite sets are recursive and this fact is exploited in chapter 5 where we continue our study of co-ordinals. In particular, co-ordinals have unique successors, and if their classical ordinals are successor numbers then the co-ordinals have unique predecessors too. However, when we deal with limit numbers we get violent deviations from the classical theory. Thus there are uncountably many co-ordinals of type o and these are all pairwise incomparable with respect to 5 . But although we have so many unreasonable co-ordinals to choose from most of our later counter-examples are based on just one and that a recursive one. We let V be the co-ordinal of a r.e. non-recursive set (of non-negative integers) ordered by magnitude. On the other hand the co-ordinal W of all the non-negative numbers ordered by magnitude is “natural” and is employed in many situations to mirror the classical results. However, we can still make use of the classical theory and a striking example is given in section 5.4 where we characterize co-ordinals which commute under addition. Multiplication is introduced in chapter 6. We imitate the standard settheoretic definition but code ordered pairs by single numbers in order to remain within our definition of ordering. Quords and co-ordinalsare both closed under multiplication and a sort of converse (as in the case of addition) holds (with the usual trivial exceptions). Multiplication is associative and distributive over addition on one side but, as in the classical case, not commutative and not distributive on the other side. In section 6.2 we establish a weak infinite generalization of the separation lemma and 8 6.3 is devoted to a proof that B + A = A if, and only if, B* W S A when A is a co-ordinal and a slightly weaker version for quords. The results of these two sections are used to obtain analogues of a number of Tarski’s theorems in TARSKI (1956). However, we have not been able to obtain analogues of Tarski’s theorems for C.0.T.s in general

INTRODUCTION

15

but only for quords. In particular we do not know whether the equivalence above holds for all C.0.T.s; we suspect not. Gandy has shown (May 1968) that a weaker version, namely, 2 + A = A implies 1+ A = A is false for arbitrary C.0.T.s. Previously HAMILTON (1968) had established the result for the definition of C.0.T.s given in CROSSLEY (1965). Exponentiation is introduced in chapter 7, again by the classical settheoretic definition plus coding of certain finite functions. Co-ordinals are closed under exponentiation but PARIKH(1966) showed that this is not true of quords and we include a version of his proof. The basic laws for exponents, e.g. A B - A C = A B + Cgo , through just by imitating the classical proofs. The chapter ends with a proof that if A 2 1, then B. A = A if, and only if, BW divides A thus mimicking the result at the end of the previous chapter though the proof here is more messy though structurally less sophisticated. Arithmetic laws are developed in chapter 8 and here we get significant divergence from classical results. Thus half of the analogues of the monotonicity laws for ordinals break down in the cases of addition, multiplication and exponentiation. However, consideration of principal numbers for addition (or multiplication or exponentiation) i.e. co-ordinals which absorb smaller co-ordinals additively (etc.) allows us to salvage linearly (and therefore well-) ordered sets of co-ordinals which obey the analogues of the classical laws. Even without this restriction we have two techniques which yield interesting and useful results. First we can use our results of the previous chapters to obtain TARSKI(1956)-style results and secondly we use the (classical) fact that there is a unique (classical) isomorphism (onto) between two well-ordered sets of the same ordinal so that if we have a recursive isomorphism it must be an extension of that classical one. This gives us cancellation theorems for co-ordinals and we extend this same technique in chapter 16. Cantor normal forms are treated in chapter 9, first in the weak sense of decomposition into sums of principal numbers for addition and then in a straight analogue of the classical case for a large class of co-ordinals. In order to effect the proof we prove the basic Euclidean algorithm, C< A B implies C = A Q R where 0 IR
-

+

16

CONSTRUCTIVE ORDER TYPES

cation, respectively, though in chapter 12 we shall show that not all such principal numbers are of this form. A lot of the chapter is taken up by computations which from time to time appear to require interjections of classical results to give all the information we need. It would be interesting to know if these classical interjections could be avoided. The bounds o"'and wowcited above seem to tie in directly with the results of EHRENFEUCHT (1957) since here we are essentially dealing with the universal formulae of his theories. E-numbers, our analogues of the classical &-numbersare introduced in chapter 11 and here we give the Aczel-Crossley proof that E-numbers can be characterized in an exactly analogous fashion to &-numbers.Thus, for example, E-numbers are the solutions of W A = A .As an immediate corollary of our proof we obtain the &,-uniqueness for principal numbers for exponentiation. ACZELin his thesis (1966) introduced recursive sequence types and we introduce a related concept, constructive sequence types, in chapter 12. Because the infinite sequences we consider have uniformity properties we are able to define sums so that the co-ordinals obtained are unique. We also prove analogues of the separation lemma and the directed refinement theorem in two versions. Next in this chapter, we consider upper bounds of sets of co-ordinals. The general situation is that upper bounds exist only for countable linearly ordered sets of co-ordinals and least upper bounds exist only trivially, i.e. as maxima. On the other hand there is an abundance of what we call minimal upper bounds for any bounded linearly ordered set of co-ordinals. Finally, we prove the existence of linearly ordered sets of co-ordinals (paths) which contain one co-ordinal for each countable ordinal and are closed under addition, multiplication and exponentiation so that they are true likenesses of the classical countable ordinals. Part two of the monograph commences with a survey of Dekker and Myhill's (and others') results on isols and recursive equivalence types. These latter are the cardinal analogues of C.0.T.s so we set out to find the analogues of isols. This we did and they rejoice in the name of losols. These are C.0.T.s of linearly ordered sets which are not recursively infinite (in the sense that they contain an infinite r.e. subset). In chapter 13 where we treat isols we only sketch proofs in general since either they are available elsewhere or can be reconstructed easily from their order

INTRODUCTION

17

analogues in other parts of the monograph. $ 13.1 contains the basic definitions, $ 13.2 exhibits several different ways of characterizing isols and $13.3 considers the cancellation laws. Quasi-finite C.0.T.s are introduced in chapter 14. These are C.0.T.s which are quords and whose converses are also quords (though we establish many equivalent definitions). These C.0.T.s get their name from the fact that classically finite ordered sets may be defined as those with no descending or ascending chain and in our case we insist that there should be no recursive ascending or descending chains. In section 2 of this chapter we show that there exist recursive quasi-finite linear orderings of type w + w * . This was originally shown by Stanley Tennenbaum and the proof given here is based on a construction by C. G. Jockusch. However, quasi-finite C.0.T.s are not closed under exponentiation though they are closed under addition and multiplication and we have not been able to establish the full “converses” of the closure results. Losols are introduced in chapter 15. All losols are quasi-finite but the converse is false. We give the Hamilton-Nerode proof that there are 2”’ losols of type z (where z is any countably infinite order type) and that these may be chosen to be pairwise incomparable under E. We change our definition of 5 slightly for convenience. Next we show that losols are closed under addition, multiplication and exponentiation and again the pseudo-converses hold with trivial exceptions. Parikh‘s construction is defeated here because there being no infinite r.e. subsets implies that there are no recursive ascending or descending chains. Chapter 16 is an extended exploitation of the fact that there is a unique order isomorphism between any two finite linearly ordered sets of the same cardinal. This culminates in a general cancellation metatheorem to the effect that P(X)=P( Y ) implies X = Y whenever P is a non-trivial function constructed from taking converses, addition, multiplication and exponentiation using losol parameters. This brings us to our final question: Does there exist a Myhill-Nerode-style theory (see NERODE (1961)) of extensions of functions to losols? The answer is “Yes” and a detailed treatment of this (rather complicated) subject by Nerode and the present author is in preparation. There are two appendices: Appendix A by P. H. G . Aczel and the author has as its main theorem: If C is a quord and 1 IC then CA = CB implies A =B for arbitrary C.0.T.s A , B.

18

CONSTRUCTIVE ORDER TYPES

Appendix B contains Hamilton’s proof that there are 2N0principal numbers for multiplication not of the form W” thus resolving a question left open in chapter 12. He also introduces the notion of an infinite product. We have relegated the section on notation and terminology to the end on the grounds that almost all of it is standard and the section will probably only be used for reference.

CHAPTER 1

RECURSIVE I S O M O R P H I S M

1.1 By a relation we mean a set of ordered pairs of natural numbers, i.e. a subset of N2. We use upper case Gothic letters (A, B, ...) for relations. A relation A is said to be reflexive if <x, Y)EA*<X,

(i)

X)EA&(Y, Y>EA.

By thefield of a relation A we mean

{x:(3y) (<x, Y> EA v .A)}

*

We write C'A for the field of a relation A. If A is a reflexive relation, then

C ' A = {x:(x,x)EA}.

A partial ordering is a reflexive relation A such that (ii)

(x, y) E A & (y, x) E A * x = y

(Antisymmetry)

and (iii)

(x, y ) ~ A & ( y ,z ) ~ A = ( x , Z)EA

(Transitivity).

A linear ordering is a partial ordering which also satisfies the trichotomy law (iv)

X,

Y E C ' A * : X = y v (x, Y > E A v (y, x ) E A .

In this monograph we shall be exclusively concerned with linear orderings.

1.1.1 DEFINITION. A relation A is said to be isomorphic to a relation B

22

CONSTRUCTIVE ORDER TYPES

[Ch. 1

if there is a one-one (possibly partial) function f such that

C ' A G ~and ~ f(C'A)=C'B

0) and

(x, Y > E A

(ii)

--

(f(X)Y

If there is such an f we write f:A

f (Y)>E B -

By or even simply AN B.

In this definitionand in definitions 1.1.4 and 1.1.5 below,f is to be one-one on the whole of its domain (and not merely on the field of A). This condition ensures that in all three casesf is well-defined on pf, though this means that ff-' may be only a restriction of the identity function in certain cases. Clearly isomorphism is an equivalence relation. We write OT(A) = { B: B A} and if A = OT (A) then A is said to be an order type.

-'

-

1.1.2 DEFINITION. A relation A is said to be recursive (r.e.) if there is a recursive %(a, b) (%(a, by c ) ) such that ( a , b)EA+'ZC(Q, b ) , (
%(a, b, 4).

1.1.3 THEOREM. If A is a recursive (r.e.) relation, then C'A is recursive (r.e.) provided A is reflexive. PROOF. C'A = {x:'u(x,x)} if A is recursive ( = { x : (32) %(x, x, z)} if A is r.e.). In this monograph we shall only deal with sub-relations of r.e. relations and although in confining our attention to r.e. linear orderings we shall not be treating C.0.T.s in the generality we did in our earlier papers (CROSSLEY, 1965,1966; ACZELand CROSSLEY, 1966) nevertheless it appears that the presentation can be made more coherent and more natural by this strategy.

1.1.4 DEFINITION. Suppose A, B are linear orderings. Then a map p ( x ) is said to be a recursive isomorphism from A to B if (0) There exist r.e. linear orderings A , B' such that A s A and BE B', (i) p is a partial recursive function, (ii) p is one-one,

ch.11

RECURSIVE ISOMORPHISM

23

(iii) C ' A = 6 p , p ( C ' A ) = C ' B ' andp(C'A)=C'B, (iv) < x , Y > E A o ( P ( x ) , P ( Y ) ) E B ' . A is recursively isomorphic to B if there is a map p which is a recursive isomorphism from A to B. We write p : A- B if p is a recursive isomorphism from A to B and A- B if there is a recursive isomorphism from A to B. We claim that recursive isomorphism is an equivalence relation. The identity map is recursive, hence recursive isomorphism is reflexive. If p is a one-one partial recursive function, then p-' (defined, of course, only on pp) is also partial recursive (see e.g. MCCARTHY, 1956, p. 177). It then follows easily that if p : A- B, then p - l :B- A. We show that recursive isomorphism is transitive. We first observe that if p : A- B, then A is r.e. if, and only if, B is r.e. Suppose ~ : A ' B', N q: B " - C ' , q: B-C and A 3A, B' 2 B, B" 2 B, C' 2 C where A', B', B", C' are r.e. Define an r.e. Bo as follows: Let B- be the r.e. set of ordered pairs ( x , ~ ) E B n ' B" with x + y and let Bo = B- u { ( x , x ) : x € C ' B - } . Clearly Bo is r.e.. Bo is linear, for suppose ( x , y ) Bo ~ and x + y , then(x, y ) B'n ~ B" and therefore (y , x ) $ B' u B" z B' n B" 2 B-. (That Bo is a partial ordering is obvious.) Clearly B o 2 B so let p' be p with range restricted to (the r.e. set) C'BO and q' be q with domain restricted to C'B'. Then p ' , q' (and therefore q'p') are one-one partial recursive functions and it is now trivial to check that q'p' :AN C. We can now introduce our fundamental definition. 1.1.5 DEFINITION. If A is a linear ordering and A = { B : B =A}, then A is said to be a constructive order type (C.O.T.) and we write A =COT(A).

1.1.6 DEFINITION. A function f is said to be a recursive permutation if f is recursive and maps .N one-one onto itself. We write f:A r B if A is recursively isomorphic to B by the recursive permutation f. We have no significant results comparable to DEKKERand MYHILL (1960 theorem 5) connecting the and N relations though they are easily shown to be distinct (see also corollary 1.2.5). (In fact R. I. SOARE (1969) has now shown that the analogue of that theorem 5 is false for linear orderings.)

1.1.7 THEOREM. (i) A r B a A - B*A- B. There exist linear orderings A, B such that A- B but A + B, (ii)

24

CONSTRUCTNE ORDER TYPES

[Ch. 1

(iii) There exist linear orderings C, D such that C N D but C $ D. PROOF. (i) Clear from definitions 1.1.1,4 and 6. (ii) Let A be a r.e. non-recursive set. Then A is infinite with an infinite non-r.e. complement A. Let

A

= {(a, a'):a,

a'EA&a I a'}

and B = {(b,b'):b,b'EA&b
Then ANB, since A, B are both well-orderings of type w, contained in the natural (r.e.) ordering of A". A- B implies A=f(A) for some partial recursivef'with domain including A. This implies A is r.e. which contradicts the choice of A. (iii) Let

c = {(c,

c'):O I cI c'}

and

D = { ( d , d ' ) : 1 I d Id ' } . Then iff(n)=n+l,f:C-D. But g : C r D implies g-'(O) is undefined which is in contradiction with g being a recursive permutation. 1.1.8 COROLLARY. There is a linear ordering C such that OT(C)=, COT(C). PROOF. Let C be as in the proof of the theorem and let

E

= {(x,

y ) : x I y &x, YEA},

where A is as in the proof above. Then EEOT(C)-COT(C). This corollary shows that C.0.T.s give a finer classification of (denumerable) relations than do (classical) order types.

1.2 An innovation (as opposed to CROSSLEY, 1965) is a uniformization of the orderings we employ. Here we shall only consider orderings which can be embedded by a recursive isomorphism in a r.e. ordering of the type of the rationals. Cantor's original paper (see CANTOR, 1915) virtually shows that every r.e. ordering is embeddable in a standard ordering R and secondly that any two standard orderings are recursively isomorphic. However we first of all show that up to recursive isomorphism it does not matter whether we consider r.e. or only recursive orderings.

Ch. 11

25

RECURSIVE ISOMORPHISM

1.2.111 THEOREM. Every r.e. linear ordering is recursively isomorphic to a recursive linear ordering (and conversely). PROOF. Let A be a r.e. linear ordering, then there is a recursive predicate %(x, y , z) such that (x, y ) ~ A o ( 3 z %(x, ) Y,z ) . Let A = C'A. If A is finite there is nothing to prove since then A is finite and hence recursive. Otherwise A is infinite r.e. and there is a one-one recursive function g which enumerates A without repetitions. Since A is linear (1) (VX) (VY) (34 ( ' 1 1 ( 9 ( 4 S(Y), 2) v '11MY)Y S(Xh z ) ) . Since A is antisymmetric, if(U(g(x), S(Y), Z0)&'11(S(Y), S(X>Yz,)theng(x)

= g(Y)Y

and hence x = y and %(g ( y ) , g (x), zo). Now let B be the relation defined by (x, Y> E B *1 ' 1(9 (49 ( Y ) , P z {'11(9 (XI, 9 (Y>Y z ) v 1 ' 1(9 (Y>Y 9 (XI, z)l) Then B is recursive, by (l), and clearly g: BE A. The analogue of theorem 1.2.1 is false for arbitrary r.e. relations (see CROSSLEY, 1965, theorem 1.4.5).

A linear ordering A is said to be dense if x

+ y &(x, y ) ~ A * ( 3 z ) (x + z & z + y &(x,

Z ) E A &(z, Y)EA);

A is said to have a first element (or minimum) a if yEC'A*(a,y)EA;

wewrite a =min(A).

First elements are clearly unique if they exist. Similarly we may define last elements. We now introduce our standard dense linear ordering with a first element, which is 0, and no last element. We depart a little from KLEENE (1955) to get the right ordering. Given a finite non-empty sequence of natural numbers a(O), ..., a(n - 1) with a ( n - 1) 9 0 ,

26

[Ch. 1

CONSTRUCTNB ORDER TYPES

we can uniquely assign to it the sequence number

n(n) =

n pf(i).

i
We may also use the notation12 d ( n ) = ( a ( O ) , ..., a ( n - 1 ) ) .

We denote the (number of the) empty sequence by 0. Clearly the set of all sequence numbers is recursive. We call this set Seq and we write Seq(n) if n is a sequence number. Note that, unlike Kleene, we insist that a non-empty sequence shall not end in zero. We next define our ordering of Seq as follows: If x, y ~ S e q say , x = d ( m ) = < a ( O ) , ..., a ( m - l ) ) , y = 6 ( n ) = ( b ( O ) , ..., b ( n - 1 ) ) ,

(2)

then let x + , y + be the infinite sequences obtained by putting zeros after a ( m- l ) , b (n- l ) , respectively. Then, if x + y , we set x +
X+

= {a(i)}im,o, y+ = {b(i)};"=o

and u ( O ) = b(O), ..., a ( k ) = b ( k ) , a ( k + l ) + b ( k + l ) ,

t h e n x + < y + justincasea(k+l)
for ( x , y ) ~ R

x(y

for ( x , y ) ~ R & x f y .

and

1.2.2 THEOREM. R is a primitive recursive dense linear ordering of Seq with first element 0 and no last element. PROOF.The only unclear part concerns the density. Suppose x&eq then clearly x < x * ( l ) so R has no last element. Suppose x
..., c ( r -

l)),

Ch. I]

RECURSIVE ISOMORPHISM

where

27

+

r = max{m, n} 1 , c ( i ) = a ( i ) if i < m, =O if m < i < n , =1 if i = r - 1 . Clearly x+
so R is dense.

1.2.3 THEOREM. Let A be any r.e. linear ordering (of a subset of Jv) then A is recursively isomorphic to a linear subordering of R. If A has a first element then we may take the recursive isomorphism to map this element onto 0. PROOF. By theorem 1.1.3, C'A is r.e. If A is finite then the result is trivial. Otherwise there is a (general) recursive function a which enumerates C'A without repetitions and such that a(0) is the first element of A (if there is one); similarly let r enumerate Seq. We fist define a recursive function p. Let p (0) =O and suppose p (0), ...,p (n) have been defined. Then putting X
p(n+l) is chosen to be r(m) where m is the least integer such that (i) r (m)is distinct from p (0), ..., p (n), and (ii) either (a) if a(i)
then for some B, q :A= B s R since recursive isomorphism is symmetric and transitive. 1.2.4 THEOREM. If A, B are r.e. dense linear orderings with first elements but no last elements then A is recursively isomorphic to B. PROOF. Since A, B are dense and r.e. there exist recursive functions a, b which enumerate C'A, C'B, respectively, without repetitions and such

28

CONSTRUCTIVE ORDER TYPES

[Ch. 1

that a(O),b(0) are the first elements of A, B. We define the graph of a recursive isomorphism p : A- B. We write u0 for a(O),bo for b(0) and suppose

(2,bO), ...)(a", b") have been defined. If n is even we let an+' be a(m) where m is the least integer such that

a(m)${aO,...)a"> and let b n f lbe b ( k ) where k is the least integer such that (i) b ( k )${bo,..., b"} and (ii) either (a) if ui
CHAPTER 2

ADDITION

2.1 Since all (suborderings of) r.e. linear orderings can be embedded in our standard ordering R by a recursive isomorphism we shall restrict our attention from now on to C.0.T.s of suborderings of R unless otherwise stated. If there is an AEA such that A E R then we write2I A s R where we set

R

= COT(R)

and B = { A : AE R } .

We call B the collection of all standard C.0.T.s when we wish to distinguish 9 from the collection of all C.0.T.s as defined in CROSSLEY (1965). Consider any subset A of Seq, then A generates the linear ordering A defined by (x, y ) ~ A - x , y ~ A & x
We shall therefore adopt the convention of conflating a subset of Seq with the ordering it generates. But if there is the possibility of confusion then we shall write [A] for the ordering the set A generates. If A and B are arbitrary orderings, then the ordinal sum of A and B is defined by A 4 B = A u B u (C'A x C'B). If the ordinal sum of two order types is defined as the order type of the ordinal sum of arbitrary representatives of the given order types, then

30

CONSTRUCTIVE ORDER TYPES

[Ch. 2

this definition is not, in general, unique. This is because the fields of the representative orderings may have a non-empty intersection. But if we define the order type of the sum in terms of representativeswhich do have disjoint fields, then the definition is unique (cf. RUSSELL and WHITEHEAD, 1927, pp. 341, 345 *160.48). Two relations are said to be strictly disjoint if their fields are disjoint. However, we observe that if A, B are reflexive, then A n B = 0+CAnC'B=0, or using the notation introduced above [A] n [B]

= @*A n B = 0

when A, BcSeq. Now, in order to define a recursive version of ordinal sum we require "recursive disjointness" i.e. C'A and C'B must be contained in sets which are effectively disjoint. If this is not the case then the following situation arises: Let A be a r.e. non-recursive set and let B be a r.e. set containing A. Then there is no (partial) recursive function defined on N which agrees with the functionsf, g on their domains of definition wheref(x)=x if XEA, otherwise undefined and g ( x ) = x + l if XEB, otherwise undefined. Hence there can be no (partial) recursive function defined on C' ( A T B) if C A = A and C'B=A, even though A and B are strictly disjoint. 2.1.1 DEFINITION. A is separable from B if there are disjoint r.e. sets A', B' such that C ' A r A' and C'BcB'. If A is separable from B we write

A) (B and we also write

A) (B for CAI) ( P I

-

2.1.2 THEOREM. If A, Bc R and (i) every element in C A precedes (under <) every element of C'B and (ii) either (a) B has a first element or (b) A has a last element then

A) (B.

PROOF. We treat only (a) and leave (b) to the reader. Let b be the

a.21

31

ADDITION

first element of B, then the sets

A'= ( x : x < b }

and B'= { x : b < x }

are recursive and disjoint and clearly contain C'A and C'B since if any element of A precedes every element of B then it certainly precedes b. This theorem will be used frequently in the sequel, especially when we are dealing with C.0.T.s of well-orderings. The following criterion is also useful.

2.1.3 THEOREM. A) (B if, and only if, there is a partial recursive function p such that if A = C'A and B = C'B then (i) 6 p z A u B , p p s { O , l } and (ii) if x € A u B then x ~ A o p ( x=) 0 and x ~ B e - p ( x=) 1.

PROOF.If A) (B, then by definition 2.1.1 there are r.e. sets A', B' such that

A ' z A , B ' 2 B and A ' n B ' = 0 . For an arbitrary r.e. set S let cs be the partial recursive function such that cs(x) = 1 if XES,

otherwise undefined.

Then XEA' imples 1-2-cAP(x)=O and XEB' implies cB.(x)=1. Let TA be a Turing machine which calculates 1 c., (x) and let T B be aTuring machine which calculates c,. (x). Further, let T(m,n) = the number (represented) on the tape of the Turing machine T at the mth step22in the calculation for argument n. Now let a new machine To be defined such that To(m,n) is as follows: (i) If TA, T B have not halted before the mth step for argument n, then To(2m+1, n ) = T ~ ( m + 1 ,n), (ii) If T A has not halted before the (m+ 1)th step and T B has not halted before the mth step, then To(2m 2, n) = TB (m 1, n), (iii) If T A halts at the mth step and TB has not halted before the mth step, then To halts at the (2m+1)th step, (iv) If TB halts at the mth step and T A has not halted before the (m+ 1)th step, then To halts at the (2m 2)nd step.

+

+

+

32

[Ch. 2

CONSTRUcIlVE ORDER TYPES

Let p ( x ) be the function defined by the machine To. Then p is partial recursive and satisfies the conclusion of the theorem since, for an argument in A u ByTAhalts if, and only if, TB does not. Conversely, let A'= ( x : p ( x ) = 0} and B'= { x : p ( x ) = 1).

Then A and B' are r.e. and disjoint and by hypothesis A S K and B E B'. This completes the proof. It is very convenient to be able to produce separable representatives of given C.0.T.s and we therefore introduce some notation to this end. We recall that if x , y ~ S e qthen x =((40,

where

*.*Y

(.Im>

and Y = <(Y>OY

me.7

(Y),),

m = Zh(x) + 1 and n = Zh(y) + 1 and X * Y = <(.)OY

-

*.

9

(&

b)OY

- *.

Y

(Y)") *

For each natural number i we let Z be the one-one partial recursive function defined by i ( x ) = ( i ) * x if xESeq, x =t= 0, O(0) = 0, i(x) is undefined, otherwise. We also set FA = ( t ( x ) : x ~ A )if A s Seq, and iA=[iA] if A=[A] c R . We also require the inverses of these maps; we shall denote them by I, so I(x) = 42' if XEZ Seq, x = O if x = O ,

+ 0,

is undefined otherwise. The orderings we are particularly interested in are those of the form ZR and we therefore introduce the notation ~ < ~ y o ( y x ), ~ i R

so

x < i y o ( 3 x ' ) (3y') ( x = i(x') & y = i(y') &x'

4 y').

Ch. 21

33

ADDITION

2.1.4 THEOREM. (i) x
2.2. 2.2.1 LEMMA. Let A,NA,, B,N B,, A,) (B, and A,) (Bl, then A,? B, N A, ? B, . PROOF. Let A,=C'A,, Bi=C'B, ( i = O , 1). By hypothesis there exist recursive isomorphisms p , q and linear orderings A;, BI such that p:Ab=A;

and q :Bb = B ;,

where Ai G A:, Bi c BI ( i = 0, 1) and the A;, B: are r.e. Further there are r.e. orderings A;, B; such that

Ai c A;, Bi c BY and

Ah n A;

=

Bg n B;'

= 0.

Let p' be the partial recursive function defined by p ' ( x ) = p ( x ) if x E 6 p n A b n A E and p(x)EA;nA;, is undefined otherwise

(where Ai = C'A,, etc.). Similarly let q' be the partial recursive function defined by q' ( x) = q (x ) if x E 6 q n B b n B g and q(x)EB;nB;, is undefined otherwise.

34

[a. 2

CONSTRUCITVE ORDER TYPES

Let r be the partial recursive function defined by ~ ( x= ) p ( x ) if p'(x) is defined, = q ( x ) if q ' ( x ) is defined,

is undefined otherwise. Since p' and q' have disjoint domains and ranges r is well-defined. Clearly r:A;$ B: -A: B: and r : A o $ Bo=Al 4 B,,

+

where A:,

A:,

B,:

8: are the r.e. relations

(w)2, r

A0 n

(A019

-

Bo n(&i)2, rP o )

This completes the proof.

Now 6 Seq) (f Seq since these two sets are recursive and disjoint hence we can define for A, 6 c R, A

+B=

6 ~ TB +

and we have the following strengthening of lemma 2.2.1. 2.2.2 THEOREM. There is a partial recursive functional @ such that if p:Ao-Al and q:Bo=B,, then

@ ( p , q):Ao

+ B,

A1

+ B,.

XES Seq, =Tq(i(x)) if xeT Seq, is undefined otherwise.

PROOF. Set @ ( p ,q) ( x ) = Q p @ ( x ) ) if

Theorem 2.2.2 allows us to make the following definition.

DEFINITION. A+B=COT(A+B), where R ~ A E A and R ~ B E B . We note that if A) (B then A$ BeCOT(A)+COT(B) by lemma 2.2.1. Notation: 0 = COT (0).

2.2.3

THEOREM. (i) A +0 =0f A = A , (ii) A +B = O o A =B=O. PROOF. Left to the reader.

2.2.4

a.21

35

ADDITION

+ is associative, viz. for all A, B, CE W, 2.2.5 THEOREM. (A

+ B ) + C = A + ( B + C ).

PROOF. We need to show that if A, B, C G R and A, B, CEA, B, C, respectively, then

6(6A T TB) T TC N QA4 T(6B IC). So set if if = i i ( y ) if is undefined

x = QQ(y)& Seq(y), x = 6I(y) & Seq(y),

p ( x ) = Q(y) =

x = i ( y ) &Seq(y), otherwise.

We leave the reader to verify that p is the required recursive isomorphism. We know classically that 1+ o + w + 1, so addition of C.0.T.s is not commutative.

2.3 In the previous section we were concerned with putting C.0.T.s together by means of sums, in this section we prove a useful lemma and an important theorem showing that we can, so to speak, take the C.0.T.s apart again. 2.3.1 LEMMA. (SEPARATION LEMMA) If A =B+ C and AEA, then there exist BEB and C E C such that B)(C and

BTC=A.

PROOF. By definition 2.2.3, there exist B’, C’ c R with B’E B and C’EC such that B + C ’ E A , say B C‘ = A ‘ .

+

Since A‘EA there is a recursive isomorphism

p:A’ N A . Let

B = p(6B’) and C = p(TC’). Since QB‘T TC’=A’ we have at once

B+C=A.

36

[Ch. 2

CONSTRUCTIVE ORDER TYPES

Finally, B) (C since

This completes the proof. Notation. A [B = { (x,

r) :(x, y ) E A &x, YE B}

I

2.3.2 THEOREM. (DIRECTED REFINEMENT THEOREM) If A there is an E such that either A = B + E and E + C = D , or A + E = B and C = E + D .

+ C = B+ D, then

+

PROOF. If D =0, let E= C. Otherwise we may assume D 0. Let AEA and C E C where A) (C and AT C E R. Then by the separation lemma (2.3.1) there are relations BEB and DED such that

B)(D and A $ C = B $ D . Let A = C'A, B = C'B, C = C'C and D = C'D. Then

AuC=BUD. We consider two cases separately. Case I . If D n A + & let

E=A-B

and

E=D[E.

By construction, A E B u E and E E D, therefore

B)(E and

E)(C.

Now B s A $ C and E c A $ C ; further, xEBandyEEimplyyEDand(x,y)EB x D E A T C .

Thus

B$ Es(ATC)[A=A. Conversely, if (x, y ) E A, then one of x, ~ E B ;x, ~ E E ;X E B & ~ E E

Ch.21

37

ADDITION

must hold since x ~ E & y e B contradicts (DxB)n(B+E)=8&EcD. ~ E. Hence AG B+ E and therefore In all three cases in (1) (x, y ) B+

A=B+E. Similarly,

E+C=D. Now put E=COT(E) and the theorem follows for this case since we are using the identity (on R) as our recursive isomorphism. Case 2. If D nA =8, then B = 0 or B nC 8. In the former case put E= 0 (since A , B, C, D are then all 0) and in the latter case the proof of the existence of an E such that A E= B and C = E+ D is merely a literal variant of that for case 1. This completes the proof.

+

+

2.3.3 COROLLARY. If E + A , + . . - + A , , + F = B o + . . . + B , , then there exist C, D such that for some I, m, 0 < 1, m 5 n

B, = C

+ A, + D .

PROOF. The assertion is trivial if n = 0. Suppose it holds for all numbers less than n, the hypothesis is true but (3C, D ) ( B , = C + A , + D ) is false for all I , m l n . Let A , E A , and A,=C'A,, etc. Then A,$B, by assumption so by the proof of the preceding theorem B , + . . - + B , , = =D' + A , .-.+A,, F for some D'. The result now follows at once by the induction hypothesis.

+

2.4 2.4.1

+

Notation: A*={<~,X):(X,~)EA},A*=COT(A*) for any A E A .

DEFINITION. A* (A*) is said to be the converse of A (A).

THEOREM.(i) ( A + B)* =B* + A * , (ii) A * * = A . PROOF. Left to the reader.

2.4.2

2.4.3 2.4.4

DEFINITION. A + * B = (A* +B*)*.

THEOREM. A+*B=B+A. PROOF. Left to the reader.

38

[Ch. 2

CONSTRUCTIVE ORDER TYPES

As a consequence of theorems 2.4.2 and 2.4.4 we have the following principle of duality: If P is any property of C.0.T.s involving only * and +*, variables for C.0.T.s and logical connectives then P is true for C.0.T.s if, and only if, P* is, where the description of P* is obtained from that of P by replacing +, by +*, respectively. We call P* the dual of P.

+,

+*

+

We now introduce our orderings of C.0.T.s. 2.4.5

DEFINITION. A is said to be an initial (final) segment of B if XE

C'A & YEC'B - C'A * (x, Y)E B(* ( y , X)

E B).

In such a case we write

A s B (As*B). We also write A S B (A<*B), if there exist AEA and BEB such that A s B ( A s * B).

LEMMA. A S B ( A s * B ) if, and only if, given AE A and BEB there is a recursive isomorphism from A onto an initial (final) segment of B. PROOF. Left to the reader.

2.4.6

2.4.7

THEOREM.(i) ( 3 C ) ( A + C = B ) * A S B , (ii) (3C) ( C + A = B ) = > A < * B .

The converse is false in general and this will be shown to be so later (chapter 14). The next theorem is very useful indeed. First we need a lemma. Notice that we do not require the A, B, C in the lemma to be separable, merely disjoint. 2.4.8

LEMMA. (M. MORLEY'S LEMMA). If p :B A $ C NA, then

B $ A N A N A 7. C. PROOF. We assume B 9 09 C since otherwise the assertion is trivial. We also assume, without loss of generality, that A E R and B A$ C E R.

+

Ch.21

39

ADDITION

By assumption there exist beB (=c‘B), CEC.Let {x,}~’,, be a recursive enumeration of { y : b 4 y < c ) . We define q: BTATC-BTA as follows: q(x,n)=x =p(x) =p(x> =x

if if if if

x ~(x,)), x = x, & (3m)(m In & x m + x, & p ( x ) + x),

+

is undefined otherwise; q ( x ) = lim q ( x , n). n- m

Clearly q is partial recursive. If a <xb and hence q (x) =x for some x by clause 4. Moreover, if q(x)=x and y<x, then q(y)=y by clauses 1 and 4. Thus B T A = D, T D,

(**)

where D1={x:q-l(x)=x}, D,={x:q-l(x)=p-l(x)) and D,nD,=0. But then it follows that q maps B $ A C one-one onto B +A. It remains to prove that q is order preserving. This is clear except when xz where by(**) x x = q ( x ) . This completes the proof that q : BTAT CI: BT A and the rest of the lemma follows by symmetry.

+

The next theorem may be regarded as a recursive analogue of the following theorem attributed to Lindenbaum and Tarski (given in SIERPINSKI, 1958, p. 248). “If an order type A is an initial segment of an order type B and the order type B is a final segment of the order type A, then A = B.” 2.4.9

THEOREM. If A S B and B I *A, then A =B.

40

[Ch. 2

CONSTRUCTIVE ORDER TYPES

PROOF. By hypothesis, if BEB there exist A E A and C such that A f C= B (by the proof of the separation lemma 2.3.1) and similarly there exist B'EB and D such that A = D $ B' where p : Brr: B'. Hence A N Df p ( A )$ p ( C ) and therefore

P(A)= D T P(A) $ P(C)Y since 6 p 2 A. By Morley's lemma we therefore have

P(A) = P(A)T P(C> = P ( B ) and the result follows by taking C.0.T.s. A -0=O, A . ( n + 1) = A 2.4.10 DEFINITION.

a

n

+A.

2.4.11 THEOREM. (i) If m>O and A . m l B . m , then A I B , (ii) If m>O and A . m l * B - m , then A 5 * B . PROOF. By duality it suffices to prove (i). If m= 1 there is nothing to prove. Otherwise A.(m + 1) = A.m + A 5 B - m + B = B . ( m

+ l),

and therefore by the proof of the directed refinement theorem (2.3.2) A.m IB - m , or there exist BEB,A E A and a C such that A.rn2B-m and A f C < * B. In the former case the result follows by the induction hypothesis. In the latter the induction hypothesis yields B S A . Hence there exist D and A (E A) such that B f D =A. By theorem 2.4.9 we obtain A =B.

2.4.12 COROLLARY. If A*m=B*m,and m>O, then A=B. PROOF.Immediate from theorems 2.4.9 and 2.4.11.

CHAPTER 3

QUORDS

3.1 We now commence our study of proper subcollections of 9.

Classically, a linear ordering A is said to be a well-ordering if either of the equivalent conditions (va) or (vb) below holds.31 (Val

(Vf) ( 3 4 ( < f ( n+ 1 ) Y f(4)E A => f ( 4 = f(. + 1)) * “There are no descending chains.”

(vb)

+

(VB) (0 B E C‘A + ( 3 b ) ( ~ E &(Va) B ( U E Bs-(b, u)EA))).

“Every non-empty subset has a least element.” In the sequel we shall principally be concerned with such well-orderings but first of all we consider recursive analogues of (va) and (vb). A linear ordering A is said to be a quasi-well-ordering 3.1.1 DEFINITION. of type (a) (type@)) if it satisfies (va) above with the quantifier (Vf) ranging over recursive functions only (if it satisfies (vb) above with the quantifier (VB) ranging over r.e. sets only).

LEMMA. (a) If B is a quasi-well-ordering of type (a) and A=B, then A is a quasi-well-ordering of type (a). (b) Similarly for type (b). PROOF. (a) Suppose p: A = B and {f(n)):=, is a recursive descending chain in A. Then pf is a partial recursive function which is everywhere 3.1.2

42

[Ch.3

CONSTRUCTIVE ORDER TYPES

defined and hence is recursive. Further p is order preserving, hence {pf(n)}:=,, is a recursive descending chain in B. (b) If S is a r.e. set then p (S) is r.e. for partial recursive p. Hence if S is a r.e. subset of C'A without first element thenp(S) is a r.e. subset of C'B without first element when p: A- 8.

3.1.3 THEOREM. A linear ordering A is a quasi-well-orderingof type (a) if, and only if, it is a quasi-well-ordering of type (b). PROOF.Since we are only dealing with linear orderings which can be embedded in R (or equivalently in some r.e. dense linear ordering) it follows from lemma 3.1.2 and the symmetry of N that we may assume A c R. Clearly if A contains no r.e. subset without a first element then A contains no recursive descending chain since {f(n)> as a set is r.e. for recursive f. Conversely, suppose A contains an infinite r.e. subset S without fist element. Let f ( 0 ) be an arbitrary element of S and define f(n

+ 1) = ps{sES & s < f ( n )

&s

>f ( n ) ) .

(1)

<

Thenf(n) is defined for all n since S has no least element under and is clearly partial recursive. Hence {f(n)}:= is a recursive descendingchain in A. This completes the proof. This theorem was first proved in a slightly different form by PARIKH (1962,1966).

3.1.4 DEFINITION. A linear ordering is a quasi-well-ordering if it is a quasi-well-ordering of type (a) or type (b). We observe that in definition 3.1.1 for type (b)we could have restricted our attention to recursive rather than just r.e. sets,for the subset of S enumerated by f in (1) above is a r.e. set of natural numbers enumerated in order of magnitude by a recursive function and hence is recursive. (The other way round is trivial since every recursive set is r.e..)

THEOREM. There exist recursive quasi-well-ordering which are not well-orderings. PROOF.Conditions (i)-(iv) (4 1.1) and (va) are clearly arithmetic conditions, hence the predicate Q ( x ) ( x is (a Gadel number of) a r.e.

3.1.5

Ch. 31

QUORDS

43

quasi-well-ordering) is arithmetic. By SPECTOR (1955, theorem l), the predicate m(x):x is (a Godel number of) a r.e. well-ordering is a complete predicate of the form

where ( V f ) ranges over all functions of one variable. Hence by KLEENE (1955) '2l3 (x)=.D(x), but not conversely. Finally, by theorem 1.2.1 and lemma 3.1.2 there is a recursive quasi-well-ordering. We shall give a direct proof of this result in chapter 14and it also follows from theorem 7.2.1. That there exist non-recursive quasi-well-orderings is trivial. Let A be an immune set and let A be defined by (x, y) E A if, and only if x, Y E A& y s x ;then A is not a well-ordering since A is of type o*. But since A contains no infinite r.e. subset, A can contain no recursive descending chain. (Clearly, A is embeddable in R since Iis recursive.)

3.2. 3.2.1 DEFINITION. A C.O.T. A is said to be a (recursive) quord if there is an AEA which is a (recursive) quasi-well-ordering. We write 1for the collection of all quords and ~21 for the collection of all recursive quords. 3.2.2 THEOREM. A is a (recursive) quord if, and only if, every AEA is a (r.e.) quasi-well-ordering. PROOF. Immediate from lemma 3.1.2, theorem 1.2.1 and the fact that if A is recursive and p is a recursive isomorphism then p ( A ) is . r.e.. Quords, and in particular recursive quords, possess many of the additive and multiplicative properties of C.0.T.s of well-orderings; we shall therefore develop part of this theory before restricting ourselves entirely to C.0.T.s of well-orderings. We note, however, and shall later give the proof that, unlike well-orderings, quasi-well-orderings are not closed under exponentiation (PARIKH, 1962,1966). THEOREM.(i) 0 ~ 9 , (ii) A , B E9 * A + B E9, (iii) A = B + C & A E ~ * B , C E ~ . PROOF. Left to the reader.

3.2.3

44

[Ch. 3

CONSTRUCTWE ORDER TYPES

It is often useful to consider descending chains of the form {f'(u)) so we give the following definiti0n.3~ 3.2.4 DEFINITION. A recursive infinite descendingchain {g(n)}:=o is said to be a splinter if there is a one-one partial recursive function,f, such that g (O)=c, g (n+ l)=f(g (n)) where c is some fixed number. We write 9 (4=f" (4. 3.2.5 THEOREM. A linear ordering is a quasi-well-ordering if, and only if, it contains no splinter. PROOF. Let A be a linear ordering and suppose that {g(n)},"=, is a recursive descending chain in A. Definef, c as follows: c = g(O),

f(n

+ 1) = g(1 + P&(Y)

=n

f

11)

(cf. ULLIAN,1960, p. 33). Then { g (n)) is a splinter in A. Conversely, suppose that {f"( c ) ) is a splinter in the linear ordering A. Let g be the function defined by

Then g is totally defined and computable, hence recursive; i.e. {g (n)) is a recursive descending chain in A. This completes the proof. The next lemma is essentially proved (in a different context) by SIERPINSKI (1958, p. 264). 3.2.6 LEMMA. If A is a quasi-well-ordering and f is a one-one, partial recursive, order preserving map of C'A into C'A then

(x, fb)>E A for all XEC'A. PROOF.Since ~EC'A=~(X)EC'Aand A is linear either ( x , f ( x ) ) ~ A or (f(x), X)EA (or both). Suppose ( f ( x ) , X)EA and ( ~ , f ( x ) ) $ A then, since f is order preserving we easily establish by induction that for all n ( f " " ( x ) , f . ( x ) ) ~ A , i.e. {f"(x)},"=, is a splinter in A. The conclusion now follows by theorem 3.2.5. THEOREM. If A is a quord and A =B+ A + C then C=O. PROOF.Assume the hypothesis and C 0. Then there exist quasi-well-

3.2.7

+

Ch. 31

45

QUORDS

orderings AEA, BEB and C E C and a recursive isomorphism f such that f : B $ A 7. C

=A.

By lemma 3.2.6 if XEC'A,(x,f(x))~A. Since C S 0 , 3xeC'A with f - ' ( x ) ~ c ' C hence ( x , f - l ( x ) ) ~ B $ A $ C and (f(x), x)EA, .if(.). But this implies (x,f(x))$A which is a contradiction so we conclude C=0 whence C=O.

3.2.8 COROLLARY. If A is a quord, then A + B = A implies B=O. 3.2.9 COROLLARY. If A is a quord, then

+

A + B = A C o B = C. PROOF. Suppose A + B = A + C , then by the directed refinement theorem (2.3.2) there is an E such that either A = A + E and E+ B= C or A + E = A and B = E + C . In either case, by corollary 3.2.8, E=O and B= C. The converse is trivial.

3.3 We know very little about the order structure of recursive quasiwell-orderings and have only the following result. But we shall show in chapter 15 that for each countably infinite order type there exist 2K0 non-recursive quasi-well-orderings of that type which are not recursively isomorphic. 3.3.1 THEOREM. If A is a r.e. quasi-well-ordering then the order type of A is either finite or of the form w+z,

where z is some (countable) order type. PROOF.Since A is r.e. A(u = { ( x , ~ ) : ( u , x ) E A & ( x , ~ ) E A & u + x } is also r.e. for every ueC'A. Hence by lemma 2.1.2 each A (u has a least element and hence every element in C'A has an immediate successor if it has any element succeeding it. Similarly, A has a first element since C'A is r.e. The theorem now follows at once.

3.3.2 COROLLARY. If A is a recursive quord, then every AEA has a least element.

CHAPTER 4

THE ORDERINGS

4.1 We denote the classical denumerable ordinals (and variables ranging

over the denumerable classical ordinals) by lower case Greek letters a, b, y, .... We regard classical ordinals as order types which contain a sub-ordering of R which is a well-ordering. Clearly any well-ordering whose type is a denumerable ordinal can be embedded by a (possibly non-recursive) isomorphism in R; but we shall exhibit a well-ordering (of type 0 . 2 ) which cannot be embedded in R by a recursive isomorphism. 4.1.1 DEFINITION. If A is the C.O.T. of a (recursive) well-ordering, then A is said to be a (recursive) co-ordinal. We let ‘3(WU)denote the collection of all co-ordinals (of all recursive co-ordinals). If a is the classical ordinal such that A Eu then u is said to be the (classical) ordinal of A and of any

AEA and we write

IAI = a and IAl = a . 4.1.2

COROLLARY. A c IAI.

4.1.3 THEOREM. A is a (recursive) co-ordinal if, and only if, every A E A is a (r.e.) well-ordering. PROOF.Left to the reader. (Use theorem 1.2.1.) 4.1.4

THEOREM. Any two linear orderings with fields of the same finite

cardinal are recursive well-orderings which are recursively isomorphic by a recursive permutation.

Ch. 41

47

THE ORDEIUNGS

PROOF. That finite linear orderings are well-orderings is well-known. Any two finite well-ordered sets of the same cardinal can be mapped onto each other in a one-one order preserving manner by a recursive permutation since they can be so mapped by some permutation and every permutation of the integers which interchanges only finitely many numbers is necessarily recursive. It follows that any two finite linear orderings with fields of the same cardinal are recursively isomorphic by a recursive permutation. 4.1.5

COROLLARY. If A is finite (i.e. if A E A ~ C ‘ is A finite), then IAl = A .

We denote finite co-ordinals by 0, 1,2, ... in the obvious way since it will be clear from the context whether n denotes a co-ordinal or a natural number. We also use A” to denote the collection of all finite co-ordinals. We observe that our notation hitherto has been consistent with this usage. THEOREM.(i) OEV, (ii) A, B E V - A + B E V , (iii) A = B + C & A E V - B , PROOF. Left to the reader.

4.1.6

CE%.

+ IBI.

4.1.7

LEMMA.IA+ BI = IAI

4.1.8

THEOREM. (i) N c 9 V c 99c 9 n

n

V c l (ii) The cardinalities of A”, W%‘,99 are all K O and of V, 9,W are c (the cardinal of the continuum). PROOF. (i) Every finite sub-ordering of R is a recursive well-ordering hence JVCWVGV. N 9 9 V since there exist infinite recursive wellorderings and W V =#V by part (ii) below. Since every co-ordinal is a quord,by theorem 3.1.5 we have 9 V c W 9 , and by theorem 1.2.3, 99s9.9999 and 99922 by part (ii). Finally V G 22, clearly, and Vg22 by the example following theorem 3.1.5. (ii) Since M yWV,99 contain only equivalenceclasses of r.e. orderings they are at most countable. Since the cardinal of A” is KO,they all have cardinal KO. There exist c distinct sub-orderings of Seq. Since there are only K O

48

CONSTRUCTIVE ORDER TYPES

[Ch. 4

recursive isomorphisms these sub-orderingslie in x classes where KOex= c. Using the axiom of choice we conclude x=c.

4.2. 4.2.1 THEOREM. If B E %then A IB o ( 3 C ) ( A + C = B ) ,

and A S*B-+C)(C

+ A = B).

PROOF.Immediate from theorem 2.1.2. 4.2.2 THEOREM. If B is a co-ordinal then the following statements are equivalent: (i) A I B & A B, (ii) there is a (unique) co-ordinal CPO such that A+C=B. PROOF. Suppose A + C, = A C, =B, then by corollary 3.2.9, C, = C,. By theorem 4.2.1 there is a C such that A + C = B and since A 9 B we must have C90.The converse is trivial.

+

+

4.2.3 THEOREM. (i) A < A, (ii) 0 1A, (iii) A
(i)* (ii)* (iii)* (iv)* (v)*

AI*A, 01 * A , A I * O o A =0, A I * B & B I * C*A I * C, A < *B*A+ C<*B+ C.

4.2.4 THEOREM. If A is a quord, then (i) A C A (i)* A C * A , (ii) A< B & B I A*A = B, (ii)* A < * B & B< * A*A = B. PROOF.Immediate from lemma 2.4.6 and lemma 3.2.6. We shall now write A < B for A s B & A+B when B is a quord and A< B for AI B and A+ B.

4.2.5 LEMMA. If B is a quord, A S B E B and A= COT(A), then A< B. PROOF.Left to the reader. We now give an easy proof of 2.4.9 for quords.

Ch. 41

4.2.6

49

THE ORDERINGS

THEOREM. If A or B is a quord, then A S B &B<*A*A=

B.

PROOF. Assume the hypotheses and that A I B , then there exist

A, A E A , B E B and recursive isomorphisms p , q, r such that p : A 2: B' IB , q : B N A" < * A

and r : A N A', where both of the inequalities (l), (2) are strict since A + B. Hence there exist C, D I 0 such that

A

= C $ r-'qp(A) $

D.

But this is impossible by the proof of theorem 3.2.7 and we conclude A = B. 4.2.7

DEFINITION. A partial ordering 5 is a tree ordering if A,B
or B I A .

It is a rooted tree ordering if it is a tree ordering and there is an element 0 such that 0 I A for every A . A partial well-ordering is a relation I satisfying (i), (ii), (iii) (0 1.1) and (va) (0 3.1). The following theorem sums up all the important properties of the orderings that we have so far obtained and adds some more. 4.2.8

THEOREM. (i) Iis a rooted tree partial well-ordering of %, (ii) I *is a tree ordering of 9 (and hence of %), (iii) Iis a rooted tree ordering but not a partial well-ordering of 9. PROOF. (i) Since the (classical) ordinals are well-ordered it follows from lemma 4.1.7 and theorem 4.2.3 that there are no descending chains under 5.The other required properties follow immediately from theorems 4.2.3 (i), 4.2.4 (ii), 4.2.3 (iv). By theorem 4.2.1 and theorem 2.3.2 (directed refinement theorem) it follows at once that I is a tree ordering and from theorem 4.2.3 (ii) that Iis a rooted tree ordering. We leave the details of the proofs of (ii) and (iii) to the reader except for the proof that I is not a partial well-ordering of 9. Let A be a

50

CONSTRUCTIVE ORDER TYPES

[Ch.4

recursive quord which is not a C.O.T. of a well-ordering41 and let {u (n)},”= be an infinite (necessarily non-recursive) descending chain in an AE A such that A c R. Let

A,=A[{x:x$a(n)}, then, clearly, each A, determines a quord A, and by theorem 2.1.2 we easily obtain m < n =5 A,, > A,, . THEOREM. If CE C22 then

4.2.9

C I*A+(3B)

(B + C = A ) .

PROOF. Since A c R there is an AEA such that A E R. Further since C I *A there is a C E Csuch that CI*AER. By theorem 3.3.1, C has a least element co or is empty. In the latter case take B = A ; in the former case let

B = A[{x:x
B) (C and clearly B 4C = A . Taking C.0.T.s yields the required result. The converse is trivial.

4.3

We know that (for quords) A =B is equivalent both to AIB&BIA

and to A s B &B I*A,

but we shall later show, using the fact that there exist incomparable co-ordinals, that I * is not anti-symmetric even for co-ordinals. One might wonder whether the analogue of the classical theorem “If a well-ordered set A is similar to a subset of a well-ordered set 6, then A is similar to an initial segment of B ’ is true for quords or co-ordinals. However, we showed in CROSSLEY (1965, Appendix) that the analogue is not true for co-ordinals.

Ch. 41

THE ORDERINGS

51

Finally, we give an example of a well-ordering of type 0 . 2 which cannot be embedded in R by a recursive isomorphism.

EXAMPLE. Let S be a r.e. non-recursive set; then U be the well-ordering of type w - 2 defined by

4.3.1

s is infinite. Let

(x,y)€UoX€S&yEs v x, y € s & x I y v x, y € & x I y .

s

Further let r be the least element of p:U

N

s. Suppose U’ E R,

then the initial w segment of U ’ r { x : x < p ( r ) } = A , say, and the final segment of type w is contained in { x : x + p ( r ) ) =Bysay. Hence S5p-’(A)

and s c p - l ( B ) ,

i.e. S and are contained in disjoint r.e. sets. This contradicts the fact that s is not recursive and we conclude that there can be no such recursive isomorphism p.

CHAPTER 5

CO-ORDINALS

5.1 We have already introduced co-ordinals and in this chapter we

investigate their basic properties. 5.1.1

DEFINITION. A co-ordinal A is said to be a successor number if

(3B)( A = B + 1) and a limit number if it is neither a successor number nor zero.

THEOREM. If A is a successor number (limit number), then IAl is a successor number (limit number). PROOF. Obvious. 5.1.2

If A = B + C , then we put C = A - B (when A , B, C are quords). By corollary 3.2.9, if A - B exists it is unique.

THEOREM. (i) A-A=O, (ii) ( A + B ) - A = B , (iii) if A is a co-ordinal (or a quord), then B ( A - B) =A , (iv) if A is a co-ordinal (or a quord) and B+ C I A , then A - (B+ C )=( A - B) - C. Proof of (iv).

5.1.3

+

B

hence

+C

A

(E!D ) ( A = B

A-(B+C)=D.

+ C + D),

Ch. 51

Also,

53

CO-ORDINALS

A -B = C +D (A-B)- C =D .

and therefore

+

THEOREM.C < A 1 e C s A . PROOF. Take AEA and suppose C < A + l ; then there is a C E C such that C < A + I where 1 = { ( o , O ) ) ~ l . 5.1.4

Clearly, C l Q A and hence C I A . The converse is trivial.

COROLLARY. If A is a quord, then A + 1 is the unique quord B such that A
5.1.5

By virtue of this corollary we call A + 1 the successor of A .

THEOREM. If A is a successor number where IAl = l + n and 1 is a limit ordinal then the sequence

5.1.6

A - n,..., A - 2 , A - 1, A , A

+ 1, A + 2, ..

of length o,is uniquely defined. PROOF. Since A is a successor number AEA implies A has a finite final segment. Now every finite set is recursive and hence separable from its complement in C'A, hence A - 1, A -2 , . .., A -m are well-defined provided A - (m- 1) is a successor number. The rest is clear.

By the same argument that finite sets are recursive and hence separable from their complements in an ordering we leave the reader to prove

5.2

5.2.1

LEMMA.A co-ordinal A is infinite if, and only if, n
5.2.2

THEOREM. A co-ordinal A is a limit number if, and only if,

B
54

CONSTRUCTIVE ORDER TYPES

[Ch. 5

if jl is not a successor number then

5.2.3

COROLLARY. If A is a co-ordinal, then A
+ 1< B .

PROOF. Left to the reader. We know from theorem 1.1.7 that constructive order types give a finer classification of well-orderings than do (classical) order types and the next theorem gives a measure of this fact. Most of the effort in the latter half of part one of this monograph will be devoted to trying to single out one co-ordinal for each denumerable ordinal, a problem that is not yet solved except for relatively small ordinals. 5.2.4 THEOREM. There are c distinct co-ordinals of ordinal o. PROOF. There are c distinct infinite subsets of JV - (0). Consider such a subset S and assign to it the sub-ordering s of R given by (x,

Y ) E S O X ,

YES

<x>< .

Then these c sets are spread among, say, x equivalence classes. Each equivalence class contains at most K O members since there are only K O recursive isomorphisms. Hence K O- x = c and therefore (using the axiom of choice) x = c. Notation.S1 W'=(((i), ( j ) ) : ( i ) = ( ( j ) &i, j > O } , W=COT(W). Let V E N-{O} be a fixed but arbitrary r.e. non-recursive set, then we put

V = {(
<j)): i , j e V & i 5 j}

and V= COT (V). Clearly,

W' = { ( ( i ) , ( j ) ) : i s j } N {(i, j ) : i < j } = W. DEFINITION. If WEW then w" is said to be a natural wellordering of type w and W is said to be the natural co-ordinal of type o. We call V, V the basic counterexamples of well-orderings, co-ordinals, respecti~ely.5~

5.2.5

Ch. 51

CO-ORDINALS

55

THEOREM. (i) 1+ W = W, (ii) 1+ V + V. (i) Take 1 E 1 as in the proof of theorem 5.1.4 and let p be the PROOF. map defined by p(x)=o if x = 0 ( 0 ) , = i + 1 if x =I(<+), is undefined otherwise. 5.2.6

Then it is easy to verify that

p : l +w'-w'. (ii) Suppose 1 + V = V. Since V is r.e. non-recursive, 0 is non-empty, say aoEV. Therefore there is a recursive isomorphism p such that p : Kao, a,>>

4v N v.

Hence

v = {
g(x)=p"(ao)

then But then g enumerates V in order of magnitude and it follows that V is recursive, contradicting our choice of V. Co-ordinals A, B are said to be incomparable if A $ B and B $ A. COROLLARY. V and W are incomparable. PROOF. By the theorem V + W. But V < W or W < V implies V or W is finite, which is impossible. 5.2.7

5.3

and

If A is a linear ordering then we write

56

CDNSTRU(;TNE ORDER 'TYPES

[ch.5

If UEC'A,then a determinesr c z d ) i n i t i a l segment a ) A = { < x , y ) : x ~ A
a n d { ~ ~ segment ~ ~ ~ ~ a l A (U = {(x, y) :a < A x IA y} A [ a = {(x, y ) : a S A X I A y}. Clearly, a is the first element of A [a. If A is a well-ordering of ordinal ci, then a) A is a well-ordering of ordinal /3, and we write a=IAl, B=la)Al. (1) Conversely, if /3
LEMMA.If IAI = a and /?
PROOF. Take AEA, then there exists UEC'Asuch that (1) holds. Let B = COT(a)A)

then clearly, B< A. 9 ( A ) = ( B :B < A ) We write 9' ( A ) = { B :B IA } and and we call B(A) and 8+(A) paths. We let B denote the collection of all paths of co-ordinals. We shall later show that there exists a path right through the countable ordinals but here we only show that paths "look like" initial segments of the classical ordinals. 5.3.2

I I

THEOREM. If IAI =ct, then 9")is isomorphic to a under the map

and 9 ' (A) is isomorphic to a + 1 under the same map.

PROOF. Immediate from lemma 5.3.1. Later we shall extend this theorem to certain paths which are closed under given co-ordinal operations and shall show that the same mapping

Ch. 51

51

C0-0RDIN ALS

is an isomorphism with respect to the corresponding classical operations. The next theorem is useful when we wish to employ classical results to get results about co-ordinals. In fact, in the next section we shall so employ it. 5.3.4

THEOREM. If A , B I C and C is a co-ordinal then

and

PROOF. By theorem 4.2.8, A and B are comparable. By theorem 5.3.2, since I I is an isomorphism on B ( C ) to ICI, A I B. The second assertion follows from the anti-symmetry of I . 5.3.5

COROLLARY. If A , B are comparable, then

and

5.3.6

COROLLARY. If IA I = c1 and p c a, then

( E ! B )(E!C) ( B

+ C = A & IBI = B).

PROOF. Immediate from lemma 5.3.1, theorem 5.3.4 and theorem 4.2.2.

This section is devoted to a characterization of commuting coordinals. We have no corresponding result for the multiplicative case.

5.4

+

5.4.1 THEOREM. If A , B are co-ordinals then A B = Bf A if, and only if, there exists a co-ordinal C and finite co-ordinals m, n such that

A=C.m

and B = C . n .

PROOF.Classically we know p+c1

(see, e.g. SIERPINSKI, 1958) that a+p= if, and only if, there exists y and finite ordinals m, n such that

a=y.m

and P = y * n .

Clearly we may assume (myn)= 1, i.e. my n are co-prime.

58

[Ch. 5

CONSTRUCTIVE ORDER TYPES

Now suppose IAl=a, IBI=/? and A + B = B + A . If c1 or /? is 0 there is nothing to prove, so we assume c 1 2/? >0 from which it follows that m 2 n. By corollary 5.3.6 there exist C;,..., CL- D;, ..., DA- such that

,,

lCfl = y - i ,

lDJl = y - j

( i = 1,..., m

- 1 ; j = 1,..., n - l ) ,

where C; < A , D; < B .

Since the Ci are bounded by A they are all comparable and so by applying corollary 5.3.6 ( m - 1 ) times we obtain C,, ..., C, where C , = C ; , C2 = C ; - C ; ,..., C , = A - C g - 1 .

Similarly we obtain D,, ..., D, since the D; are less than B. Set C,+ =D j f o r j = l , ..., n and m + n = r . Then lCil = y

for i = 1, ..., r ,

and

A+B=C,

+...+ C , + C m + 1 + . . . + C , = B + A = C , + 1 + . . .

+ c, + c1

+ - . a +

c,.

Again by corollary 5.3.6 we obtain

cj= Ca(i)

(i = 1 , ..., r ) ,

where 0 is the permutation53 1 m+l

on r letters. Let z be the cycle (12

...n n + 1 ... r ...r 1 ...m

...r ) so that z(i)=i+l =1

if if

i
then a=zm and z is of order r. Now (m,r ) = 1 so there exist integers u, v such that mu + rv = 1.

Ch. 51

59

C0-0R DINA LS

Hence 6u

= Tmu =

-ru

- 2.

It now follows from (2) that

C i= Cou(i)= C i + l if i < r , =C,

if i = r ,

so Ci= C j = C, say, for i, j = 1, ..., r and we conclude A=C.m,

The converse is trivial.

B=C.n.

CHAPTER 6

MULTIPLICATION

6.1 We define multiplication of orderings in the usual way except that we code the ordered pairs we obtain by means of the functionj(x,y) defined by j ( x , Y ) = +(x + Y ) .( + Y + 1) + x. We recall that there exist k, I such that

and that j , k, I are all primitive recursive.

6.1.1 DEFINITION. A.B={(j(a, b),j(u', b ' ) ) : ~u'EC'A& , (b<,b' V . b=b'~C'B&a
Ch. 61

MULTIPLICATION

61

where a,EC'A and b,EC'B. Let A'={a,:n>O} and B'={b,:n>O}, then there are four cases to consider: (i) A and B' are both finite, (ii) A' is infinite and B' is finite, (iii) A' is finite and B' is infinite, (iv) A' and B' are both infinite. (i) is impossible since then {j(a,, b,)} would contain only finitely many elements. (ii) Since B' is finite, there is at least one number ~ E Bfor ' which { j (an,b ) :an €A'}

is infinite. Let the distinct a, in this set be a(ni) i=O, 1, ... where i< j*ni
= a,

a(ni + 1) = a,

, 9

where

r = p , { s > t.&.u < S J U , =k a,} and a, = a(ni). It follows at once that { a (nil]:

0

is a recursive descending chain in A which is a contradiction. (iii) This case is dealt with in a manner very similar to (ii). We omit the details. (iv) Let { b ( n i ) :i 2 0 } be the set of distinct bi where i<j*ni
2. If i<j and E B then, since ( b (nj), b (ni)> E B

62

CONSTRUCTIVE ORDER TYPES

[Ch. 6

by the definition of A. B and since B is anti-symmetric, b(ni)=b(nj),

which contradicts our assumption that i<j. Hence ( b ( n i ) } g 0is a descending chain in B; but it is also recursive, since b(n0) = bo b(ni + 1) = br

Y

9

where

r = ~ , { s > t . & . u c s ~ b , * b , } andb,=b(ni).

This is impossible since B is a quasi-well-ordering. The second assertion in the theorem is clear from definition 6.1.1. There is a partial recursive functional Y such that if 6.1.3 THEOREM. p:Ao=A, and q:Bo=B, then !Pup, q):Ao*Bo1: A,*B,.

PROOF. Set P ! (p, q ) ( x ) = j ( p k ( x ) , ql(x)). Theorem 6.1.3. permits the following definition. 6.1.4

DEFINITION. A.B=COT(A.B) for any AEA and BEB.

We shall often write AB for A - B and AB for A-B.

6.1.5

THEOREM. If A, B G

R, then

A . 8 = R.Rnj(A,B)’, where A=C‘A, B=C‘B and j(A,B)= {j(a,b):aEA & ~ E B } .

PROOF. Left to the reader.

6.1.6 THEOREM. R * R N R. PROOF. R - R is a recursive linear ordering by theorem 6.1.2. Further, if x
and x = j ( a , b ) ,

then either b
y=j(a’,b’),

R is dense, there is a c such that b
Ch. 61

63

MULTIPLICATION

or b=b' and a i a ' and there is a d such that a
Hence either x


Rj(a9

x


~ j ( b) 4
c)


R

Y,

or and so

R

e

R

Y

R is dense. Clearly j(0,O) is the first element of R - R and if R and

j ( a , b)EC'R.

b =
then j ( a , b) < R .

d ( 4 (b(O), ..., b ( m ) 4-I)),

so R - R has no last element. By theorem 1.2.4 it follows that R. R- R. Since R - R and R are recursive linear orderings there is a minimal61 recursive isomorphism (i.e. with domain C ' R . R and range C'R)

m: R . R

N

R.

We shall regard this m as fixed from now on. Now suppose we are given A, B c R, then by theorem 6.1.6 there exists a C such that m:A*B N C c R, where C'C=m(j(C'A, CB)). We denote this (unique) C by m(A, B) whenever A, B E R. Clearly, bytheorem6.1.3,ifm': R- Randm'(A, B) is defined analogously to m (A, B) then for some recursive isomorphismp R

p : m(A, B)

N

e

m'(A, B),

where p may be taken to have range and domain Seq and to be independent of A, By in fact take p

= m'm-'

,

64

[Ch. 6

CONSTRUCTWE ORDER TYPES

6.1.8 THEOREM. Multiplication is associative, i.e. ( A * B ) * C= A - ( B . C ) .

PROOF. Immediate from theorem 6.1.3 and lemma 6.1.7. Multiplication is not commutative since we know classically w *2 9 2 o =o.This theorem allows us to omit brackets in writing products of several C.0.T.s.

-

6.1.9 THEOREM. A*B=O*A=OvB=O. PROOF. j ( A , B) = O o A =0 v B =0.

6.2. 6.2.1 THEOREM. A (B+ C ) = A B + A C . PROOF. Take A, 6, C E R, then p:A.(B + C) N A - B + A - C , where p is defined as follows

We leave the reader to verify that p is one-one; the other properties are obvious. The other distributive law does not hold since classically we have (o+l) o=W2*02+W.

Ch. 61

6.2.2 (ii) (iii) (iv) (v) (vi)

65

MULTIPLICA’ITON

THEOREM. (i) A =A - 1,

A .n+A = A + A - n=A * (n + l), A rn A * n = A (m+n), A - W = A + A . W = A * n + A . W foralln, A (mn)= ( A am) n = ( A n) * rn , A . W = ( A - n ) .Wfor all n. PROOF. (i), (ii) Left to the reader. (iii) By theorem 6.2.1. (iv) By (i) and theorem 6.2.1 we have

+

-

A.1

+ A.W = A * ( l + W )= A - W

by theorem 5.2.6. The other part follows by induction on n and is left to the reader. (v) By theorem 6.1.8. (vi) Let AEA and let d, r be the (primitive recursive) functions such that x = nd(x) + r ( x ) and 0 I r(x) < n and let P (XI = j ( j (k (x), rl (XI), dl (4). then p is one-one and (primitive) recursive. We assert that p is order preserving between A . W and (A- I,)*W where

I, = { ( i , j ) : i lj < n} and

W = {(i, j ) : i 5j } , for

(A.1,J.W

u),j(j(b, t), v ) ) : s , t < n.&. u = v & s < t ) & a , bEC‘A v . u = v & s = t & ( a , b)EA}

= { ( j ( j ( a , s),

(u

.


and (x, y ) ~ A - W e x = j ( a , nq

where

+ r) & y

= j ( a ’ , nq‘

+ r’),

0 I r, r’ < n & nq + r < nq’ + r’ & a , a’EC‘A . v . n q + r = nq’ r’ & ( a , a’)EA.

+

Condition (2) is equivalent to

0 1 r , r ’ < n.&.(q < q’ v q = q’ & r < r’) & a , a’EC‘A . v . q = q‘ & r = r‘ & (a, a’)EA.

66

CONSTRUCTIVE ORDER TYPES

[Ch. 6

Comparison of (1) and (3) immediately shows that p is order preserving. This completes the proof.

+ +

6.2.3 THEOREM. (i) ( A B) - n+ A = A + (B+A ) -n, (ii) ( A B). (n 1)=A (B+ A).n +B, (iii) ( A+B) W =A + (B+ A ) . W. PROOF. (i), (ii) Proof by induction using the associativity of addition. (iii) Let AEA, BEB where A, B E R, then

+

p:(A

+

+ B)*W

N

A +(B +A)-W,

where if x = j (0 ( Y ) , (0)) , P(X>= W Y ) if x = j ( q Y ) , W), = j (70 ( y ) , (2)) =j ( T T ( y ) , ) & z 2 I , is undefined otherwise. We leave the reader to check that p has the required properties.

+

+

6.2.4 THEOREM. If A B =B A , then (i) A am +Ban =Ben + A . m, (ii) ( A + B ) . m = A . m + B - r n . PROOF. By induction using theorem 6.2.2.(ii). 6.2.5

THEOREM. If B . W= A + C and C40, then there exist n, D, E such

that A=B.n+D,

D+E=B

and E + B - W = C .

PROOF.We consider only the non-trivial case when A , B, C are all non-zero. Let BEB, then by the separation lemma (2.3.1) there exist A E A and C E Csuch that A)(C and B . W = A $ C , where W is as in the proof of theorem 6.2.2.(vi). By assumption C90+4,

B some n'20. hence there is a C E C Cwhere c = j ( b , n')for some ~ E Cand Therefore F = {2(a):a~CA}

Ch. 61

67

MULTIPLICATION

is a set of natural numbers bounded by n'. F therefore has a maximum, let it be n. Further let

D = A[j(B, n) and

E = C[j(B, n ) ,

where B = C ' B and j(B, n ) = { j ( b , n): b E B } .

Then, as in the proof of the directed refinement theorem (2.3.2)it is easily verified that

D)(E, A = B - n T D, D T E = B and E B-W[{x: l ( x ) > n } = C. We observe that B-W[{x: l ( x ) > n} N B-W under the map p : x + j ( k ( x ) , l(x) A (n

(1)

+ 1))

defined only on {x: I(x) > n}.

Taking C.0.T.s of both sides of the equations in (1) completes the proof. The above theorem is not an immediate consequence of the directed refinement theorem for the following reason. We know that B.n+B.W=B.W

forall n .

Suppose B . W = A + C with C+O, then it follows easily from the directed refinement theorem (2.3.2) that for each n either

If A 5 B - n for some n, then we are through, but A 2 Ban for all n does not implyA>B-WfortakeA=V,B=l.

6.2.6 COROLLARY. If A is a co-ordinal and A < B . W then A S B - n for some n.

6.2.7 THEOREM. If C is a recursive quord or a co-ordinal, then

68

[Ch. 6

CONSTRUCTIVE ORDER TYPES

PROOF.By theorem 4.2.1, C<*B*W*(3A)(A+C=B.W).

Hence by theorem 6.2.5 there is an E such that

+B-W = C ,

E

hence B - W I *C

COROLLARY. If A+O, A I B - W and A I * B . W , then A = B . W . PROOF. Immediate from theorem 4.2.6 and the theorem above.

6.2.8

We now complete our demonstration that I * is not antisymmetric.

I.'and W are incomparable but V<(V+W).W

and W < W + ( V + W ) . W ,

hence since I is a tree ordering (V

Now V

+ W ) * W* W + (V + W

) K

+ w + (V + W).W = (V + W ) * W ,

hence W

+ (V f W ) * WI *(V + W ) * W .

But (V+ w)*WI*W+(VfW)*W,

hence if I * is antisymmetric (2) is false, which is impossible.62

6.3 We now present a lemma which we shall use in giving existence proofs later on. 6.3.1 LEMMA. If A is a quord, then B

+ A = A e-(3C) ( B - W + C = A ) .

If A is a recursive quord then B.+ A = A e B * W I A .

PROOF. The second statement follows immediately from the first. Suppose Be W+ C = A , then B

+ A = B + ( B - w+ c ) = ( B + B - W )+ c = B - W -t C = A .

Ch. 61

MULTIPLICATION

69

Now suppose B+A =A. If B=O, then the assertion is trivial. Similarly if A =O, so we assume neither of these occurs. By hypothesis there exist quasi-well-orderings A, B E R and a recursive isomorphism p such that

p:B+A=A where AEA and BEB and A) (B. Let A = C'A and B = C'B. We introduce the following notation for this proof only: rn

B, =

u P"+

(B) Y

n=O

B, = A[B,, A0 = (x:(Vn) ( p - " ( x ) ~ A ) } , A, = A I A o . Before giving the formal details we sketchthe idea of the proof. Classically, if a, /? are ordinals then

p + a = a*fi.n c a for all n . Hence 1irnp.n I a so

forsome y .

p.o+y=a

Now, firstly (as we shall show in chapter 12) in general we cannot uniquely define limits (though in this case we could) and secondly even if a limit had been defined then we should still have to establish separability of the orderings representing pew and y. In this particular case, however, we can establish both fairly easily by an ad hoc device. We observe that

B + (pew + y )

=p.0 + y

and in the case of any particular ordering of type c1. the elements in that ordering belonging to the final segment of type y all remain fixed under any map p such that p : B A N A.

+

So in order to separate the j3-w and y orderings we simply look at the

sets of elements which remain fixed under p and those which do not. These two sets include A, and B,, respectively. To effect the proof we shall establish 1) A o n B , = 8 , 2) A , u B , = A ,

70

[Ch. 6

CONSTRUCTIVE ORDER TYPES

3) B,EB-W, 4) XEA,*~(X) = X, 5) x ~ B * p ( x ) x,

+

6) Bw) (A0 7) B,$A,=A. 1) If XEB, then x = p " ( y ) for some n>O, some ~ E BHence . p - " ( x ) is defined and $A; so x$Ao. 2) Since p maps B u A onto A, XEA implies either 9

(Vn) [P-"(x)EAl

Y

or

( 3 4 CP-"(x)EBI1.e. XEA-XEA, v X E B . Conversely, XEA,*X = ~O(X)EA

and X E B , + X = p"(y) for some ~ E B ,some n > 0,

so in either case XEA. 3) Since B) (A there is a partial recursive function f such that if XE B u A then x ~ A @ f ( x )= 0 & X E B * ~ ( X ) = 1 by theorem 2.1.3. We now usefto calculate a function q such that q : B,

N

B-W.

Let 4 ( x ) =j(P-"(X), n - I), where n=pr{r> 1&f(p-'(x))=~&(Vs)(s
I)}.

(n is undefined if any conjunct is undefined.) Clearly, q is partial recursive. Suppose q (x) =q (y), then q (x) = j(x', n) = q ( y ) for some x' = p-"-' (x) = p-n-

(y).

But p-' is one-one, therefore x = y and q is one-one. We now show q maps B, onto C'BsW. Suppose XEB,; then both conjuncts in the definition of n, above, are defined and hence q ( x ) is defined. Clearly, q(x)ECB.W. Now suppose j ( x , n)eC'B*W; then p " + l ( x ) ~ B and , q ( p " + ' ( x ) ) = j ( x ,n); hence C'B-WCq(B,).

Ch.61

71

MULTIPLICATION

Next we show that q is order-preserving between B, and B.W. It suffices to show that if

(4

(xo, Y o ) E B m

xo = pm

Y o = P" ( Y )

Y

where x, ~ E and B 0 < I < m*p'(x)EA

and 0 < s < n * p ' ( y ) ~ A ,

then

lsrn
or l l r n = n & ( x , y ) E B .

If m >n, then since p is one-one and order-preserving, But therefore, by the antisymmetry of B$ A, Y = pm-"(x),

which contradicts B) (A. Thus m l n . If m=n, then

( X , J J ) E B + A where x , y ~ B . We conclude ( x , y ) E B. This completes the proof of 3). 4) Since A is a quasi-well-ordering and A, c A, A, is also a quasi-wellordering. Now p maps A, = C'A, onto A, since XEA, * p - l ( x ) ~ A , & p ( x ) ~ A , which implies AO~p(Ao)_cA,. Now p is order preserving, hence by lemma 3.2.6 p, p-' are both increasing on A, so we conclude p is the identity on A,. 5) XEB* x =p" ( y ) for some n > 0, some ~ E BSince p is one-one

.

x = p ( x ) implies p-"(x) = p-"+'(x).

But ~ - " ( x ) E B , ~ - " + ' ( x ) E Aand B n A = 0

since B)(A.

We conclude p (x) 9x. 6) Since p is partial recursive, 6 p is r.e. Hence by 4) and 5) A, c { x : x ~ d &p p ( x ) 9 x>

12

[Ch. 6

CONSTRUCTIVE ORDER TYPES

and B,

c { x : x ~ b p& P(X)

= X}

and the sets on the right hand sides are r.e. Thus B,) (A,. 7) By 6), B,$ A, is well-defined. By 2) C'(B,p A,)=A. By definition B,GA and A,sA. It therefore suffices to prove that B, x A.

c A and A E B, .7. A,.

If XEB, and YEA, then

(In) [p-"(x)~B] and (Vn) [ p - " ( y ) ~ A ] . Hence ( p - " ( x ) , p - " ( y ) ) ~ B x A E B p A for some n ,

and since p is order preserving we have (x, y)eA. If ( x , Y)EA, then either (i) x, ~ E B or , (ii) XEB,, YEA, or (iii) x, YEA, or (iv) X E A , , ~ E B @ since A =A, v B, by 2). Hence in order to complete the proof of 7) we only need to show (iv) is impossible. If (iv) holds, then there is an n such that p - " ( x ) ~ A and p - " ( y ) ~ B , which is impossible since p is order preserving and

(A x B ) n ( B T A) = 0. We now complete the proof of the lemma. By 3) B,EB. W. Let C = COT(A,),

then by 7),

B.W

+C =A.

6.3.2 THEOREM. If C is a quord, A +B+ C = C and ( A + B ) . W=(B+A). W, then A+C=B

+ c = c.

PROOF.^^ By lemma 6.3.1 and the first part of the hypothesis there is a D such that ( A + B )* w f D = C . Hence by the second part of the hypothesis, with the same D, ( B + A).W

+D =C .

Now by theorem 6.2.3.(iii), (A+B).W= A

+ (B + A ) * W ,

Ch. 61

MULTIPLICATlON

hence A +(B

73

+ A ) * W + D = ( A + B ) * W+ D = C ,

and therefore A+C=C.

Similarly, from (1) we conclude B+C=C.

6.3.3 THEOREM. If A m +B I B (or equivalently, A m +B =B ) and m >0, then A . n + B = B f o r all n. PROOF. Clearly, B< * A * m + B ,hence by theorem 4.2.6, B = A . m + B . From lemma 6.3.1 we obtain (3C) [ ( A - m ) .W

+ C =B],

but by theorem 6.2.2.(vi), ( A - m ) .W = A . W

and therefore by lemma 6.3.1, A+B=B.

An easy induction now establishes A.n+B=B

forall n.

6.3.4 THEOREM. If A - W + C = B . m for some m, where B is a co-ordinal, then A - W + C' =B for some C'. PROOF(by induction on m). If m=O or 1 then the assertion is trivial. Now suppose m>O and the assertion holds for m. Then A -W

+ C =B.m + B .

Hence by the directed refinement theorem (2.3.2) there is an E such that either A-W +E =B.m, or A . W = B - m + E & E + C = B & E +O. In the former case we are through by the induction hypothesis; in the latter E<*A*W

74

CONSTRUCnVE ORDER TYPES

and hence by theorem 6.2.5, since B - m is a co-ordinal, F

+A .W =E

for some (co-ordinal) F .

Now E+ C = B implies F + A*W+C=B,

hence A*W

However,

+ C I*B .

B IB . m IA - W IA - W + C

and hence by theorem 4.2.6, A.W+C=B.

[Ch. 6

CHAPTER 7

EXPONENTIATION

7.1 In this chapter we define exponentiation and give PARIKH’S proof (1962, 1966) that the collection of (recursive) quords is not closed under exponentiation. The classical method of defining exponentiation of orderings is by means of an ordering of finite sequences of elements from the given orderings. Since we are dealing only with quasi-well-orderings which can be embedded in R by recursive isomorphisms we give a definition in the spirit of theorem 6.1.5. Convention. We shall assume from now on that if AE R and A has a first element then this first element is 0, unless otherwise stated. We write min(B) for the first element of an arbitrary ordering B if it exists. As in the proof of theorem 1.2.4 we can always assume that if

p:A

N

A’ E R then p(min(A))

= 0.

We shall also need to code finite sequences of ordered pairs of numbers so we introduce bracket symbols. 7.1.1 DEFINITION. A symbol of the form

where n 2 - 1 and the ai, bi ( i =0, ..., n) are natural numbers is said to be a bracket symbol. If the symbol is simply ( ) (i.e. n = - 1) then we call this the empty bracket symbol and denote it by 0. We use upper case

76

[Ch. 7

CONSTRUCTIVE ORDER TYPES

bold face letters (A, B, C,...) for bracket symbols. We write (A7 B ) for the collection of all bracket symbols (including the empty one) of the form (1) where aieCCA, b,eC'B, a , + min(A) i = O ,..., n, bi+bi+l and ( b i + l ,bi)EB, i = O ,..., n - 1 . We define our coding by e(0) = 0,

and if A = ( ba0o *... * *a, "n),

ai c 0,

where p i denotes the ith prime ( p o=2).'1 7.1.2 THEOREM. e is a one-one primitive recursive function from the set of all bracket symbols with the ai+O into N.Further, pe is recursive. PROOF. The a,, bi and n are uniquely recoverable from any element x in pe since if x =!= 0 then

ai = k((x),), bi = l((x),) and n = Zh(x) + 1 . The other parts of the assertion are left to the reader to prove. We shall write bo > b l * * * > b, for bo> b ,

& a * - &

bn-l

> b,

and similarly write bo > B bl

"*>B

b,

in the obvious way.

7.1.3 DEFINITION.

R R = RexpR=

ri ...'{) a,

= { ( e ( K ) , e ( K ' ) ) : K = r o ' * * ' I ? Z ) & Kl = a. ... a,,, a,

'*'

Ch. 71

77

EXPONENTIATION

& K , K'E(R, R) & { K = O v [b,> b , > - - * > b, & bb>b;>--*>bA & [K

+ 0 & {(m I n & (Vr) ( r I m =s a, = a: & b, = b:))

v ( 3 r ) ( V t ) [(t < r o ( V s I t ) (a, = a: & b, = b:))

8c (b, < b: v [ b , = b: 8c a,

< ~:1)1)1>).

If A, B r R and min(A)=O A B = A e x p B = R e x p R n ( A , B)'. 7.1.4

THEOREM. There is a partial recursive functional 9 such that if and q: Bo= B,, A,, BizR, and min(A,)=O ( i = O , 1). Then

p:A,-A,

3 ( p , 9): A, exp Bo N A, exp B,

.

PROOF. Let Z(P, 4 ) (0)

= 0,

is undefined otherwise. That 3 is well-defined follows at once from theorem 7.1.2 and the rest of the proof is left to the reader. DEFINITION. If p:AoNA, and q: 8,- B,, where A,, B,c R, and min(A,)=O ( i = O , 1) then we set

7.1.5

A, exp B, = 3 ( P , 4) (A0 exp B,) * This definition does not define A, expB, uniquely, though we could easily arrange this by requiring, e.g. that p , q be recursive isomorphisms with least Godel numbers; however, this is unimportant as up to recursive isomorphism the definition is unique as the next theorem shows. THEOREM. If p: A, N A, and q: B, N B, where Ai, B, are arbitrary quasi-well-orderings (embeddable in R by some recursive isomorphisms) and min(A,)=O (i=O, 1) then

7.1.6

A. exp B, PROOF. Left to the reader.

N

A, exp B,

78

CONSTRUCTIVE ORDER TYPES

[Ch. 7

7.2 7.2.1 THEOREM. (i) If A, B are well-orderings then A exp B is a wellordering. (ii) (PARIKH, 1962, 1966). If B is a well-ordering and A is a quasiwell-ordering, then A exp B is a quasi-well-ordering. (iii) (PARIKH,1962, 1966). There is a recursive quasi-well-ordering A such that T exp A is not a quasi-well-ordering where T E ~ . PROOF. (i) is well-known. (ii) Assume A is a quasi-well-ordering, B is a well-ordering and there is a recursive descending chain in A exp 6, namely { c ~ } ~where ~ ,

ci= e

a,,

... ain*

Now let bir,be defined for each i by

r, = ps { bi+1 , < bi,,}

or n, if there is no such 5 .

Let ei=biri and let {A} be the sequence of distinct e,. Now {A} is finite, = {f,, ...,f , } , say, since otherwise {A} is a descending chain in B which is impossible. It follows that if f , = e , then we have e,=fn for t 2 s . Now for such t we have c, = e ( b O . . . b S 4 es a20 *. . at, s- 1 at, s

),

where s and b,, ..., b s - l , es are fixed. Since the c, form an (infinite) recursive descending chain one of the sequences {arO}Zt,

*.*)

{ars}:=t

must be infinite, say {arq},"'. This sequence is clearly recursive since it only depends on two fixed numbers t, q and it is the sequence

{k (r"=t

*

Hence A contains a recursive descending chain which is impossible. (1952) theorem IV we can obtain all the sets Enwhich (iii) By KLEENE can be expressed in X2 form {a:

( 3 4(VY) R(aY

XY

Y)j

in a uniform way by setting

E n = { ~ : ( 3 x ) ( ' d ~ ) T z ( an,, x , Y ) } ,

Ch. 71

79

EXPONENTIATION

where T, is a fixed (primitive) recursive predicate. Now we can define a set S which is a Z2 set but whose complement, though infinite, contains no infinite X, subset, i.e. S is a “Z,-simple” set. We shall show that any recursive descending chain in an ordering 6 we define below would compel an infinite Z, set to lie in S which is impossible. Let S = { a :( 3 n ) (3x) { ( V y ) T , ( n , a, x, y ) & 2n < a &(Va’)(Vx’) [ j ( a ’ , x’) < j ( a , x ) &2n < u ’ e ( 3 y ) i T,(n, a’, x’, y)]}},

i.e. aES if, and only if, a is the “first” element of some E , (as determined by the ordering from the j ( a , x)) and a>2n. Clearly, if aeS, then aEE, for some n such that a>2n. Secondly, if En is infinite, then EnnS + 8.For if En is infinite, there is some element b E E, with b >2n and we choose the smallest, i.e. take

a = k ( ~ , { k ( c ) ~ E&, k ( c ) > 2 n & ( V y ) T Z ( n , k ( c ) , ~ ( c ) , Y ) } ) * Then a is well-defined and by the definition of S, aES. Hence s contains no infinite Z, subset. Next s is infinite for of the integers 1, ..., 2n+ 1, at most n are in S so at least one of n+l,..., 2n+1 is in S. Now let C = S so C = { a :(Vx) ( 3 ~Q) (a, x, Y ) } for some (primitive) recursive Q. Let us call a k-attained by n if (vx)
(3Y)<m Q ( a , X, Y )

and call a k-attained if, for some n, a is k-attained by n. So XEC-X is k-attained for every k. Now we define a subset B of j ( M , 2’* Seq) by X E B ~ =Xj ( m , (x,,

..., x,)) & m = pr { ( V i ) (0 < i I r * xiis k-attained by r)} & x , = o & ( V i ) < , ( X i < X i + l -< 2x, 1).

+

We also write

x i = [.Ii

for i = O,..., k.

Now we order B by stipulating ( x , y ) ~ B o x , y ~& B (Z(x)+l(y)

or I ( y ) = 2 ’ ) .

Clearly 6 is a recursive linear ordering. Suppose {a,} is a recursive descending chain in 6. We shall prove that this implies C contains an

80

[Ch. I

CONSTRUCL'NE ORDER TYPES

[anlo=0 for all n. Suppose that for

infinite Z, subset. By definition of B, n>N,

Canlo,

Canlr

*.*,

are constant. Now

,

an > B an+,

hence

[anlr+ I [a,,

by the definition of

Jr+

<

and for n > N [anlr

< [ 4 r + 1 5 2 [anlr + 1

9

so { t a n l r + 11 is a bounded non-decreasing sequence of integers and so is constant for n > N', where N' is some integer > N. Therefore

*..' C 4 r + 1

[anlo,

are constant for n > N ' . Hence for each r, [.,,Ir is eventually constant. We conclude that the sequences represented by sequence numbers Z(a,) must be unbounded in length, i.e. each [a,], is k-attained for arbitrarily large k and hence for all k, provided that n is sufficiently large. Hence E' = { a ' : ( 3 i ) (3n) (Vm)[rn > n => [aJi = a ' ] }

is an infinite Z, set contained in C. This contradicts the definition of C and we conclude that B is a (recursive) quasi-well-ordering. By theorem 6.1.2 it follows that A=W.B is also a (recursive) quasiwell-ordering. Now let

T = ( ( 0 , o>, ( 0 , I>, (1,1>>, then T E and ~ if XEC'TexpA then x is of the form x=e(

a0

... a k

1 ... 1

)'

where a.

>A

a,

* a *

>A

uk.

We abbreviate the notation to x = (a09

'a.3

ak).

Now let g be the one-one recursive function which enumerates the

Ch. 71

81

EXPONENTIATION

(infinite recursive) set B in order of magnitude. Suppose j ( m , <xo, ..., x k - J ) , j ( q , (

..., x k ) ) ~ B ,

~ 0 ,

then j ( m , <xo,..., x k - J ) < d ( q , ( ~ 0 ,

xd),

(2) since if xo, ...)x k are k-attained by q then xo, ..., xk-l are (k- 1)-attained by some m s q . So in particular the smallest number in B must be of the form g (0) = j(N, (0)) for some N .

We now define a function f which we shall show yields a recursive descending chain in T exp A. Set f(o) = j (2, j (N, (0))) and suppose for n 2 0, f(n) = (ao, 4, g(n) = j ( m , <xo,...,xl)). *.*Y

+

We describe a procedure for obtainingf ( n 1) and only afterwards show that f (n + 1) always exists. It will be clear that f is partial recursive. Let ui be the first term uj in f ( n ) of the form j ( p , j ( q , <xo, ..., x ~ - ~ ) ) ) with P 2 1 .

Let a; = j ( p

- 1 , j ( q , <xo, ...,

and replace ui by a; to obtain

(3)

a2

[Ch. 7

CONSTRUCTIVE ORDER TYPES

so

j(xl

+ 2,j(m, (xo,...,

~2)))


- 1,.j(q,

(xo,...,

.,-I>))

(6)

and

f(n

+l>
It remains to show thatf(n) is always defined. Clearly, f ( 0 )is well-defined and contains a term of the form (3), namely j ( 2 , j ( N , ( O ) ) ) , since Zg(1) is a proper extension of the sequence number Ig(0). Now suppose thatf(n) contains such a term, then by (2)

s(n) = j ( w <xo,..., xJ), where for some aiinf(n) ai = j ( p , j ( q , ( ~ o , . . . ,xi-,))>.

Now the p in this ai was obtained in one of two ways : either (i) as xi +2 in the construction of the a' in the previous step or (ii) as p - 1 in the construction of a:. In case (i), clearly p > 1 and in case (ii) we observe that xi satisfies x,-,<x,<2x,-,+l,

+

so there are at most xi-l 1 choices for x,. Thus case (ii) could only have 1 times in the calculations of values offwhich used arisen at most this particular p. In either case, therefore, p 2 1. It follows that f(n + 1) is well-defined and contains a term a' of the form (2) by (6). Hence by induction on n , f ( n ) is defined for all n sofis partial recursive and total, hence recursive. This completes the proof.

+

7.3. 7.3.1 DEFINITION. If A 2 1 and B+O then A B = A exp B=COT(A exp B) for AEA and BEB. OA=O if ,440; O o = l , A o = l if A21. By theorem 7.1.6,A expB is uniquely defined. 7.3.2 THEOREM. (i) If A+O and B are co-ordinals, then A expB is always defined and is a co-ordinal, (ii) There is a recursive quord A such that 2A is defined and is not a quord.

Ch. 71

EXPONENTIATION

83

(iii) If A expB is a quord (co-ordinal) and A 10,1 and B i O then A and B are quords (co-ordinals). PROOF. (i), (ii) are immediate from theorem 7.2.1.(i), (iii). (iii) Since A 4 0 , 1, if AEA, then there exists UEC'A such that a+min(A). Hence the map

is a one-one partial recursive map of C ' B into C'A exp B for BEB. Hence if {d,} is a (recursive) descending chain in B, then {b(d,,)}is a (recursive) descending chain in A exp B. Hence if A exp B is a quord (co-ordinal), then B is a quord (co-ordinal). Similarly, if B i0 and BE B then there exists b E C' B and the map 4x1=e

(a>

if x

+ min(A),

is undefined otherwise,

is a one-one partial recursive map of C'A(min A into C'A exp B for AEA. Hence if {d,} is a (recursive) descending chain in A, then ( ~ ( d , , )is} a (recursive) descending chain in A exp B. Hence if A exp B is a quord (co-ordinal) then A is a quord (co-ordinal). We recall that AB is defined only if A =0 or A 2 1.

7.3.3 THEOREM. (AB)' =ABC. PROOF. If A=O, B = C = 1 then both sides of the equation are 0; in all other cases for A, B, C=O or 1, both sides are 1. We therefore assume A, B, C+O, 1. Let AEA, BEBand CECwhere A, B, CC R, thenassuming that a, b, c below (with or without sub- or superscripts) always denote elements belonging respectively to C'A, C ' B and C'C,

84

CONSTRUCIlVE ORDER TYPES

[Ch. 7

&: D = 0.v . D =i0= & [(n I n' & (Vr) ( r 5 n * c, = c: & 4 , = 4:)) v (3r) ( ~ t{ () t < r-(Vs

I t ) (c, = c: & 4, = 41))

& (c, i C: . v . C, = c: & (q,, 4 : ) E AB))]) .

Now (q,, q:)EABo(t,

I t: & (Vs) (s 5 tr* b,, = b:, & a,, = a:,)> v (3u) ( V U ) {(u < u * b,, = b:, & a,, = a:,)

.

& [bru < b:, v .b,, = b:, & a,, < a:,]}

.

On the other hand

Now let p be the partial recursive function defined only on the recursive set

(eo ...

{e(G): G = io

by p(O)=O,

where

*'*

ik)

ek

& (Vi) ( e , E p e ) ) ,

Ch. 71

85

EXPONENTIATION

Using the definitions of (AB)' and ABcgiven above the reader will readily verify that p is order preserving. That p is one-one and partial recursive follows easily from theorem 7.1.2. It follows that p: (AB)'

N

ABc.

Taking C.0.T.s completes the proof. THEOREM. A exp(B+C)=AB*AC. PROOF. Let AEA, BEB and C E C where B) (C. Then BCC is welldefined and by definition 2.1.1 there exist disjoint r.e. sets B, C such that

7.3.4

C ' B c B and C ' C c C . Now

C'(A exp(B T C)) = { e ( D ) :DE(A, B C C)} and

82 eO

> B + C el . " > B + C

This last clause is equivalent to e, > B ... e, v e , > '... e, v (3s)(s < r & e, >'... es & e,+,

>B

... e,).

On the other hand C'(AB-AC)= { j ( e ( D , ) ,e ( D , ) ) : D , E ( A , B) & D,E(A, C)) and

& e,+,

>B...

e,

and

Recalling that if r=s, then D, = O and e(Dl)=O, and that if s= - 1, D , =O and e(D,)=O, it follows that the map p defined by

P (4=

(.I*

k (4

86

[Ch. 7

CONSTRUCTIVE ORDER TYPES

+

is order-preserving between AB*A' and A exp (B C). Now let

D = { j ( e ( D l ) ,e ( D , ) ) :D , =

(bo ...... b,,,) a.

a,,,

& D , = ('y

& (Vi) (ai,a:

a0

".

":)

...a,

+ min (A) & b i E B & c ~ E C ) }

and let q be the map p with domain restricted to D. Then q is clearly partial recursive since D is r.e. Further q is one-one, for suppose q(x)=q(y)=z, say. Then z = p j ( e ( D , ) ,e ( D , ) ) = e ( D ) for some bracket symbols D,, D,, where D is uniquely determined since e is one-one. Now

D = (eO*''~:), a,

...

where a,$:min(A) and e , e B u C . Further, there is precisely one number s such that - 1I i < s * e , E C & s < i I r * e i E B by the definition of the domain of q. Therefore D,, D , are uniquely determined and q is one-one. Thus q is a recursive isomorphism from AB-AConto A exp(B$ C); taking C.0.T.s completes the proof. 7.3.5

COROLLARY. A"=A. ....A.

n factors PROOF. By induction on n is left to the reader. THEOREM. (i) A " - A w = A W , (ii) (A")W=AW. PROOF. By theorem 6.2.2. (iv), (vi).

7.3.6

We now prove an analogue of lemma 6.3.1.72 Since, if A+O is a recursive quord, then 1IA (corollary 3.3.2) the following lemma holds for recursive quords and co-ordinals.

7.4

7.4.1

LEMMA. If A 2 1 is a quord, then

B A = A o ( 3 C ) (BWC = A ) .

Ch. 71

87

EXPONENTIATION

PROOF. BwC= A+BA= B1'IyC= BWC=A by theorems 7.3.6 and 5.2.6.(i). Conversely, suppose BA =A. We may assume A > 1 since otherwise we may take C = 1 (= B). By hypothesis there exist quasi-well-orderings A, B c R and a recursive isomorphismfsuch that A E A , B E B and f : A - B A . Let A = C'A, B =C'B. Since 1< A we may assume min(A) = min(B) = 0 . Now we put (even if &A, though then a, may be undefined for all r greater than some s) a0

=a3

ar+l

=

U (a,).

Then, if UEA, f(a)=j(b,,a,)

forsome b,€B, a 1 € A

(7)

and in general f ( a , ) = j ( b , , ~ ~ + ~forsome ) ~,EB,~,+,EA.

We first of all show that ai+zi+l for all i if aeA. Suppose a+,

then

j ( 0 , a) < B A j ( O , a l )

and hence f-'j(O,n)=a'<

f-'j(O,a,)
by(7).

Therefore j (0, a') < BA j (0, a )

and a" = f - ' j ( O , a ' ) < f - ' j ( O , a ) = a' and so on. Since, for aeA, a', a", ...are all defined, {a, a', a", ...>is a recursive descending chain in A which is impossible since A is a quord. Now suppose ai=ai+lfor some UEAand some i, then f(ai)=j(bi,

ai+l)

and j ( 0 , a i + l ) s B A j ( b i , a i + l ) *

+

Now if b, 0 then (say) a'=f-'j(O,

a,+ 1) < A f -

' j ( b , ai+1) =ai + 1.

Hence, as above, we get a recursive descending chain sincefis order preserving. This is impossible so we conclude bi=O. Thus we have a = a o > a l ~ ~ ~ ~ > a r = a and , + l b,= = ~~ 0= ~ b,+l = . . a .

88

[Ch. 7

CONSTRUCTIVE ORDER TYPES

Let n be the partial recursive function defined by ( a ) = P, {ar = ar+ 1

I

9

then n is always defined if a e A and moreover, lai a, and Ibi b, are partial recursive functions since the above process of calculating the a, and bi is clearly effective. Let C=A[{x:If(x)=x} and let D=g(BW) where g is the partial recursive function defined only on pe, which maps only bracket symbol images of the form nOnl ... n,

where n , b,,E.N

and no > n, >.-->n, 2 0

onto no n o - 1 ... n , + 1 n , n , - 1 . . . n, n , - 1 . . , 0 e ( b , o 0 ... 0 b,, 0 ... bnS 0 ... 0

(Note. Although this last expression is not technically a bracket symbol since we are allowing zeros in the bottom row, nevertheless no confusion will arise since all natural numbers S n , occur if the bracket symbol is not 0 and if it is g gives the value 0. Further, n,>O or b,,>O since no=bn0=O only if we are dealing with the minimum element and this maps to 0.) 1.e. g (x) inserts the missing positive integers in the top row of e-' (x) and in the columns where an integer was missing inserts min(B) in the bottom row and finally g takes the image under e of the resulting bracket symbol (g (O)=O). It is clear that g, and hence D, is well-defined. We shall now show that A z D - C from which it follows at once that A=BWC where C=COT(C). Let

(') =

'(.

n(x)- 1 ... i (kf (lf)"'"'-' (XI ... kf ( l f y (x)

... kf

(x)

Clearly, h is partial recursive. Suppose h (x) =h ( y ) , then kh (x) =kh ( y ) and lh (x) = lh ( y ) . Hence

).('">fI(

= (lf)"'Y'(y).

Now, since e is one-one we have nCx) = n ( y )

(8)

Ch. 71

89

EXPONENTIATION

and therefore, for 0 Ir
= kf(U)’(Y)*

(9)

Putting r =n ( x ) - 1 in (9) and using (8) we have f(Zf)”‘”’-’(x) = f(zf)”‘”’-’(y). But f is one-one, hence

and by induction it follows that x=y, i.e. h is one-one. We next prove that h is order-preserving. Suppose u < ~ u ‘and let n = maximum of n (a) and n(a’)(which are both defined since a, u’EA). Since f is order-preserving between A and BAY ai < A a l e f ( a i I < B ~ f ( a l ) O U ~ + <~A U ; + ~

ai+l = a l + l

or

&bi+l


where bi + =kf (ai+ J. Hence u

or a, = U: & b, < A b; or U , = a; & u , - I = uL-1 & b,-I


< A U;

.... ....

....

or

a,=a:&...&a,=a;&b,<,b;


bk-l

or

90

[Ch. 7

CONSTRUCTIVE ORDER TYPES

.... . . . . or a, = a: & b, = b: &.-. & b2 = b; & bl x Bb; ,

Sincef(ai)=j(ai+l, bi+l). Now h ( a )
(12)

(11) and (12) are equivalent since, as we observed above, if n>n(a), then u, =an(a)and b, =min (6) =0.

Finally, h maps C'A onto C'DC for suppose

( b , ,Then U,EA where C E C C(and b,eB). Let uo=c, ~ ~ + ~ = f - l ( j a,)). and if a, EA, then a, + E A ; in particular a,,EA and a,,=h- (x). We have therefore proved h:AczDC and the lemma is established.

'

THEOREM. nw = W if n >0. PROOF. It sufficesto prove that if N~ n then , W N NW.Let

7.4.2

N = ( ( x , y ) : 0 Ix I y < n} , then clearly N en.

If SEN, then s is expressible in the form s = nrar

+ nr-lar-l +...+ a,,

where for all i, O l a i t n . Let f be the (partial) recursive function defined by

where columns with lower entry 0 have been omitted. E.g.

f(n2-3

+ n.0 + 2) = e

(i '1)

Ch. 71

8xpoNE"I1ON

91

for n > 3 . Then, if u , w N and u=nrur+...+uo and u=n'b,+-.-+b, (where some of the leading coefficients may be zero), u

< v o ( 3 t I r ) ( ( w > t = s a , = 6,) & a , .c 6,)

(13)

and u = ~ e ( V t ) ( a= , b,).

(we remark that the fact that ar, ,... and b,, br-l,... may be zero does not affect the ordering.) But the ordering 5 given in (13) is precisely the ordering in NWof the bracket symbol images

when columns with bottom row zero have been omitted in the bracket symbols. Clearly, f is one-one and hence f:W N NWand the theorem is established.

CHAPTER 8

ARITHMETIC LAWS AND PRINCIPAL NUMBERS

8.1 We now investigate the arithmetic laws of C.0.T.s. Some laws for addition were established in chapter 4 and we first of all strengthen these. 8.1.1 THEOREM. If A , By C are quords, then B < C-A + B < A

+ c.

PROOF.Immediate from theorem 4.2.3.(v) and corollary 3.2.9. However, the other obvious analogue of a classical additive law

BIC*B+AIC+A

(*I

does not hold in general even for co-ordinals. For let A = W, B=O and C = V, then (*) implies W and V are comparable which contradicts corollary 5.2.7. However, (*) does hold if A , B, C are less than what we call a principal number for addition. Classically, a principal number for addition (otherwise called a y-number (BACHMANN, 1955, p. 67) or a prime component (SIERPINSKI, 1958, p. 279)) may be defined as an ordinal number y such that any of the following (equivalent) conditions holds: a
or P = y ,

P
y =0 or y = ma for some ordinal u .

(1") (2') (3") (4")

Ch. 81

ARITHMETIC LAWS AND PRINCIPAL NUMBERS

93

We leave consideration of a recursive analogue of (4') until we have dealt with Cantor Normal Forms, only remarking that we shall show that every co-ordinal of the form W Ais a principal number for addition but that these are not the only principal numbers for addition.

8.1.2 THEOREM. If A is a quord 2 1 then the following implications hold between (l),(2), (3) below:

-

z

(1) (2) - I (3) B < A*B + A = A , B+C=A*C=O or C = A , B, C < A*B + C < A .

(1) (2) (3)

PROOF. (1)*(2). Suppose (1) holds and B+ C = A. Then if C+O, B< A and by (l), B+A=A, hence by corollary 3.2.9, C=A. (2)*(1). Suppose (2) holds and B c A , then there is a C 9 0 such that B + C = A and by (2) we have C=A, hence (1) holds. (1)*(3). Suppose (1) holds and B, C
8.1.3 DEFINITION. A co-ordinal (quord) A is said to be a principal (9-principal) number for addition if A > 1 and B
8.1.4 THEOREM. If A € # ( +), then IAl is a limit number. PROOF. Clearly no finite co-ordinal is a principal number for addition. Suppose IA1 = 1+n where 1 is a limit number and n >0. Then by theorem 5.1.6 B=A-niswell-defined, JBJ=AandBO,B+A=A and 1+n +1= 1 which is impossible.

94

[Ch. 8

CONSTRUCTIVE ORDER TYPES

8.1.5

THEOREM. If P is a principal number for addition and A , By C < P

then A
PROOF. By theorem 8.1.2, A , B, C < P imply A + C < P and B + C t P . Therefore by theorem 4.2.8, A + C and B + C are comparable. Now classically, aa+ysp+

y,

hence by theorem 5.3.4 A + C < B+ C. COROLLARY. If A, B < P and P is a principal number for addition, then B I A + B. Conversely, if A + B < P , then A , B t P . PROOF. Trivially A < P if A + B < P . Since P is a principal number A + B < P implies A + B + P = P . But A < P so A + P = P and therefore A + ( B + P ) = A + P . Hence B + P = P and B I P . Finally we cannot have B =P since then A +B 43'. 8.1.6

8.2 We now give a series of results for quords. All the results in this

section were in a sense obtained in the context of ordinal algebras 1956) by Tarski. Many of the proofs are highly derivative from (TARSKI, Tarski's proofs. We do not know which, if any, of these results can be extended to arbitrary C.0.T.s though a start on this problem was made in 9 2.4 using Morley's lemma. 8.2.1

THEOREM. If B g 0 is a quord and there exist C, D such that then B = A * W .

B+C=A.W=D+B,

PROOF. By theorem 6.2.5. D + B = A - W implies A . W I * B. B+ C = A . W implies B< A . W, hence by theorem 4.2.6, B= A . W. 8.2.2

THEOREM. If C is a quord, then A + B + C = C and ( A + B ) - W = ( B + A ) . W A+C=B+C=C.

PROOF. By lemma 6.3.1, using the first condition, ( A + B).W

+D =C

for some quord D .

iff

Ch. 81

ARITHMETIC LAWS AND PRINCIPAL NUMBERS

95

By theorem 6.2.3.(iii), (B

+ A ) - W = B + ( A + B).W = ( A + B ) . W by the second condition.

Hence by lemma 6.3.1, B * W + E = ( A +B).W

forsomequordE.

Therefore B

+ C = B + ( A + B ) * W+ D , = B + (B.W + E ) + D , =B*W+E+D, by theorem 6.2.2.(iv) = ( A + B)*W D = C .

+

Thus B + C = C and therefore A + C = A + B + C = C . Now suppose A + C = B+ C = C,then A + B+ C = C and B+ A + C = C . Hence ( B + A ) . W + D = C and ( A + B ) . W+E=Cfor some quords D,E, by lemma 6.3.1. By the directed refinement theorem 2.3.2 it follows that for some F either (B + A)+W F = ( A B ) * W , or ( A + B)-W F = (B + A ) . W . (5) Wesuppose the latter holds. Now (A + B ) . W = A + ( B + A ) . W by theorem 6.2.3.(iii), hence (B + A ) . W I * ( A + B ) . W and hence by ( 5 ) and

+ +

+

theorem 4.2.6, (B + A ) . W = ( A + B ) . W . This completes the proof.

8.2.3 THEOREM. If A, C (or B, C) are quords, then A + C.n = B + C.n*A + C = B + C . PROOF. By the directed refinement theorem 2.3.2, there exists an E such that either A + E = B or B + E = A and E + C - n= C - n . Suppose, without loss of generality, that A + E = B . By lemma 6.3.1 we have, for some F, E . W + F = C * n = C + C . ( n - 1) by theorem 6.2.2.(ii). Now by the directed reiinement theorem 2.3.2, there exists G such that E * W + G = C and G + C - ( n - l ) = F

(i)

96

[a. 8

CONSTRUCTIVE ORDER TYPES

or C+G=E.W

and G + F = C . ( n - l ) .

(ii)

If (i) holds then

E+C=E+E*W+G =E*W+G =

c.

by theorem 6.2.2.(iv)

Now B+ C = ( A +E ) + C = A + ( E + C ) = A + C and the theorem is established for this case. If (ii) holds, then by theorem 6.2.5 there exists H such that H + E . W = G. But then C - ( n - 1)= G + F = H + E . W+F=Hf C - n = H + C * ( n- 1)+ C (last step by theorem 6.2.2.(ii)). By lemma 2.4.8 (proof) we therefore have C . ( n - l ) = C * ( n - l ) + C . But then by corollary 3.2.9 we have C=O which reduces the theorem to a triviality. 8.2.4

THEOREM. If A, C, D (or B, C) are quords and A + C - n + D = B + Cqn, then A + C + E = B + C for some (quord) E .

Before we proceed to the proof we observe that if one side (and hence the other) of the k s t equality is a recursive quord or co-ordinal then the theorem may be stated thus: A

+ Can 5 B + C.n=+A + C IB + C .

PROOF(by induction on n). If n = 1, then the assertion is trivially true with B= D. Now suppose the implication holds for n, then A

+ C * ( n+ 1) + D

=B

+ C.(n + 1)

implies

A + C - n + ( C + D ) = B + Can + C . Therefore, by the directed refinement theorem 2.3.2, there is an F such that either A + C - n + F = B + C - n and C + D = F C , or A + C - n = B + C - n + F and F + C + D = C . In the former case, by the induction hypothesis

+

A+C+E=B+C

for some E. In the latter case, by theorem 3.2.7, D=O and hence

A

+ C.(n + 1) = B + C * ( n+ 1)

Ch. 81

ARITHMETIC LAWS AND PRINCIPAL NUMBERS

97

and by theorem 8.2.3, A+ C=B+ C.

This completes the proof.

8.3 By analogy we now introduce principal numbers for multiplication (cf. BACHMANN, 1955,p. 66). 8.3.1 DEFINITION. A co-ordinal (quord) A is said to be a principal (.%principal) number for multiplication if A > 2 and 0
We write X ( . )[3(.)]for the collection of all principal (9-principal) numbers for multiplication. Later on we shall show that A E3(-)implies A is divisible by or divides Wwnfor every n. As in the classical case O < B
A

is a stronger condition than BC=A*B=A

or C = A .

But also, for co-ordinals, the former condition is stronger than 0 < B, C < A* BC < A .

For V satisfies this last condition but is not a principal number for multiplication since 2< V but by lemma 7.4.1,2V = V implies 2w = V and by theorem 7.4.2,2w = W + V. 8.3.2 THEOREM. If A is a co-ordinal in X ( -),then IAJis a limit number. PROOF. Left to the reader (cf. theorem 8.1.4).

+

8.3.3 THEOREM. (i) If B 0, then A I A B when B is a recursive quord, a co-ordinal or B 2 1. (ii) If B > 1, then A < A B whenever ABO for an arbitrary quord B. (iii) If A divides B and IAI =lBl, then A = B . PROOF. We prove only (ii) and (iii) leaving (i) to the reader.

98

mNsTRum ORDJiR TYPES

[Ch. 8

(ii) B > l = > ( E !C ) (B=l+C&C+O).HenceAB=A(l+C)=A+AC, where A C l O i f A 9 0 ; thus A c A B . (iii) By (i) A divides B implies A S B hence by theorem 5.3.4, A =B. 8.3.4 THEOREM. If A and B are isomorphic well-orderings then there is a unique isomorphism f such that every isomorphism between A and B is an extension off. (SIERPINSKI, 1958, p. 264, corollary 3).

We now prove a left cancellation law for co-ordinals using this classical theorem 8.3.4. Later on we shall use the same technique to obtain a cancellation law for exponentiation for co-ordinals. 8.3.5

THEOREM. If A 9 0 and A , B, C are co-ordinals, then AB = AC* B = C .

PROOF. Let AEA, BEB and C E C and suppose p : AB N AC.

Then AB-AC and since AB and AC are well-orderings it follows from the preceding theorem that p is an extension of the unique minimal isomorphism, p,, say, between AB and AC. Now, classically, u 0 &a/? = ay-p = y ,

*

hence there is an isomorphism q, (not necessarily partial recursive) such that

q,:B-C. Now the map r , : j ( a , b)=+,

q,(b)),

defined only on C A B is an isomorphism between AB and AC and therefore, by theorem 8.3.4, p is an extension of r,. Since A+O, there is an element, say a,, in C'A. Let p , be the map p with domain and range restricted to (j(ao,n):nENZ,

then po is partial recursive. Further, if poj(ao,x) is defined, then its value is j(u,, y) for some y.

Ch. 81

ARITHMETIC LAWS AND PRINCIPAL NUMBERS

99

Now let qo be the map 40 :x + IPOj (ao, x)

,

then clearly qo is partial recursive and agrees with qc on C'B (again by theorem 8.3.4). qo is one-one, since 40

(XI = 40 ( Y ) * k o j (ao, x) = b o j (ao, Y ) * P o j (ao, x) = j (UO? c ) Lk P o j (a05 Y > = j (a03 c>

(since ppo E {j(uo,n ) : n e M }by construction) * j ( a 0 , x) = j ( u o , y)

(since p is one-one)

*x=y.

Thus qo is partial recursive and one-one and also agrees with qe on C'B, i.e. q0:B=C from which the theorem follows. We have a stronger version of the above theorem which derives from SIERPINSIU (1948), however, we shall leave the proof of this theorem until appendix A.

8.3.6 LEMMA. If M is a principal number for multiplication, and B, C 0, then

+

BC< M o B < M & C < M .

PROOF. Suppose BC<M, then B<M or C = l by theorem 8.3.3.(ii). In the former case B M = M and in the latter trivially, C<M. Now BC < M

=> BCM = M

,

and therefore by theorem 8.3.5, CM=M.

Using theorem 8.3.3.(ii) it follows that C<M.

Conversely, C <M*CM

=M

and B < M a B M = M .

100

CONSTRUCTIVE ORDER TYPES

[Ch. 8

Hence (BC) M

= B ( C M ) = BM = M

and by theorem 8.3.3.(ii) BC<M.

8.4. 8.4.1 THEOREM. If A, C are quords then (i) If A =I= 0, then B < C* AB < A C ,

(ii) BI; C-ABS AC. PROOF.Let AEA and CEC, then there exists BEB such that B
+@

C=B$D. Now the reader will readily verify that AC = A(B $ D) = AB $ AD. But AD

+0,since A, D=k8, hence AB

-= AC

and the theorem follows by taking C.0.T.s. (ii) follows at once from (i).

8.4.2 THEOREM. There exist co-ordinals A, ByC+O such that A < B but AC $ BC.

PROOF. Let A = 1, B= V and C = W, then AC=W

and B C = V W .

By theorem 8.3.3.(i), VIVW.

Hence if

w 5 vw, V and W are comparable by theorem 4.2.8 which contradicts corollary 5.2.7.

Ch. 81

ARITHMETIC LAWS AND PRINCIPAL NUMBERS

101

8.4.3 THEOREM. If there is a principal number for multiplication, My such that B, C < M (or equivalently B C < M ) then A < B*AC

BC

PROOF. If B or C=O there is nothing to prove. Similarly if A=O. Otherwise, by lemma 8.3.6, AC<M

and B C < M .

Hence, by theorem 4.2.8, A C and BC are comparable. Now classically, for arbitrary ordinals a, /I, y a < P*ar

S

Pr,

hence ACI BC by corollary 5.3.5. 8.4.4 THEOREM. If A, B, C are co-ordinals, then AC < BC* A < B .

PROOF. If C=O, then the assertion is trivial. If C+O, then by theorem 8.3.3.(i), A I A C and B S B C . Hence by the transitivity of Iand theorem 4.2.8, A and B are comparable. By the classical theorem UY


P,

we have IAl .c IBI

and hence by corollary 5.3.5 A
8.4.5 THEOREM. There exist co-ordinals A, B, C such that O
but A $ B .

PROOF.(As in the classical case.) Let A =2, B= 1, C = W. 8.4.6 THEOREM. If By C are comparable quords, then A B
PROOF. Immediate from theorem 8.4.1 .(i).

102

[Ch. 8

CONSTRUCTIVE ORDER TYPES

THEOREM. If A, ByC are co-ordinals and AB< AC then B < C ; similarly with “ s ” replacing “ < ” at both occurrences. PROOF. AB
8.4.7

So A C = A D + A E by theorem 6.2.1. By theorem 4.2.1 we have ( 3 F ) ( A . B + F = A . C ) and again using corollary 5.3.6 ( E ! G ) ( E ! H ) ( G + H = A C& IGl=a/3). We must have then G = A B = A D , and by theorem 8.3.5, B = D . D < C by (*) so the result follows. The other part of the theorem follows from the above and theorem 8.3.5.

It is well-known that, classically, for arbitrary order types

b, z,

0 - n = z - n & n > O*o = z (and similarly with ‘‘5” replacing “ =” at both occurrences, see (1948) for proofs). For quords the analogues are also true SIERPINSKI, provided we use the strong interpretation of I ; thus we have: 8.4.8

THEOREM. If A or B is a quord, then A - n + C = B - n & n > O* (30) (A

+ D = B).

PROOF^^ (by induction on n). If n=1, then the assertion is trivial. Now assume the theorem holds for n and that A-(n + 1) + C = B . ( n Then

A * n+ ( A

+ 1).

+ C ) = Ban + B

and by the directed refinement theorem 2.3.2, there is an E such that either A-n + E = B * n ( & A + C = E + B), or A * n = Ben + E & E + A + C = B . In the former case the assertion follows by the induction hypothesis. In the latter, by the induction hypothesis we have, for some F, A=B+F,

Ch. 81

ARITHMETIC LAWS AM) PRINCIPAL NUMBERS

103

whence E+B+F+C=B

and by theorem 3.2.1, E+B=B,

F=C=O.

We conclude A =B.

8.5 We now introduce principal numbers for exponentiation and investigate the arithmetic laws for exponentiation.

8.5.1 DEFINITION. A co-ordinal (quord) A is said to be a principal (9-principal) number for exponentiation if A > 2 and

1 < B < A - B~ = A . We write &' (exp) [1(exp)] for the collection of all principal (%principal) numbers for exponentiation. Later on we shall obtain an explicit description of all principal numbers for exponentiation (chapter 11). THEOREM. If AE&'(exp), then IAl is a limit number. PROOF. Left to the reader.

8.5.2

The condition in definition 8.5.1 is stronger than the condition 2 5 B, C < A*BC < A

This will be shown later by proving (lemma 11.2.2) that

2A = A *

W divides A ,

whereas V satisfies (6) but not (7). 8.5.3

THEOREM. If A > 1, B, C are co-ordinals, then A~=A~*B=c.

PROOF.^^ Let AEA, BEB and CEC and suppose p : A~

A=.

104

CONSTRUCTIVE ORDER TYPES

[Ch. 8

Then A'-AC and since A' and A' are well-orderings it follows from theorem 8.3.4 that p is an extension of the unique minimal isomorphism p e , say, between A' and A'. Now, classically, u>l&aS=aY+B=y,

hence there is an isomorphism qc (not necessarily partial recursive) such that qc:B C . Now the map N

defined only on 83 e (A, B) is an isomorphism between A' and A'. Hence by theorem 8.3.4, p is an extension of rc. Since A > 1, there is a non-minimum element, say a,, in C'A. Let p , be the map p with domain and range restricted to

then p , is partial recursive. Further if

is defined then its value is

for some y. Now let q, be the map

then clearly q, is partial recursive and agrees with qc on theorem 8.3.4). qo is one-one, since

C'B (again by

Ch. 81

ARITHMETIC LAWS AND PRINCIF'AL NUMBERS

for some u, v , m,n, xl, ..., x,, y,, image of p , is of the form

...,y,,,. But by the definition of p,,

105

any

2j(oo. b )

and hence n =m =0,u = v = a, and

and

Therefore, from qo(x) = qo (Y), we have

from which it follows, sincep, and e are one-one, that x=y. Thus we have shown that q, is a recursive isomorphism between B and C and the proof is complete. 8.5.4

THEOREM. There exist co-ordinals A, B, C, all > 1, such that A C = B C but A + B .

PROOF(as in the classical case). Let A = 2 , B = 3 , C = W, then by theorem 7.4.2, 2" = 3" = W. 8.5.5

THEOREM. (i) A > 1 & B
(for arbitrary quords A,

B, 0. (ii) If C is a co-ordinal, then A > 18z B < C+AB
d>,$,b, hence any bracket symbol in (A,

BT D) is of the form

106

CONSTRUCTIVE ORDER TYPES

[Ch. 8

where the d&'D, b j ~ C ' Band the a,, aj.eC'A ( i = O , . . . , m ;j=O,..., n). Hence from the definition of exponentiation we easily obtain

A0 5 AO'D= AC. (ii) By theorem 7.2.1.(ii), assuming A E A ,etc. as above, A' (and hence AB)is a quasi-well-ordering. Hence it suffices, by lemma 4.2.5, to prove that A~ A'.

+

A a+min(A). Since D+0 there Now A > 1, hence there exists ~ E C with also exists dsC'D. Therefore

(")

e a

CAC

- CAB,

from which we have the required result.

LEMMA.(i) If C is a co-ordinal and A, C > 1 then A < A'. (ii) If C r 1, then A
8.5.6

Now, by the definition of exponentiation we have at once 1 IA D since A 9 0.

Hence

A' = A ( l + E) for some E, by theorem 6.2.1. = A + AE

The required result now follows at once. 8.5.7

THEOFCEM. There exist co-ordinals A , B, C such that A
but A C $ B c .

PROOF.Let A = 2 , B= Vand C= W,then by theorem 7.4.2, A'=C=W and by lemma 8.5.6.(i), v < VW=BC. Now,if IB ~ ,

Ch. 81

107

ARITHMETIC LAWS AND PRINCIPAL NUMBERS

then by theorem 4.2.8, V and W are comparable which contradicts corollary 5.2.7. Thus we see that the analogue of one of the classical laws for exponentiation breaks down in a very similar way to one of the multiplicative laws (theorem 8.4.2). However, the similarity also extends to the cases where the analogues do go over.

LEMMA.If E is a principal number for exponentiation, then A , B<E+AB< E and conversely if A, B > 1. PROOF. The assertion is trivial if A , B I 1. Otherwise, if E is a principal number for exponentiation, then A < E*AE= E and similarly for B ; moreover, we must also have that A and B are co-ordinals since E is a co-ordinal. It follows that A(BE) =E.

8.5.8

Since B < E it follows by theorem 2.1.2 that there is a (co-ordinal) C such that B+C=E. Therefore E = A ( B ~ )= A ( B + C ) = A B . A C , But A'> I , since C + O ; hence A'= 1 + D forsome D + O and it follows that i.e.

+

E = A B ( l + D) = AB + ABD where ABD 0 ,

A~ < E . Conversely, suppose A,B>1

and A B < E ,

then by lemma 8.5.6.(i), A<E.

Since E is a principal number for exponentiation,

E=

= ( A ~=) ~

By theorem 8.5.3 it follows that BE=E

108

CONSTRUCIWB ORDER TYPES

[Ch. 8

and hence by theorem 8.3.3.(ii) we have B<E. 8.5.9 THEOREM. If there is a principal number for exponentiation, E, such that B, C < E, then A
PROOF. By the transitivity of 5 and lemma 8.5.8, A C < E and B C < E .

Hence by theorem 4.2.8, AC and BC are comparable co-ordinals. Now, classically, for ordinals a, fl, y, hence

8.5.10 THEOREM. If A, B, C are co-ordinals, then

c B~ =+ A .C B . PROOF. If C=O, there is nothing to prove. Otherwise, by lemma 8.5.6(ii), A I A C and B I Bc

and therefore by theorem 4.2.8 and the transitivity of I, A and B are comparable. Hence by the classical theorem for ordinals ay < pY*u

< 8,

we have IAI < IBI

and hence A < B .

8.5.11 THEOREM. There exist co-ordinals A, B, C such that

1 < AC IBC but A $ B.

PROOF(as in the classical case). Let A=3, B = 2 and C=W, then by theorem 7.4.2, A ~ = B ~W =.

Ch.81

ARITHMETIC LAWS AND PRINCIPAL NUMBERS

8.5.12 THEOREM. (i) If B, C are comparable and A > 1, then A*

=.

3

B
PROOF. By theorem 8.5.5., since C I B means C = B or C < B.

109

CHAPTER 9

CANTOR NORMAL FORMS

9.1 In this chapter we prove the existence of Cantor Normal Formsg1 for a large class of co-ordinals, but first of all we prove a general decomposition theorem for all co-ordinals less than some principal number for addition. We restrict our attention from now until the end of Part One of this monograph to co-ordinals, returning to the consideration of quords in Part Two. Since we are dealing with co-ordinals we shall repeatedly use the fact (theorem 4.2.2) that A < B e ( 3 C )( A + C = B & C

=+ 0).

9.1.1 LEMMA. A co-ordinal P is a principal number for addition if, and only if, A
9.1.2 THEOREM. If P E S (+), then P. WE&( +) and there is no principal number Q such that P
PROOF. Suppose P E X (+) and A < P -W ,then by theorem 6.2.5, there exist n, D such that

A = P * n + D , where D < P.

Ch. 91

111

CANTOR NORMAL FORMS

Hence A + P . W = A +(P + P - W ) =(A +P)+P-W =(Pen D +P)+P-W = P . ( n + 1) + P a w =P.W.

since 1 + W = W ,

+

sincePsX(+)

and D < P ,

Hence P.WE*( +). Clearly PCP-W since P 9 0. Now suppose Q s X ( +) and P < Q , then by lemma 9.1.1, P*W
hence Q


9.1.3 THEOREM. g2 If O c A < P E X ( +), then there exist principal numbers for addition Ply ..., Pnsuch that P > P,,2 . . * 2Pl

and A = P,,

Pl.

A = Q,

Q1

+ a * * +

Further, if + * * a +

is any other representation of A as a sum of principal numbers Q, (i=l,..., m) such that

Q,

>***

2

Q1,

then m = n & P i = Qi for i

=

1, ..., n.

Conversely, if A is expressible in the form (2) with P,, 2

2 PI where the

Pi(i= 1, ..., n) are principal numbers for addition, then there is a principal number P e.g. P,,. W,such that P > A and P > Pi for i

=

1, ..., n.

PROOF(by transfinite induction on the partial well-ordering I). We assume O < A < P E H ( + ) and take as induction hypothesis: If O c B c A

112

CONSTRUCTIVE ORDER TYPES

[Ch. 9

then B is uniquely expressible in the form (2) where the P, are principal numbers satisfying (1). If A is a principal number for addition then the theorem follows at once from theorem 9.1.2. Now suppose A is not a principal number, then there exist B, C such that B+C=A,C+O,A

and B + C < P .

(3)

By corollary 8.1.6, C
Let P, be the least C satisfying (3). PI exists since { C :c < A }

is well-ordered by I .We show that P, is a principal number for addition. Suppose Pi = D + E , then by corollary 8.1.6, E c P and hence by theorem 4.2.8, PI and E are comparable. But

(El5 lP1l

I

hence E IPi

and by the minimality of P,, E=P,.

Therefore PIis a principal number by theorem 8.1.2. Now let B, be the least B such that B + P, = A . If B, = 0, then A is a principal number which contradicts our assumption. Hence B, + O

and by the induction hypothesis B, has a unique decomposition where

Ch. 91

113

CANTOR NORMAL FORMS

and all the Pi(i = 2, ..., n) are principal numbers. Therefore A = Pn +**.+Pl

and, since p, < p , all the Pi (i = 1, ..., n) are comparable by theorem 4.2.8. Suppose pz < PlB

then, since Pl is a principal number for addition,

Pz +PI =PI; hence A=(Pn+-.+Pz)+P1 =(P.+--+P,)+P,.

Now

*

Pz 0 3 Pn+

9.-

+ P3 < P, + - + P2 = B , , **

whereas Bl was chosen as the least B such that B + P , = A . This contradiction shows that Pz 2 PI and we conclude that A has a decomposition of the form (2). As regards uniqueness, let A

= P,, +..,+PI

and A = Q,

+-.+ Q ,

be two decompositions of A as a sum of non-increasing principal numbers. By theorem 4.2.8, P,,and Q, are comparable. Suppose Pn>Qm,

then, since P,,is a principal number for addition,

Q, Therefore

+ P,,= P,.

+ P,, Pi = Q , + Q , + P,, - ... = Q,.(m + 1) + A .

A = Q,

+ a * * +

+ * a * +

Since A+O, we therefore have Q,.(m

+ 1) < A .

114

[Ch. 9

CONSTRUCTIVE ORDER TYPES

On the other hand, if i Im, then or Qm.2,

Qi+Qm=Qm

according as Qi

< Qm

or

= Qm.

Qi

Consequently, we have Q,.(m

+ 1) + A = A < A + Q;m

5 Q;2m

and by corollary 3.2.9, it follows that A
(5)

which contradicts (4) above. We conclude Q, 4: P,, and similarly P,, 4: Q,,

but since P,, and Q, are comparable it follows that

P,,= Q, and P,-,

+...+ P, = Q,-, +...+ Q1

(by corollary 3.2.9). Proceeding thus we obtain after s( = minimum of m, n ) steps P,,-r = Qm-r r = 0, ..., s and either P,+ - . , + P I= 0 or Q, Q, = 0 , + . e m +

where t = maximum of m - s , n-s. By theorem 2.2.4.(ii) it follows that t =0 and hence that m =n and

Pi= Qi for i

= 1, ..., n ( = m )

For the converse: if A=P,,+**.+P,,

then by the argument which produced ( 5 ) we have A I P;n

and consequently A < P,,. W .

By theorem 9.1.2, P,,. Wis a principal number for addition. This completes the proof.

Ch. 91

115

CANTOR NORMAL FORMS

9.2.

LEMMA. C
9.2.1

The next lemma is a strengthening of theorem 6.2.5. 9.2.2

LEMMA.If A 90,then

c < A B o ( 3 D ) ( D < B & A D 5 c < A ( D + 1)). PROOF.The implication from right to left is clear from corollary 5.2.3, theorem 8.4.1 and the transitivity of < for co-ordinals. Suppose now that C
then there exists FPO such that C+F=AB.

Given AEA and BEB, then by the separation lemma 2.3.1, there exists CEC and F E F such that [C)(Fand]

C $ F=A.B.

Let

f = min(F), d = l(f), D = d) B and D

= COT(D).

We shall show that for this D, ADIC
Now AD E C E AB,

since j ( a , x ) ~ e A D = . xsB d =>j(a7 x )

SABf

=.j(a, X)EC'C.

Further, since AD and C are both initial segments of AB, we must have ADsC. By corollary 5.2.3,

D
116

[Ch. 9

CONSTRUCTIVE ORDER TYPES

and hence by theorem 8.4.1.(ii) it follows that A(D

+ 1) IA B .

From theorem 4.2.8 we now have or C I A ( D + l ) ,

A(D+1)1C

but

c c A * ( DT ( ( 4d ) ) ) and so we have

c IA ( D + 1).

Finally, since

f E C j ( A , d ) - C ' C , C + A ( D + 1) and the proof is complete. 9.2.3

LEMMA.1 1 C < A ~ o ( 3 0 ) ( D < B & A D s C < A D f l ) .

PROOF. The implication from right to left follows at once from corollary 5.2.3, theorem 8.5.5.(ii) and the transitivity of <. Now suppose lIC
C +F

=AB.

Further, there exist A, B such that

A = [AIEA and B = [B]EB and by the separation lemma 2.3.1, there exist C E Cand FEFsuch that

c~

F=A~.

Let f = min(F),

then f

=

( b o ... b:) a,

where bo>

... > b,,,b,eB

... a,

(i=O,...,n), n+-1

(since C21) and93

O+U,EA( i = O , . . . , H).

Also let

D = bo)B and D = COT (D) ,

Ch. 91

CANTOR NORMAL FORMS

117

then we claim ADs C. Now XEC'A" implies

where do
f,

which implies XEC'C. Further, if y€C'ADand < y , x)eAD, then by the same argument Y < A exp B f It follows at once (from corollary 4.2.5) that AD Ic.

Now let E = D T ((bo, bo)} ,

then XEC'CJX=O

or x = e

where bb>-..>b;, biEB and 0 8 a ; e A . Further, by the definition of bo, (bb, bo)

E

B

and hence x€C'AEand

C5AEdD+'. But C 8 AEsince

f E C'AE - C'C and it follows that

AD 5 C < A D + ' . LEMMA. A T W & B< C*AB+ AC= A'. PROOF. Since B < C , (30)( D + O & B + D = C ) .

9.2.4

Hence + A =~ A +~A ~ . A ~ = AB(l + A D ) = AB(l + A E)

+

by theorem 7.3.4, by theorem 6.2.1, by lemma 8.5.6.(i) , where A + E = A D ,

118

[Ch. 9

CONSTRUCTIVE ORDER TYPES

+ W + F + E)

= A'(1

= AB(W

+ F + E)

=A ~ A D =A

9.2.5

+

since W < A* W F = A for some F , by theorem 5.2.6.(i),

~ .

LEMMA. If C < AB, then there exist unique Q, R such that C = AQ

+R ,

where 0 I Q < B & O I R < A .

PROOF. By lemma 9.2.2, C < AB*(3Q)(O I Q < B & AQ I C < A ( Q

Hence AQ I C < AQ

+ 1)).

+A .

Since A , C, Q are co-ordinals, R=C-AQ

+R as required. Now suppose C = AQ + R = AQ, + R , ,

is well-defined, 0 5 R < A and C = A Q

where 0 1 Q,
and O I R < A ,

by theorem 4.2.8 Q and Q, are comparable (and so too are R and R,). By corollary 5.2.3, if Q + Q, then either Q 1 + l I Q or Q + l S Q , .

Suppose the former holds, then AQl I C < A (Q1

+ 1) I AQ < C ,

which is a contradiction. Similarly we cannot have Q

+ 1s Q,, hence

Q = Qi and by corollary 3.2.9, R=R,. 9.2.6

LEMMA.If 1I C< A* then there exist unique D , Q, R such that

C = A D Q + R , where D < B ,

O
and O j R < A D .

Ch.91

119

CANTOR NORMAL FORMS

PROOF. By lemma 9.2.3, 1 IC < AB* ( 3 0 ) ( A DIC < AD+' & D < B ) , hence C < AD+' = A D . A .

Now, if A < C, D + 0. By lemma 9.2.5, there therefore exist Q, R such that C = A D Q + R , where O < Q < A

and O I R < A .

Suppose that we also had C = AEQ, + R , ,

where E < B ,

O
and O I R , < A .

By theorem 4.2.8, D and E are comparable and hence by corollary 5.2.3, if D =kE, then D + l I E or E + l l D .

Suppose the former holds, then by theorem 8.5.5.(i), and using the fact that A E is a co-ordinal, we have A ~ cS < A ~ + I '

C,

which is a contradiction. Likewise we cannot have E + l s D and we conclude D=E.

Now suppose A 2 C, then either trivially A = C or C = A o Q + R , where O < Q < A

and R = O

and Q, R are unique by the preceding lemma. Thus for all A satisfying the hypothesis we obtain D, Q, R uniquely as required. 9.2.7

THEOREM. If 1 IC < AB then C is uniquely expressible in the form C = A B 1 * D+ 1 .-.+ ABr.D,,

(1)

where B > B , > - . . > B , and l l D , < A f o r l l i l r . PROOF. By lemma 9.2.6, 1 Ic < A~ 3 c

=

D,

+ c, ,

where O I D , < A , O I C , < A B ' and B , < B and B,, D,, C, are unique. Since Cl < AB' < A B and AB' I C < A B

120

CONSTRUCTWE ORDER TYPES

[Ch. 9

it follows that C, < C < A B .

Since {E:E< A B } is well-ordered by Iwe obtain, by a finite number of repetitions of the above process, the decomposition (1). Suppose that C is also expressible as C = AF'.H1 +-.*+AFs*Hs, where B> F, > >F, and 1IHi
+

C = AB'*D1 C1 = AB1*Hl+ El ,

where 0
or H,ID,+J

D,+IIH, Suppose the former holds, then

C=A~'.D1+C1
which is a contradiction. Similarly, H, $ D, + 1 and we conclude

D, = H , . By corollary 3.2.9 it follows that C, = El

and hence by induction on the maximum of r, s that Bi = Fi and D i = H i for 1 Ii Ir = s .

This completes the proof. As an immediate consequence we obtain our main theorem of this section. 9.2.8 THEOREM (CANTORNORMAL

FORM).

If O
Ch. 91

CANTOR NORMAL FORMS

121

uniquely expressible in the form

c=~ where A > A , >

~ * +...+ - n , WAp*n,,

(2)

> A , and the ni are finite, non-zero co-ordinals.

COROLLARY. A necessary and sufficient condition that a co-ordinal C+O should have a Cantor Normal Form (2) is that there should exist a co-ordinal A such that C < WA. PROOF. The condition is obviously sufficient. Now suppose C has the given Cantor Normal Form and let A be greater than A , (say, take A =A, + 1); then by n, ...+n, applications of lemma 9.2.4, we obtain C + WA=W Awhence C < WA.

9.2.9

+

C H A P T E R 10

UNIQUENESS RESULTS

10.1 We showed earlier (theorems 4.1.4, 5.2.4) that the finite co-ordinals are unique but that there exist c mutually incomparable co-ordinals of classical ordinal o.Shortly we shall show that this latter result holds for all infinite denumerable ordinals. However, we shall go on to give criteria for some collections of co-ordinals to contain precisely one representative for each member of a given collection of classical ordinals. In this chapter we shall give simple criteria for sufficiently small well-orderings to be 'natural' in the sense that if two well-orderings of the same (classical) type are natural then they are recursively isomorphic. Clearly, it is sufficient to show that the co-ordinal associated with such a natural wellordering is uniquely determined. In fact, it will turn out that natural wellorderings are recursively enumerable. Here we shall only deal with addition and multiplication and shall treat exponentiation in chapter 11. The results can be extended further by considering additional functions (see e.g. CROSSLEY and PARIKH, 1963) but at present no uniform procedure is available for obtaining natural well-orderings of larger ordinals and indeed it is not known whether natural well-orderings exist for all the recursive ordinals (CHURCH and KLEENE, 1936). 10.1.1 DEFINITION. l01 If at is a collection of co-ordinals, then at is said to be a-unique if A , B E d & & A l=IBI < a * A = B . is said to be strictly a-unique if a ' is a-unique but not /&unique for any p>a. a t '

Ch. 101

UNIQUENESS RESULTS

123

10.1.2 COROLLARY. % is strictly o-unique. PROOF. Immediate from theorems 4.1.4, 5.2.4.

We have the following strengthening of this result. 10.1.3 THEOREM. If a is any infinite denumerable ordinal then there exist c mutually incomparable co-ordinals of ordinal a. PROOF. If a is infinite, then a=w+b for some b. Let B be a fixed co-ordinal of ordinal b. (Such a co-ordinal exists since there exists a wellordering of type b and hence a well-ordering embeddable in R; though, of course, the embedding may not be partial recursive.) Let V,, V2 be two incomparable co-ordinals of ordinal o then by corollary 5.3.5, if V, B is comparable with V2+ B then V, +B= V, B and by theorem 4.2.8, Vl and V, are comparable which contradicts the choice of V, and V2. Since, by theorem 5.2.4, there are c mutually incomparable co-ordinals of type o,the theorem now follows at once.

+

+

We now proceed to prove that the collections of principal numbers for addition and multiplication are strictly omand owo-unique,respectively. In order to do this we prove certain closure conditions for such principal numbers and then we establish necessary conditions for “small” co-ordinals to be principal numbers. 10.1.4 THEOREM. If Pl, P,E&‘(+), then

PROOF. P,

+P,€X(+) *PI

=. P ,

+ (PI + Pz)= P, + Pz + P, = Pz by corollary 3.2.9

* P , < Pz. 10.1.5 THEOREM. If PI,P,E&‘(+), then P,.P,E%(+). PROOF. By lemma 9.2.5, A
(3Q)( 3 R ) ( A = P1Q + R & 0 I Q < P2 & 0 I R < P i ) .

124

[Ch.10

CONSTRUCTIVE ORDER TYPES

Hence A

+ PIP2 = (PIQ + R ) + PIP2 = P,Q + ( R + Pi + Pip2) = PiQ = P1Q = Pi

since Pz is infinite and 1 by definition of if( +), since R < Pi ,

+ (Pi + PiPz)

+ PIP2

(Q + pz)

+ Pz = Pz

since Q < Pz .

= P,Pz

10.1.6 THEOREM. If PE&'(+) and A is any co-ordinal > 0, then P A € S ( +). PROOF. By theorem 9.2.7,0< C < PAimplies C = PA'*Dl

+ * - a +

PAr*Dr,

where A > A , > - . . > A , andO
+

+

PA'*Dr P A = PAr(Dr P') where A = A, + E , = PA'(& i- P P ) where P + F = P E , =P y P +F) since P ~ i f ( + ) ,

+

= pAr.pE

=PA.

Hence C + P A = PA'.Dl

+*..+PAr-'*Dr-l+ P A

and it follows by induction on r that

c + PA= PA and hence that PA€%'( +).

10.1.7 COROLLARY. For any co-ordinal A > 0, W Ais a principal number for addition. 10.1.8 LEMMA. If A€&'( +) then A = W" for some n, or for all n, W"< A . PROOF.If A E ~+)( then A > 1 and hence by lemma 6.3.1, l

a

w

=

W5A.

Ch. 101

UNIQUENESS RESULTS

125

If A =I= W, then W
then by lemma 6.3.1, W".W=W"+'
The lemma now follows by induction. In the same way one proves the following corollary; the details are left to the reader. 10.1.9 COROLLARY. If A , B E9(+) and B e A , then either Be W"= A or for all n, B . W"
10.1.10 THEOREM. The collection X ( +) of principal numbers for addition is strictly w*-unique. PROOF. By corollary 10.1.7, all the co-ordinals of the form W" for n > O are principal numbers for addition. By lemma 10.1.8, these are the only principal numbers for addition with classical ordinals
WW=W"*W=V, which is impossible; so &( +) is strictly ww-unique. Corollary 10.1.7 above may be regarded as the recursive analogue of the classical result that w ais a (classical) principal number for addition for any CI > 0. However, not every principal number for addition is of the form W Aas we shall show later in chapter 12.

10.2 In this section we establish the wo0-uniqueness of the collection of principal numbers for multiplication.

10.2.1 LEMMA. Every principal number for multiplication is a principal number for addition, but not conversely. PROOF. Suppose P E ~ ( . )then , 2 c P and hence 2P=P. Now by lemma 7.4.1 it follows that 2w divides P, but by theorem 7.4.2, 2W=W, whence W divides P and W I P. By lemma 6.3.1 we therefore have

1+P=P.

126

[Ch. 10

CONSTRUCTIVE ORDER TYPES

Suppose, then, that 0 c A < P, therefore P=AP=A(l+P) =A+AP =A+P

and we conclude PE%( +). The converse is false since W z is a principal number for addition by corollary 10.1.7 but m2 is not a (classical) principal number for multiplication and so W’$%(.).

10.2.2 COROLLARY. If P E X (

a),

then W divides P .

10.2.3 THEOREM. If Ply P 2 ~ X.),( then Pl

+ P2€%(*)--P1 + Pz = P Z O P , < P z .

PROOF. Immediate from lemma 10.2.1 and theorem 10.1.4. 10.2.4

THEOREM. If PlyPzE% ( .), then P , P , € % ( * ) O P , P , = Pz-P,


PROOF. The implications from right to left are obvious. Now suppose P , . P , E X ( . ) , then P,
By theorem 8.3.5 it now follows that

LEMMA. If B + A = A , then ( A + B ) . W = A . W. PROOF. ( A B) . W = A ( B A ) . W by theorem 6.2.3.(iii) , =A+A.W by hypothesis, =A.W by theorem 5.2.6.(i)

10.2.5

+

10.2.6

+ +

LEMMA. If P E ~+),( D < P and B < A , then P B + D+ P A = P A .

Ch. 101

UNIQUENESS RESULTS

127

PROOF. By corollary 5.2.3, B< A+B+ 1 I A , hence

PB.D

+ P A = P B ( D+ PA-B) = P B ( D+ P + E ) =PB(P+ E)

where A - B 2 1, for some E , since P E X ( + ) ,

=pBpA-B

=PA.

THEOREM. If P E % ( - ) , then P A e % ( * ) if, and only if, A = l or A€#(+). PROOF. If PA^#(*) and A + 1, then by theorem 8.5.5.@), 10.2.7

B < A=#

c: p A

and hence PA = pBpA = p B + A

Then by theorem 8.5.3. we have A=B+A,

whence A€&'(+). Conversely, suppose A€%( +)-and O < C < P A . By theorem 9.2.7, C

= PA1.D1

+ - . a +

P A r * D ,= P A ' . D , + B , say,

where A > A , > . . - > A , andO
B

+ P A ' * D l= P A " D 1 ,

Now, by corollary 10.2.2, W divides P , say P = WE,

therefore

128

[Ch. 10

CONSTRUCTIVE ORDER TYPES

- PAI

+A

since P E & ' ( - ) , since A E &'( +) .

=P A

This completes the proof. 10.2.8 COROLLARY. For any co-ordinal A , W W Ais a principal number for multiplication. PROOF. Immediate from theorem 10.2.7 and corollary 10.1.7, for A>O. That W is a principal number for multiplication follows at once from theorem 6.2.2.(vi). 10.2.9 LEMMA. If AE 2 then A = Ww"for some n or, for all n, WW" divides A and Ww"< A. PROOF, If A E 3(.), then 2 < A and hence 2A =A, whence by lemma (a),

7.4.1,

2"

=

W divides A .

Now suppose

Wwn< A , then (W"") A = A , (Ww")W= W""" divides A and

WW"+'5 A .

The lemma now follows by induction.

In exactly similar fashion one proves the following corollary; we leave the details to the reader. 10.2.10 COROLLARY. If A, BE^(-) and B < A , then BW"=Afor some n or, for all n, BWndivides A and BW"
w w w = W"V*

w" = wv* w = I/,

which is impossible. Hence &'( -) is strictly om"-unique.

Ch. lo]

UNIQUENESS RESULTS

129

Corollary 10.2.8 above may be regarded as the recursive analogue of the result that warnis a (classical) principal number for multiplication. However, not every principal number for multiplication is of the form W w A .This follows at once from the fact, remarked above and proved in chapter 12, that there is a P E X ( + ) such that P=k W A and theorem 1O.2.7.lo2

10.3 Since if PE%( +), P ( P ) is closed under addition, and similarly for PE&‘(.), P ( P ) is closed under multiplication and also (by theorem 10.2.1) addition we have the following theorem. 10.3.1 THEOREM. If “=@(+) then PEX( )

-

(where @ and 0 are (classical) addition and multiplication of ordinals). 10.3.2 COROLLARY, If P E X ( + ) [PE%(.)] then [PI=w” [lPl=warn] for some ordinal u.

These results, of course, also apply to any P satisfying the weaker condition (3) of § 8.1 (and its multiplicative analogue in the case of multiplication). The useful fact about taking P E X (+) [or X ( *)Iis that the path can be extended to a longer closed path. 10.3.3 THEOREM. For every denumerable ordinal ct there is a principal number for addition (for multiplication) P such that IP I 2 u. PROOF. Immediate from corollaries 10.1.7 and 10.2.8.

Later (chapter 12) we shall strengthen these results and prove that there exist paths closed under addition and multiplication (and exponentiation) which are of length K, i.e. “as long as possible”. As an immediate application of theorem 10.3.1 we have the following definition and theorem. 10.3.4 DEFIMTION. A co-ordinal A is said to be indecomposable (with

130

CONSTRUCTIVE ORDER TYPES

tch. 10

respect to multiplication) if A=BC

implies B = A or C = A .

THEOREM (UNIQUE FACTORIZATION THEOREM). If 1
10.3.6 COROLLARY. The infinite indecomposable co-ordinals c W W A are precisely those of the forms

WwB and

W c + 1 where B < A , O < C < W A .

10.3.7 COROLLARY. The only principal numbers for multiplication which are c W w *are those of the form W W Bfor some B < A .

C H A P T E R 11

E- NUMB ER S 111

11.1 The analogues of the classical arithmetic laws for addition and multiplication do not, as we have seen, carry over to co-ordinals in general. In the case of exponentiation, however, the situation is much more agreeable and in this chapter we determineexplicitly all the principal numbers for exponentiation and show that analogues of differing classical methods of defining &-numbersyield the same collection of principal numbers for exponentiation. As a corollary we get the result that the collection of principal numbers for exponentiation is strictly E,-unique.112 11.1.1 THEOREM. X(exp)cX(.). PROOF. Suppose PEX(exp), then by lemma 8.5.8,

if A < P ,

then A A < P

and hence, for 1< A < P, AAP=

= p =Ape

Therefore, for such A , AP = P by theorem 8.5.3 and hence P E X (-). Now W W r z X ( . )by corollary 10.2.8 and W < W w< W(WW)hence WW$X(exp)so X(exp)=I=X(-).This completes the proof.

We now describe a construction told us by Parikh which assigns, in a natural way, a well-ordering of type E, to a given well-ordering of type a.

132

CONSTRUCTIVE ORDER TYPES

11.1.2

[Ch. 1 1

DEFINITION. If A is a well-ordering, then we set

+

s(A) = {(x + 2 , y 2 ) : ( x , ~ ) E A )and t ( A ) = {(O,O>, ( 0 , I>, ( 1 , 1 > > T s(A). Next we define E(A) to be the smallest set E of ordered pairs of natural numbers satisfying 1)-3) below 1) If (x, Y)E t(A), then

2) Tf113x,2E.-.2Ex,, where r ~ 1 , m 1 , . . . , m r andeither ~1 r > l or m, > 1 or x1 is not of the form e

(" ') and analogous conditions hold for

Y ~ , . . . , Y ~ , ~ , ~ ~ , then ...,~,, (e

(m , ".... m,O e (i: ...n,'">) O E x1

3) If x

xrl)y

=e

(:

E

"'

"'":'> ... n 0

if, and only if ,

and (x, X ) E E and y = e

and ( y , Y ) E E ,then ( x , , y ) ~ Efor 1 I i l r , otherwise.

( x , Y ) E E if ( y , x) E E

Clearly E (A) and C'E (A) are uniformly primitive recursive in A. 11.1.3 THEOREM. If A is a well-ordering of type a, then E(A) is a wellordering of type E,. PROOF. Let cp be the map from E, = { fi :/3 <E,} into C'E ( A ) given by

Ch. 111

133

E-NUMBERS

where y < a, lxlA = y ,

/3 is not an &-number,o or 0 , cp (pi) = xi and /31 p,. > * * a >

We prove by transfinite induction that cp is well-defined and maps E, one-one onto C'E(A) in an order preserving way; from this the theorem follows at once. cp is well-defined since if B<E, then either (i) exactly one of p=O, p=o, f?=E,, for some unique y
+ +

case (ii) the pi, mi are uniquely determined if we specify > ...> /Ir. (Note that all the pi are <j? in this case.) That cp is one-one is clear from the uniqueness of the Cantor normal form. Since every ordinal < E , is expressible in the forms given above it follows that cp is onto and the order preserving property of cp is obvious from definition 11.1.2 and the observation that dl-ml

+ . . a +

m@r*rn,<&,,+pi < .sy for

1 Ii Ir .

This completes the proof. A straightforward calculation shows that A N B if, and only if, E (A) N E (B) but we prove a stronger result. Notation. We write A I B if (i) A S B, (ii) p is partial recursive, 6 p 2 C'A, p p E (0, l}, (iii) if x E C'B and p (x)is defined, then p ( x ) = O-XEC'A

.&. P(X) = I - X E C ' B - C'A

11.1.4 LEMMA.If A S B then for each B E B there exists AEA and p such that (i) 6 p 2 C'B and (ii) A I B. PROOF. Let BEB, then since A S B there exists AEA such that AI 8. Without loss of generality we may assume B = [B] and therefore, since A I B, that A = [A] for some AGB. By theorems 2.1.2, 3 there exists p such that 6 p = Seq and A I B.

134

[Ch. 11

CONSTRUCTIVE ORDER TYPES

LEMMA. A S B<=.E(A)sE(B). PROOF. By lemma 11.1.4, if A S B then there exists p such that

11.1.5

6 p z C ' B and

A S , 8.

Let q be the partial recursive function defined by

q (e

(m i ......xm,JO) ) '1

=q

( ')for all

(xl) i f r > 1 or m, > 1 or xi += e Y

y

and undefined otherwise. (q is an extension of pcpBcpA where cpA, cpe are the maps cp from .zIAl, onto C'E(A), C'E(B), respectively, defined as in the proof of theorem 11.1.3.) By transfinite induction (as in the proof of theorem 11.1.3.) one readily

verifies that 6q 2 C'E(B) and E(A) s qE(B); we leave the details to the reader. Conversely, suppose E(A)sE(B), then by the lemma for some q, E(A)r,E(B) and 6qzC'E(B). Now from the definition of E(A) we have

and similarly B may be obtained from E(B). Clearly A<,B

( (":'>I*

ifweput p ( x ) = q e

THEOREM. A- BeE(A)-E(B), PROOF. Similar to the above: left to the reader.

11.1.6

This theorem justifies the following definition.

Ch. 111

135

E-NUMBERS

11.1.7 DEFINITION. E(A)=COT(E(A)) for any AEA. A co-ordinal of the form E(A) is said to be an E-number. 11.1.8 COROLLARY. A sBoE(A)sE(B). PROOF. Immediate from lemma 11.1.5.

11.2 The following is the main theorem of this chapter. 11.2.1 THEOREM. The following statements are equivalent: (i) X is an infinite principal number for exponentiation, (ii) P=x, (iii) X = w or w'=x, (iv) X = W or X is an E-number. PROOF. (i)=.(ii) follows from the definition of principal numbers. For (ii)*(iii) we need a lemma. 11.2.2 LEMMA. 2'=X*X= WD for some D>O. We first sketch the idea of the proof. Since p :2' N X for some p, we can represent each 1x1 for XEC'Xin the form 1x1 = 21x11

+...+ p.1 + 2"' + ...+ 2"r,

where rnl>-.'>m, are the (only) finite exponents. Given such an expression then 1x1 1 is obtained by adding 2' and reducing the result so that we again get a unique expression. Clearly this addition can only affect the finite exponents and so the problem is to determine explicitly and effectively how "far in" the 2' has an effect on the exponents. The calculation of t = t ( x ) , below, does just this. The function J/ enumerates the elements of C'X corresponding to the finite ordinals. Using these two functions it easily follows that X N W - D where D is obtained from the numbers which do not represent successor numbers in X. PROOF. Let XEX, then by hypothesis there exists p such that

+

p:2%

x.

Let J/ be the partial recursive function (uniformly recursive in p) defined by J/ (0) = P (e (0)) 3

136

[Ch. 11

CONSTRUCTIVE ORDER TYPES

where m,> >m,. Since every integer >O has a unique dyadic expansion, $ is well-defmed. Ifx=C(lxo ...x,1) let

...

t(x) =pz(0

Iz s r

+ 1 &(Vi)(z Ii 5 r = > x i = $ ( r - i)]

and be undefined for all other x and, further, writing t for t (x) since no confusion will arise, let if t = O ,

if t = r + l , be undefined for x not of the above form. Finally, let

D = {<x, u>:(x, u> fdX & (VZ) .(

4 4z)

Y

* +)))

,

then the reader will readily verify that q:W*DN X, where 4 (An, 4)=P (sn(4). LEMMA. 2'= X* W x=X or X= W. PROOF. If 2'= X,then X= WD for some D by the previous lemma. Hence 2wD= WD =X.But 2wD=(2w)D= WDby theorem 7.4.2 and hence W D = WD.Now D >0, since X> 0, hence for some E, D = 1+E and

11.2.3

W D =

p + e = W.WE= WD.

By theorem 8.3.5 it follows that W E= D = 1 + E.

If E=O; then X= W; otherwise E>O and by corollary 10.1.7 we have 1+ E = W E = 1 + W E .

Ch. 111

E-NUMBERS

137

Therefore, E= W E =D and

Thus (ii)*(i.ii). Using a related sort of argument to that used to prove lemma 11.2.2 we prove (iii)*(iv). The essential point is that any number not representing an &-number(in the ordering X) is expressible in Cantor normal form in terms of numbers representing (strictly) smaller ordinals. 11.2.4 LEMMA. If W x = X , then X=E(A) for some A. PROOF.Suppose W x = X and X E X , then there exists a recursive isomorphism p such that

p : x -wx.

We may assume min(X)=O. Let

and let q be the partial recursive function defined by

and undefined otherwise.

(5)

Then it is easily proved that 6 q z C'E(A) (by transfinite induction) and the q maps C'E(A) into C'W'. q is one-one. We prove this by induction on the maximum number n of applications of the cases (1)-(4) above in the computation of the q(xi)necessary to compute q(x). If n =0, then only (1x3)arise. The only difficulty is if cases (2), (3) conflict. But e(!)

138

CONSTRUCTIVE ORDER TYPES

[a. 11

represents the ordinal 1 in Wx so the right hand side of (2) represents w. On the other hand e

(3

,if it is in C'E(A), represents a fixed point of the

isomorphism induced between the ordinals by p by the condition in (3) and therefore represents an €-number in Wx, so there is in fact no contlict of requirements. Now suppose q(x) = q(y) = e

(m, ... m, 1'

then the ai, mi and r are uniquely determined and each ai = q ( x i )for some xi by condition (5). Moreover, the xi are uniquely determined by the induction hypothesis. Thus we see that r, the xi and the miare all uniquely determined and since e is one-one we conclude x = y . Finally, we leave the reader to prove by transfinite induction that 4:E(A)

= Wx,

whence E (A) N X. Putting A =COT (A) yields the required result. Now we show (iv) =-(iii).

LEMMA.W E ( A=)E (A). PROOF. Suppose AEA, then let f be the partial recursive function defined by

11.2.5

and undefined otherwise.

Ch. 111

E-NUMBERS

139

Then it is readily verified as in the preceding proof that

f:E(A)

1:

WE(A),

from which the lemma follows. Finally we show that (iii)

(ii)&(i). This requires a number of lemmata.

LEMMA. BA=A* (AB)W=AW. PROOF. (A@" = (AB)2+W = (AB) (AB) (AB)W = A ( B A ) B(AB)" = A (AB) by the hypothesis, = A(AB)W. By lemma 7.4.1, it follows that A" divides (AB)" and, since lA"l= I(AB)wI (by e.g. BACHMANN, 1955, p. 57) it followsfrom theorem 8.3.3.(iii) that A" = (AB)". 11.2.6

11.2.7 LEMMA. B+A=A&A= WD+n*(A+B)W=AW. PROOF.We leave the trivial case D = O to the reader. We first observe that if B+ A = WD then (A

+ B)AW = ( A + B ) (WD)" = ( ( A + B ) W ) D(WD)" by theorem 8.2.2 with A W for C ,

= ( A W ) D(WD)W = ( W D W ) D(WD)"

= ( W D ) 2 + w= (WD)" = A".

By lemma 7.4.1 it follows that (A+B)" divides AW and hence (by e.g. BACHMANN, 1955, p. 57 and theorem 8.3.3.(iii)) that A"

=(A

+

Now suppose A = WD + n where n >0, then

+ ( A + B)"

(A

Hence

= (WD

+ n + B)"

= (WD

+ B')W

where B'=n+B.

140

[Ch. 11

CONSTRUCTIVE ORDER TYPES

If B+A =A, then

B+WD+n=WD+n and hence by theorem 5.1.6, B + WD= WD. Therefore B'+ W D = n + B + W D = n + W D = WD by lemma 7.4.1 (since we are assuming DPO). Putting A'= WD we immediately obtain from the first part of the proof (A' i.e.

+ B')w = A f w ,

(WD + n

i.e.

+ B)W = (WD)W,

( A + B)" = (W D ) W .

Finally n+ WD= WD and hence putting B=n, A = WD in the first part of the proof we get (WD + n)" = (WD)" . (7) Combining (6) and (7) gives the desired result.

11.2.8 LEMMA.Wx=X=+X is a principal number for addition and multiplication. PROOF. This follows at once from corollaries 10.1.7 and 10.2.8. LEMMA.W x=X& 1 c Y <X* Yx=X. PROOF. By theorem 9.2.8 the hypothesis implies

11.2.9

y = ~ ~ 1 +..-+ . n WX*-n, ~ where r 2 1, X> X,> ...>X,and the n, are finite, non-zero. I f r = l , X,=O, then Y = n , > l , hence yx = nT = n"x - "*"X = W"X = 1 -n1

x,

since W x is a principal number for multiplication and W is a principal number for exponentiation by theorem 7.4.2. In any other case we have yx = y w x = (Y")WX = ((W"1.n , ) W ) W X

=(<wx,)w )wx = WXlWWX = WXlWX = wwx =X.

by lemma 11.2.7, by lemma 11.2.6, by lemma 11.2.8,

Ch. 111

141

E-NUMBERS

Putting Y=2 we get (iii)*(ii) and the stated version of the last lemma gives (iii)*(i), so the proof of the theorem is complete. 11.2.10 COROLLARY. The collection Z(ex p ) of all principal numbers for exponentiation is strictly &,-unique. PROOF. By theorem 11.2.1 every principal number Wis of the form E (A)for some co-ordinald. By theorem 11.1.3if IE (A)I <&,then IE (A)I =en for some n and consequently IAl = n which implies A =n. So S ( e x p ) is &,-unique. Now consider E( W )and E( V); by corollary 11.1.8, E ( W ) = E ( V ) implies W= V which is a contradiction, thus X(exp) is strictly &,-unique.

+

11.2.11

THEOREM. If P E Z (exp), then

( 9(P), +, .,exp) is isomorphic to

(PI, 0 , 0,exp) by the map I I (where @, 0 , exp denote addition, multiplication and exponentiation of ordinals). PROOF. Since A, B c P imply A B < P the theorem follows at once from theorem 10.3.1.

CHAPTER 12

INFINITE SERIES

12.1 In this chapter we introduce constructive sequence types (C.S.T.s) in the spirit of A ~ ~ ~ t ( 1 9 6We 6 ) .first of all prove results for C.S.T.s analogous to those for C.0.T.s. and then we use them to prove that there are no non-trivial least upper bounds for collections of co-ordinals but also that there exist paths of co-ordinals of length equal to the first uncountable ordinal which are closed under addition, multiplication and exponentiation. If C is a linear ordering and to each iEc'C is assigned (not necessarily by any constructive rule) a linear ordering Ai, then we write

A

= (Ai: C)

(1)

for the sequence {Ai:liIc
-

12.1.2 DEFINITION. Let A, B be as in the previous definition; then a pair

Ch. 121

INFINITE SERIES

143

Iixp(i, x), I x q ( x ) is said to be a recursive isomorphism pair if (i) q : C N D , (ii) for all iEc'C, Ixp(i, x ) : A ~ N B ~ ( ~ ) . (iii) Ixp(i, x ) is one-one for all i. We write p , q : A N B if p , q is such a pair and A N B if such a pair exists and we say that A and B are recursively isomorphic. Note that since we require p to be a partial recursive function of both variables the recursive isomorphisms in 12.1.2(ii) are uniform in i. Recursive isomorphism (of S.0.s) is an equivalence relation - this is immediate from the remarks following definition 1.1.4 and the fact that if Ixp(i, x ) is one-one for all i ~ c ' Cthen the function p' defined by p'(i, x) = y o p ( i , y ) = x

is uniformly recursive in i. 12.1.3 DEFINITION. If A is a S.O. and d = { B : B = A } then d is said to be a constructive sequence type (C.S.T.) and we write d =CST (A). If A is a sequence well-ordering then d is said to be a sequence co-ordinal. In dealing with recursive isomorphism pairs we need not have assumed that Ixp(i, x ) was one-one for all i and not just for the iEc'C (where A , B , p , q are as in definition 12.1.2) for if p were not so then we could define p'(i, x ) = p ( i , x ) if x = v,,(p(i,x ) = p ( i , y ) ) , undefined otherwise, where v is Kleene's indefinite description operator (KLEENE, 1952, p. 347). Then, using the fact that p ( i , x) is one-one for iEc'C, p (i, x)

= p' (i,

x) for such i.

If p , q : A N B except that condition 12.1.2(iii) is not satisfied, then p', q:A N B. But this condition being satisfied for a recursive isomorphism pair p', q means that this pair determines a r.e. sequence ordering A = ( A :C) defined by Aj = (dIxp'(j, x ) ) n ~ R, C = (dq)' n R (note that q is one-one on the whole of its domain by definition 1.1.4).

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CONSTRUCTIVE ORDER TYPES

12.2 In chapter 2 we defined

A

+B = ~ A + - ~ B

and we now regard this as the sum of the two element sequence (A, B) in order to generalize the notion of sum to take in the infinite case. Let us define, for A a S.O.

ZA =

(Ai)

= { ( ; ( u ) , j ( b ) ) : ( i< = j & ~ E C ' & A~ bEC'AJ

or (i=j~C'C&u~~,b)}.

LEMMA. If A =(A,: C) and Ai(i€c'C) and PROOF. Left to the reader.

12.2.1

CG R, then C A c R.

12.2.2 THEOREM. There is a partial recursive functional I: such that if p , q : A z B then (2)

Z(p, q ) : C A N C B ,

where the N in (2) denotes recursive isomorphism of linear orderings. PROOF. Set C (PY 4 ) (4 = 2 (P( Y , Jj (x)) where lZ1 = 4 ((40) and Y = (XI0 * Y

Clearly I: is a partial recursive functional and A

C (P?4 ) (;

(4) = 4 (9 (P( 4 4) -

Suppose C (p, q) (x) = t , then t =i (u) for some uniquely determined i, u. Since q is one-one, there is a unique z such that q (2)= i. Since Axp (i, x) is one-one for aZZ i there is a unique u such that p (z, u) =u. But 2 (u) =x and hence x is uniquely determined by t so X ( p , q) is one-one everywhere. We leave the reader to check that C ( p , 9) satisfies (iii), (iv) of dehition 1.1.4.

If d is a C.S.T., then Zd=COT(ZA) for any 12.2.3 DEFINITION. A E d such that Ai, C c R. Theorem 12.2.2 shows that this is a consistent definition but Z d is not necessarily defined for every C.S.T. d.We shall say that a C.S.T. d is

Ch. 121

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145

standard if there is an A=(A,:C) such that for all I'EC'C, A,= R and C c R. So a C.S.T. d is standard if, and only if, Z d is defined. 12.2.4 THEOREM. d is standard if, and only if, for each A a pair of partial recursive functions p ( i , x), q ( x ) such that (i) q:C=DcR and (ii) lZxp(i,x):A,=B,~R; i.e. p maps the A, uniformly into R.

E there ~ is

We now leave the reader to prove that having zeros in a S.O. makes no difference to its sum. 12.2.5 THEOREM. Let A =(A,:C) and let D c C and suppose that if k C C - C D then A,=$. Then if A'=(A,:D) and either side of (3) below is defined then so is the other and (3) holds: Z A N XA' . (3) The only point to be borne in mind is that since only C is changed to D and we are always dealing with linear orderings embeddable in R this single change does not affect the uniform embedding of the A, in R.

Let us write 0,=(0:A) then we have: 12.2.6 COROLLARY. ZO, =0 for all A c R. since, in general, the actual choice of We shall also write 0 = CST (0,) A will be irrelevant. There are two ways (at least !) of adding S.0.s; pointwise and (what we may call) ordinally. Pointwise addition seems not to be very useful so we only treat the other kind of addition. 12.2.7

DEFINITION. A + B= (E,: C 7. D) if A = (A,:C),

where

E,=A,

B = (B,:D),

if ~EC'C, E i = Bi if iEC'D,

provided C) (D. If C) (D we write also A) (B and say that A is separable from B.

The pure theory of addition of C.S.T.s (i.e. that part of the theory not explicitly involving C.0.T.s) exactly parallels that of addition of C.0.T.s

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[Ch. 12

CONSTRUCTIVE ORDER TYPES

and since we shall be concerned with a finer treatment of sums of C.S.T.s we only give the definitions we require.

12.2.8 DEFINITION. d +9= CST ( A +B ) for any A E d ,B e 9 such that C) (D (C, D as in definition 12.2.7). That d + 9is well-defined follows from the following theorem.

12.2.9 THEOREM. If A = A ’ , B E B’, A ) ( B and A’) (B’ then A + B E A‘

+ B’.

PROOF.By hypothesis there exist p, q:A

and r.e. sets a, a’, 8,

N

A ‘ ; r , s: Bcz B’

p’ such that

a ? C‘C, a‘ 2 C‘C’, p

2 C‘D, p’

2 C‘D’

and a n B = a ’ n P ’ = 0 . Define u ( i ) = q ( i ) if i ~ c c and q ( i ) E p , = q ‘ ( i ) if i ~ a ’ and q ‘ ( i ) E P ’ , undefined otherwise

and u(i, x)= p(i, x) if i E a and q ( i ) e p , = p’(i, x) if i ~ a ’ and q ’ ( i ) E / 3 ’ , undefined otherwise.

Then the reader will readily complete the proof.

12.3 We can now obtain proofs of many analogues of the theorems we obtained for addition of C.0.T.s. We shall omit proof of most of these analogues: the essential difference is already contained in the proof of theorem 12.2.9. There the definition of u mirrors exactly the definition of u. We shall not need refinements until we deal with theorems which essentially involve (at least one of) the Ai. In this section we state some of the theorems on addition of C.S.T.s which do not involve C.

Ch. 121

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147

12.3.1 THEOREM.(i) d + 6= 8 + d = d, (ii) &+a= O o d = a = 6.

+

12.3.2 THEOREM. is associative, i.e.

+ (2+ U) = d + (2+ U). 12.3.3 THEOREM. (cf. separation lemma). If &=a+% and A e d then there exist BE^ and CEUsuch that B) (C and

A=B+C 12.3.4 THEOREM. (cf. directed refinement theorem). If d +U=2+ 9 then there is an d such that either d = B + B

and & + V = 9 ,

or d + & = 2 and

+

U=&++.

+

Notation. d - 0= 8,d *(n 1)= d e n d. 12.3.5 THEOREM. (i) d - ( r n + n ) = d - m + d - n , (ii) d.rnn=(d.rn)-n.

We conclude this section with a theorem which is not obtained in the same way as the above but is useful. 12.3.6 THEOREM. If A = ( A , : BqC), then

EA=EB+EC where

B = ( A i :B)

and

C = ( A i :C),

provided B) (C.

12.4 We now consider finer versions of the separation lemma and the directed refinement theorem than those in 12.3.3,4. In TARSKI(1956), infinite sums are considered and although these do not hold in full generality we can make translations of them using Z. Thus corresponding

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CONSTRUCTIVE ORDER TYPES

[Ch. 12

to Tarski's postulate (111) (1956, p. 8) we have theorem 12.4.1, below. Notation. If d is a C.S.T. and A E d we write d i= COT(Ai) where A = ( A i :C).

Clearly, d iis well-defined if C is given. However, when we do not specify the C it will be clear either how C is to be chosen or that the assertion is purely existential and therefore independent of the particular choice of C from the various possibilities. This is the case in the next theorem and lemma. 12.4.1 THEOREM (generalized directed refinement theorem). If X d = B+ C and C+O, then there exist C.0.T.s D, E, C.S.T.s do, d' and an integer i such that either

XdO+D=B,

E+Xdl=C

and D + E = d i ,

or Cdo=B

and X d ' = C .

Before we prove this theorem we prove the following: 12.4.2

LEMMA (generalized separation lemma). If Xd=B+C

and A

E (A=(Ai:D)). ~

Then there exist BEB, CEC, S.0.s A', A' and linear orderings E, F such that either (i) ZAoTE=BB,F7.ZA1=C and ET F=A, where A=Ao+(A,)+A1 and (Ai)=(Ai: {(i,i)}) or (ii) ZAo=B,ZA'=C and AO+A' = A . ~ PROOF. By the separation lemma 2.3.1 given C A E X and C A N B' + C', where B'e B and C' E C, there exist BE B and C E C such that 9

B)(C and C A = B 2 C. Now C'CA = {i(a):iEC'D&aEC'Ai}. Let Do = { i : ( 3 j ) (3x)(3(x)~C'B & i I D j ) } .

Ch. 121

INFINITE SERIES

149

Then, clearly Do= D [Do is an initial segment of D. Let D'

= C'D - Do

and D' = D [D'.

Now D' z { i : ( 3 j ) (3x) ( j ( x ) ~ C ' c&jsi } = DZ,say. If D' = D2 then B)(C implies Do)(D'. If D'=!=D2 thenjED1-D2 implies Aj=O and in this case we redefine DO, D' by Do=j)D and D' = D [ j for such a j . That Do) (D' then follows from lemma 2.1.2. We now have two cases to consider. (i) If (3)(3) (3c) ( i ( u ) ~ c ' B & i ( b ) ~ C ' Cand ) (ii) otherwise. (ii) In this case we leave the reader to check that A' = A [Do, A' = A [D' suffice where A [G = (Ai: D [G) = {Ai:i~C'D[G}.

In case (i) it is readily verified (cf. the proof of lemma 2.3.1) that there is a unique i for which (3b)(3c)(i(b)eCLB& i ( c ) ~ C ' c ) . Let Do = Do - {i}, D, = D' - {i}, AO=A[D,,A' = A [ D , . Then we have ZA0TAiTI;A'=B$C. Moreover, ZCA0is an initial segment of B and I;A1 is a final segment of C. Let E = B[(C'B - C'EAO) and F = C [(C'C - CIA'). Then we leave the reader to check (again compare the proof of 2.3.1) that ET F=Ai. This completes the proof of the lemma and theorem 12.4.1 follows at once by taking C.S.T.S. This lemma and theorem can be generalized to the case of Ld=XB but this (painful) exercise is left to the reader. The classical case is treated in TARSKI (1956) and the effective separability required is guaranteed by theorem 12.2.4 using heavily the fact that R is a recursive ordering.

I50

[Ch. 12

CONSTRUCTIVE ORDER TYPES

COROLLARY. If B IE d , and each (or equivalently, some) A E d contains no zeros then there is an dosuch that for some dl 12.4.3

and dois minimal with respect to these two properties. PROOF.In case (i) of the theorem take

+

do= do C.S.T.( A i :{(i,

i))),

in case (ii) simply take do= do.

12.5 From now until the end of this chapter we restrict our attention to well-orderings. The results on bounds apply almost without exception (puce the mention of classical ordinals) to quords and details of those results which can be readily adapted to the present framework may be found in CROSSLEY (1965).

DEFINITION. A co-ordinal A is said to be an upper bound for a collection of co-ordinals d if BEd implies B< A. A is said to be the Zeust upper bound if it is an upper bound and if C is any other upper bound then A S C. A is said to be a minimal upper bound if it is an upper bound and if C is any other upper bound then C $ A. 12.5.1

By the anti-symmetry of least upper bounds are unique if they exist at all. Also, it is clear that all least upper bounds are minimal, though the converse is certainly not true as we shall show. In fact we shall prove that all least upper bounds are trivial and determine when minimal upper bounds exist. Further we shall show that if one minimal bound exists then so do 2 " O mutually incomparable minimal bounds. We can define lower bounds in the obvious way but we have no interesting results concerning them (cf. 8 3 of CROSSLEY, 1965). 12.5.2 LEMMA. If a collection of co-ordinals d has an upper bound then any two members of the collection are comparable. PROOF.Let A be an upper bound for d,then B, C E implies ~ BSA and CIA whence by theorem 4.2.8 B and C are comparable.

Ch. 121

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INFINITE SERIES

12.5.3 LEMMA.A collection of co-ordinals has a least upper bound (which is a co-ordinal) only if it is countable. PROOF. Suppose rd is a collection of co-ordinals with least upper bound A then IAl is a countable ordinal, call it a. By lemma 12.5.2 if By CEd and I BI = I C I =fi Ia then by corollary 5.3.5, B = C. Hence I I is a one-one function from d into the ordinals
-

12.5.4 LEMMA.Supposep :C D is a classicalisomorphism, A = (Ai :C), B=(Bj:D), Ai=B,(,, for all i E C C and for all i, min(A,)=O, then E A N EB implies C N D. PROOF. Clearly the map defined by A

4 (i (4)= P ( i ) (4

7

restricted to C'EA is an isomorphism between X A and EB. Hence, if

r : E A N EB, then by theorem 8.3.4, r is an extension of q. Moreover, the map s defined by s ( i ) = ( r (t (O))), 7

is an extension of the (classical) isomorphism between C and D - this again by theorem 8.3.4 and the fact that 0 = min (Ai) which means that s always yields a value for &C'C. Clearly s is partial recursive and one-one everywhere it is defined. Thus, as required, we have s: C N D. 12.5.5 LEMMA.A strictly increasing (under <) sequence of co-ordinals of type o has no least upper bound but has 2'O minimal upper bounds. PROOF. Let J Z Z = { A ~ ) be ~ < a~ strictly increasing sequence of coordinals, then by corollary 3.2.9 there exist unique B i ( i < o ) such that A0 =Boy

An+1

=An

+Bn+1*

Using the axiom of choice, choose B i d i so that B i c R. Then, since the sequence ( A i } is strictly increasing, no Bi is empty. Let B'W)=(Bi:W)

andlet B'")=(Di:U),

where D l i l u = B i

and U is any well-ordering of type w not recursively isomorphic to W.

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CONSTRU(JIIVE ORDER TYPES

[Ch. 12

By theorem 5.2.4 U may be chosen in 2Ko different ways. Finally let

Bcw)= COT E(B(*)),B'" = COT Z(B'"'). We claim that B(w)and all the B(') are minimal upper bounds for d. E then ~ A = A , for some n < o and hence by theorem 12.3.6,

If A

n

A, =

n

C Bi 5 i C i= 1 =l

-p

Bi+ COT(Z(Bi: i > n)),

7 .

(4)

Similarly, one shows B(') is an upper bound for d.By lemma 12.5.4, all the B(') are incomparable with B('") and with each other and it only remains to show that if C
THEOREM. Let d be a collection of co-ordinals and let

a= sup (A1 (here we permit a=O, the first uncountable ordinal). Then A

E

~

(i) d has a least upper bound if, and only if, d (is countable and) has a maximum. (ii) If d is countable, all the A€&' are comparable and d has no maximum then d has 2K0mutually incomparable minimal upper bounds (but no least upper bound). (iii) If d is uncountable then d has no upper bound. PROOF. (i) follows at once from the preceding lemmata. (ii) Assume the hypotheses. Since d is countable, tl is countable hence there is a strictly increasing o-sequence {Ci}i
LEMMA. If A is a minimal upper bound for d then Id41 = sup IBI B E d

.

Ch. 121

INFIh'ITE SERIES

153

PROOF.Assume the hypothesis and let sup /I31=a. By corollary 5.3.6 BE&

there is a unique C < A such that ICI = a. The reader will now easily verify (using theorem 4.2.8) that C is also an upper bound for d and hence C = A (since C,A are comparable).

12.6 Now we deal with the question of paths P (cf. chapter 5) which go "right through" the classical ordinals in the following sense. 12.6.1 DEFINITION. A Zong path (of co-ordinals) is a set of co-ordinals d linearly (& so well-)ordered by I such that for each (countable) ordinal a there exists (a unique) A E d with IA I = a. 12.6.2 THEOREM. There exist 2' long paths. PROOF. We prove by transfinite induction on the classical ordinals that for each ordinal a there is a co-ordinal A, such that (i) IA,] =a and (ii) A , is comparable with all A , for /.I
12.7 In this section we prove a few theorems about principal numbers including Aczel's unpublished result that there exist principal numbers for addition, and for multiplication not of the forms W A ,W W Arespectively. ,

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CONSTRUCTIVE ORDER TYPES

[Ch. 12

12.7.1 THEOREM. If B = { P i } z o is a strictly increasing sequence of principal numbers for addition then there exist 2'O principal numbers for addition which are minimal upper bounds for 8. PROOF. Let Pcu)be constructed exactly as in the proof of lemma 12.5.5 (with "P" for "A" everywhere) then for each co-ordinal U of classical ordinal w we obtain one minimal upper bound Pcu).We now show that

PCU)E a ?( +) . BY (4) P,, 5 PCu) for all n and also B < P * ( 3 n ) ( B < P,),

(5)

as is easily verified. Hence, using the fact that P , E ~ ( + ) ,

B < Pcu) *B

+ P")

=B

+ P,, + (PCu)- P,)

+ (P(")- P,) - pCU),

= P,

where n is the number given by (5). We conclude that P'')E&'( all U with IU (= w. This completes the proof.

+) for

There are 2N0co-ordinals U (of type o)such that UE W 12.7.2 LEMMA. and W + U, similarly for well-orderings. PROOF. The former assertion follows from the latter since C.0.T.s are always countable classes of linear orderings. C'W has 2N0infinite, and therefore cofinal, subsets. Let U be such a set with the ordering induced by W. 12.7.3 LEMMA. If B= W . A and BEB then there is a function Axs(x) such that in B Is(x)l= 1x1 1, which has a one-one partial recursive extension. PROOF. Clearly it suffices to consider B=W.A, but then s ( x ) = j ( k ( x ) + l , Z(x)) is such a function.

+

Let B = {Pi}; , be a strictly increasing sequence of 12.7.4 THEOREM. principal numbers for addition; then there exist 2'O principal numbers for addition which are minimal upper bounds for B and which are not of the form WA. PROOF. We repeat the construction of theorem 12.7.1. but with one

Ch. 121

155

INFINITE SERIES

change. We choose the B l s (see the proof of lemma 12.5.5) so that min (B,) =0 for all i and we set min (Bi [(C' Bi - { O } ] ) =ui.Clearly this can be done whilst still ensuring B,G R; for example, we can take B: as in the n proof of lemma 12.5.5 and then take Bi=(ui+ 1) B: before arranging the first two elements as above. Now we construct P(') as before where U = ( is a well-ordering satisfying the conclusion of lemma 12.7.2. Suppose P(')= W*A for some A ;then by lemma 12.7.3 there is a one-one partial recursive function s such that Is(x)l= 1x1 1 in Pcu). In particular, therefore we have

+

sj(i, 0) = j ( i , u,),

whence hi is a partial recursive function, defined on the whole of the natural numbers and is one-one. It follows that u : W-U which is a contradiction. This completes the proof. COROLLARY. If d ={ Ww"i},",o is a strictly increasing sequence of principal numbers for multiplication then there exist 2'O minimal upper bounds for d which are principal numbers for multiplication not of the form WWA. PROOF. By theorem 12.7.4, there are 2K0minimal upper bounds A for { W A i } g , which , are principal numbers for addition and are not of the form W B .By theorem 10.2.7, W Ais a principal number for multiplication. We leave the reader to check that each such W Ais a minimal upper bound for d . Finally, if W Awere of the form W w Bwe should have A = W Bby theorem 8.5.3 which contradicts the construction of A. This completes the proof. 12.7.5

CHAPTER 1 3

IS OL s 131

13.1 So far we have been studying the properties of equivalence classes

of ordered sets under one-one, partial recursive, order preserving maps and have found both similarities with and differences from the classical theory. One of the initial stimuli for this study was DEKKER and MYHILL’S (1960) treatment of a recursive analogue of cardinal number theory and we now include a brief review of their (and others’) results insofar as these parallel ours in the remaining chapters of this m0nograph.l3~ None of the definitions and results in this chapter are due to the present author. Throughout this chapter we shall give references to the order analogues in this monograph and when the proofs required here differ only slightly or not at all from those in the order analogue cases we shall omit them leaving the reader to look them up and make the necessary amendments. Since some of these will be references to the next two chapters the reader who is not familiar with isols and R.E.T.s may care to regard the omitted proofs as exercises. 13.1.1 DEFINITION (cf. definition 1.1.4). A set A is said to be recursively equivalent to (a set) B if there is a one-one, partial recursive map p such that

A ~ d pand p ( A ) = B .

We write p :A N B or A N B in this case.133

As in the order case (see the remarks following definition 1.1.4) recursive

160

CONSTRUCTIVE ORDER TYPES

[Ch.13

equivalence is an equivalence relation. We set A = RET(A) = {B:B N A )

and call A a recursive equivalence type (R.E.T.). So the R.E.T.s are the “cardinal” analogues of C.0.T.s. Definition 13.1.1 is not exactly imitated by definition 1.1.4, the precise analogue would require p:A‘=B’ (in the above sense) where AEA’, BGB’ and A’, B‘ are r.e. However, since Sp and pp are r.e. it follows at once that the two definitions of recursive equivalence coincide. Moreover, since +, and exponentiation are determined by partial recursive functionals all theorems concerning these operations are unaffected by the different definitions.

-

13.1.2

DEFINITION. A set A is said to be isolated if it is finite or immune.

13.1.3

THEOREM (cf. theorem 15.1.1). If A e B and finite finite then B is immune immune

Ais

1

13.1.4 THEOREM (cf. theorem 15.1.2). The following statements are equivalent. (i) A contains an infinite r.e. subset, (ii) A contains an infinite recursive subset, (iii) A is not isolated. PROOF. (i)*(ii). Every infinite r.e. set contains an infinite recursive subset as follows: Let S be an infinite r.e. set and let a be some element of S. Set g(0) = a , g ( n 1) = V , { X € S & X > g ( n ) ) , where v, is Kleene’s indefinite description operator. Then g is clearly partial recursive and everywhere defined since S is infinite. Therefore g is recursive and enumerates a subset T of S in order of magnitude so T is a recursive set. The other implications are trivial.

+

13.1.5 D E ~ O N We. set R=RET(JY), n=RET({O, 1,..., n - 1)) for nEJY.

Ch. 131

161

ISOLS

So R is the “cardinal” of W (the C.O.T. of the standard w-ordering). We can define (cardinal) addition of R.E.T.s again using separability conditions to ensure a proper definition.

(cf. definition 2.2.3). A +B= RET(Au B) for any 13.1.6 DEFINITION AEA, B E B such that A) (B. As in chapter 2 we can show that any two R.E.T.s have separable representatives. 13.1.7 THEOREM (cf. lemma 2.2.1). If Ao=Al, Bo=B1, A,) (B, A,) (Bl, then A, u B, N A, u B, .

and

13.1.8 THEOREM (cf. theorems 2.2.4,5). (i) +(for R.E.T.s) is associative and commutative, (ii) A+O=A.

We also have a separation lemma and refinement theorem but although the separation lemma is virtually identical with that for C.0.T.s viz. if AEA and A = B + C then ( ~ B E B()~ C E C()A = B u C & B ) (C), since we have no order properties the refinement theorem becomes : 13.1.9 THEOREM (cf. theorem 2.3.2). If A + B= C+ D, then there exist E l , ..., E4such that

+ E,, C = E , + E3,

A

= El

B =E, D =E,

+ E4, + E4.

PROOF. By the separation lemma if AEA, B E B and A) (B, there exist CEC, D E D such that A u B = C u D . Let E l = A n C , E , = A n D , E3= B nC and E4= B nD. Then we leave the reader to check Ei)(Ej if i+j,

using A) (B and C) (D, and to complete the proof.

13.2.

A.B=RET(j(A, B)) for any AEA, BEB. 13.2.1 DEFINITION.

162

13.2.2

CONSTRUCTIVE ORDER TYPES

[Ch.13

THEOREM (cf. theorem 6.1.3). If Ao=A1 and BO=B1,then j(A0, BCJ

=j(A19

4).

This theorem justifies definition 13.2.1.

13.2.3 THEOREM (cf. theorems 6.1.8 and 6.2.1). Multiplication (.) of R.E.T.s is associative and commutative and is distributive over addition. 13.2.4 DEFINITION. A s B e ( 3 C )( A + C = B ) . A G B ~ ( ~ A E(3BeB)(AsB). A) 13.2.5 THEOREM (cf. lemma 6.3.1). B + A = A if, and only if, B e R S A . The proof of this theorem is rather more complicated than that of lemma 6.3.1 and we only indicate the main lines of the proof. Full details may be found in DEKKER and MYHILL(1960, pp. 75-77). Suppose we are given f : B u A - A and B)(A. Then there exist disjoint r.e. sets B, 2 B and A, =A. Let A* be the r.e. set of natural numbers xeA, such that x>f-'(x) and for some r XEA,, f - l ( x ) e A l ,

..., f - r C 1 ( ~ ) ~ Af -Il,( x ) e B 1 .

In order to ensure separability we use the fact that the natural numbers are well-ordered so the s below is finitely determinable and we set p(x)=j(s,fWr(x)) where s is the largest number of terms which can occur in a (finite) sequence (with k, f-kqX)>...>

f-ka-2

(4> S-'(x).

The rest of this part of the proof consists in checking that for every yeB and every s there is an x e A such that p ( x ) = j ( s , y ) and also that A*) ((BuA)-A*. In fact, sincef"(y) is defined for all n ifyeB and since f i s one-one x, s always exist it is clear that j ( s , x)eR.B. Separability is also easy to establish since the set of x's which are mapped is recursive in A*. The converse follows easily from 1 R = R and theorem 13.2.3 (cf. theorem 5.2.6.(i)).

+

13.2.6

THEOREM (cf. theorem 8.4.1). If B I C , then A B I A C .

Ch. 131

163

ISOLS

13.2.7 THEOREM (cf. theorem 4.2.8). Iis reflexive, antisymmetric and transitive (on R.E.T.s). PROOF. Reflexivity and transitivity are clear. If B I A and A s B then for some C, D, B + C = A and A + D = B so A + ( D + C ) = A whence by theorem 13.2.5, ( D + C ) . R I A . Now D I D + C hence by theorem 13.2.6 D * R I ( D + C ) - R I Aand therefore by theorem 13.2.5, A=D+A=B.

13.2.8 THEOREM (cf. theorems 13.1.4 and 15.1.2). The following statements are equivalent for an R.E.T. A : (i) A E A implies A is isolated, (ii) R $ A, (iii) B c A = - B+ 1 9 By (iv) B c A & B + C = B + D = C = D , (v) B G A and BEB-B is isolated. PROOF. Clearly the negation of (ii) implies the negation of (i). (iii)o(i) follows easily from theorem 13.2.5. (iv)c>(iii) is trivial. (iii)e(ii) follows from theorem 13.2.5. (i)+(v) is trivial so there remains (v)=.(iv). We shall merely sketch the proof, full details being in DEKKERand MYHILL (1960) pp. 89-91. Suppose B+ C = B + D then there exist BEB,etc. and a one-one, partial recursive map p : B u C 1: B u D , where B ) (C, B ) (D and we may assume C) (D. Consequently there exist disjoint r.e. sets B,?B, C12 C and D,I>D. Restrict domain of p to B, u C1 and range of p to B , u D,. We leave the reader to fill in the details to show that the map, q, defined below is such that q:C-D. q ( x ) is defined if, and only if, there is a finite sequence. x, P(X),

. a * ,

PYX),

(1)

such that x ~ C , , p ( x )...,p'-'( , x)EB1 andp'(x)ED,. We then set q ( x ) = p'(x). p k ( x ) cannot always be in B if XECfor then, since p is one-one, we should have an infinite r.e. subset of B contradicting (v). Moreover, we can effectively decide which of B,, C, , D, x is in if xEB, u C, u D, so

164

CONSTRUCTIVE ORDER TYPES

[Ch. 13

the finite sequence (1) can be effectively generated and it follows that q is partial recursive. q obviously has an inverse given by considering sequences like (1) with p - k for pk,C, for D, and D, for C,. The above theorem illustrates the significance of the R.E.T.s of isolated sets. 13.2.9 DEFINITION. An R.E.T. A is said to be an is01 if it satisfies (any of) the conditions in theorem 13.2.8. 13.2.10 COROLLARY. An is01 contains a recursive set if, and only if, it is finite.

There are in general KO partial recursive, one-one maps from an infinite isolated set onto itself - just exchange two points in the set - but we shall show in the next chapters (theorems 14.1.4 and 15.1.3) that if the set is ordered and the map required to be order preserving then there is only one of these maps. However, we cannot map isolated sets into proper subsets of themselves so isols are effective analogues of Dedekind finite numbers. 13.2.11 LEMMA. A is isolated if, and only if, BcA*B*A. PROOF. If A is infinite and not isolated, A contains an infinite recursive set B={b}uB,, b#B,. But (since l + R = R ) there is a one-one, partial recursive mapp:B-B, so setf(x)=p(x) if x ~ B , f ( x ) = xotherwise then clearly f :A N A - { b } . Conversely, if there exists B c A withp:ANB then if a ~ A - B , p ( a ) + a and p"(a)+p"(a) for n+m. Butp"(a)EA for all n so {p"(n):n=O, 1, ...} is an infinite r.e. subset of A. If A is finite the lemma is trivially true. 13.2.12 THEOREM (cf. theorem 15.2.2). (i) A IB implies A -c B, (ii) Every infinite is01 has KO <-predecessors and 2'O 5-predecessors. (iii) A -c B does not imply A IB. PROOF. (i) is obvious. (ii) B < A only if for AEA there exists BEB and a C such that B) (C and B u C =A. Since each r.e. set only separates off at most one B, A has at most K O <-predecessors. n IA for all n so there are exactly KO I-pre-

Ch. 131

ISOLS

165

decessors. A E A implies A has 2’O distinct subsets and these are divided into R.E.T.s containing at most K O elements each. (iii) follows trivially from (ii).

13.3. 13.3.1 THEOREM (cf. theorem 15.3.1). If A, B are isols then so too are A+B, A-B. The proof is virtually given as the proof of theorem 15.3.1 below. We can define exponentiation as follows: Since A , B are sets of integers we may regard them as embedded in Seq by considering A’ = {2“+’:a e A ) and then we can map min[A’] to 0 to obtain Al without changing the C.O.T. or R.E.T. of A. Next we define

AB = C [ A l ]exp [BJ, AB = RET(AB). However, we shall not prove any theorems about exponentiation of R.E.T.s but shall leave that to the ardent reader. As in theorem 15.3.1 below we can show that isols are closed under exponentiation. We now turn to cancellation laws for isols. We have already pointed out that for isols A +B= A + C implies B = C. In general we have results likef(X, Y ) = f ( X , Z ) implies Y = Z for isols X , Y, Z (possibly with side conditions). However, the result of replacing “=” by “I”is generally to make the implications false though we shall see that for losols the implications still go through. We shall not give proofs for these are, in general, of three kinds: (1) (for theorem 13.3.2 below) almost the same proof as DEKKER and MYHILL(1960) yields the result for isols and losols, (2) (for theorem 13.3.3, below) quite complicated counter-examples are required for isols but proofs as in (1) will work for losols, (3) (for theorem 13.3.4 below) special (and differing) techniques are required for the two theories. In the theorems below all R.E.T.s are assumed to be isols unless otherwise stated. 13.3.2 THEOREM (i) (cf. theorem 16.1.2). If A+O and AB=AC, then B=C.

166

CONSTRUCTIVE ORDER TYPES

[Ch. 13

(ii) (cf. theorem 16.3.2). If A 2 2 and AB=AC,then B = C . (iii) (cf. theorem 16.3.2). If A+O & BA= CA,then B= C . 13.3.3 THEOREM. The following statements are false. (i) (contrast theorem 16.2.2). If A=i=Oand AB
A.n =B.n*A=

B.

(ii) (ELLENTUCK, 1963). There exists an is01 X and R.E.T.s A , B such that X + 0, X A =XB and A 4=B. Finally, NERODE (1961) has shown that strong metatheorems go through for isols. We state only a very weak version which parallels our theorem 16.4.2. 13.3.5 THEOREM (NERODE, 1961). If P ( X ) is a function of X alone constructed by finite composition from functions of the forms

x f A , X . A , , (1 + X)A', A:, (where Al 2 1 and A 2 2 2) then P(X) =P(Y)*X = Y whenever X , Yare isols and all the parameters Ai are isols.

C H A P T E R 14

QUASI-FINITENESS

14.1 As we noted in the previous chapter the theory of recursive equivalence types has tended to be most interesting when R.E.T.s of isolated sets have been considered; that is, sets with no infinite r.e. subset. The question naturally arises as to what are the order analogues of isols? Originally, Kreisel proposed quords to the author, but PARIKH (1962, 1966) showed that quords are not closed under exponentiation and it was not until the Leicester Logic Colloquium in 1965 that Nerode and the author came to the conclusion that what are the most natural analogues are IosoIs. We shall introduce these in the next chapter. However, there is a larger class of C.0.T.s which the author considered in his thesis (1963) and the appendix to (1965) and we now take a brief look at the standard C.0.T.s of this type. Classically, finite ordered sets may be defined as linearly ordered sets which contain no descending or ascending chains. So we now consider (cf. Q 3.1) linearly ordered sets with no recursive descending or ascending chains. As in chapter 3, for sets embeddable in R by a recursive isomorphism, this condition is equivalent to every non-empty subset having a minimum and a maximum element. We recall that A * = { ( y , x):(x,y)~A}, A*=COT(A*) for any AEA. A* is the converse of A and A* the converse of A. 14.1.1 DEFINITION. A linear ordering A is said to be quasi-jnite if A, A* are quasi-well-orderings. A C.O.T. A is said to be quasi-finite if A, A* are quords.

168

[Ch. 14

CONSTRUCTIVE ORDER TYPES

14.1.2 THEOREM. (i) If A is quasi-finite and A E A , then A is quasifinite. (i) If some AEA is quasi-finite, then every A E A is quasi-finite. PROOF. Immediate from theorem 3.2.2. Notation. We write B E A if, for some AEA there is a BEB with 6 s A. (Since B E A in the set-theoretic sense implies B = A because A, B are equivalence classes, this notation will not cause confusion.) We also write B c A if B c A and B 4 A .

14.1.3 THEOREM. The following statements are equivalentfor a C.O.T. A. (i) A is quasi-finite, (ii) B E A implies B is quasi-finite, (iii) W$ A & W*4 A , (iv) B E A implies B+ C = B+ D*C= D.&. C + B= D B*C= D , (v) B E A*B+ 1?= B & 1 B+ B. PROOF. (i)-(ii), (iv)=.(v). Left to the reader. (ii)*(i). W = ((i, j ) :i I ~ ] E W. If W ' E A and AEA, then there exists a recursive isomorphism

+

+

p :W N W' E A .

But then {p (i)}: is a recursiveascending chain in A so A* is not a quord. Similarly if W* -c A. (iii)-(v) follows from lemma 6.3.1. (iii)=.(i). Suppose A is not quasi-finite. Let A E A then A or A* is not a quasi-well-ordering. Suppose (ai}$ is a recursive descending chain in A. Then liai is a one-one partial recursive function (which is everywhere defined) such that i <j if, and only if, a j :isj } E W*. Similarly if ( a i } g ois a recursive ascending chain in A. (i)=.(iv) follows from definition 14.1.1, corollary 3.2.9 and theorem 2.4.4. 14.1.4 THEOREM.If A is quasi-finite thenp :A N B G A impliesp extends the identity on C'A. PROOF. We may assume without loss of generality that B E AE R. Suppose p(a)=ka for some aEC'A, thenp(a)< a or a< p(a). Sincep is order preserving and 6pzC'A it follows that (p"(a)) is a recursive descending or ascending chain in A .

Ch. 141

QUASI-FINITENESS

169

We do not know whether the converse is true - we suspect it is not - but we have the following weak converse. THEOREM. If A is not quasi-finite then there exists BEA and C c B such that B-C. PROOF. Immediate from theorem 14.1.3.(v) and lemma 6.3.1.

14.1.5

It is clear that immune subsets of Seq generate quasi-finite linear orderings and therefore there are 2" quasi-finite C.0.T.s and in fact we shall show in theorem 15.2.6 that there are O 2' quasi-finite C.0.T.s of each countably infinite order type. However, there are recursive quasifinite linear orderings. This result, proved below as theorem 14.2.2, is due to S. Tennenbaum. We recall that a C.O.T. is said to be recursive if it contains a recursive (or equivalently r.e.) linear ordering. 14.2

THEOREM. If A is quasi-finite and r.e. (or recursive) then the order type of A is either finite or of the form

14.2.1

w+(w+w*).z+w*,

where z is some countable order type. PROOF. Clearly we may assume without loss of generality that A E R. Since A is r.e. and non-empty theorem 3.3.1 shows that the type of A is of the form o + o and that of A* of the form w +z where o,z are countable order types. Hence A is of a type w + p +o*.Now by the proof of theorem 3.3.1, every element in C'A has an immediate successor (predecessor) if it has any successor (predecessor). Hence the order type of A is as stated in the theorem. THEOREM (R. S. TENNENBAUM). There exists a recursive linear ordering of type w + w* embeddable in R which is infinite but contains no recursive ascending or descending chain. PROOF.^^^ By corollary 1.2.6 it is sufficient to show that there exists a r.e. linear ordering with the required properties. We first sketch the main lines of the proof. Since R has type 1 q we can identify the elements of Seq which are non-zero with rational numbers and we can identify real numbers with Dedekind sections of R. For the proof we first construct a real number A such that neither the upper nor the lower Dedekind

14.2.2

+

170

CONSTRUCTIVE ORDER TYPES

[Ch. 14

section determined by A is r.e. We then show that this real number A is the limit of a recursive sequence of rationals (in fact it is the unique limit). Finally we show that the intersection of this sequence with the lower (upper) Dedekind section of A cannot contain a recursive ascending (descending) chain. We now proceed to the details. Let wo, wl, ... be a uniform enumeration of the r.e. sets of non-zero sequence numbers. (Since Seq is a recursive predicate such an enumeration is readily obtained from any of the usual enumerations of sets of natural numbers, cf. KLEENE (1952).) We shall also use the notation Ei, 17,,to express that a predicate is expressible in prenex form with i alternations of (number) quantifiers the first being existential, universal. A i= X i n ni. XEO,

isin El- U l .

LEMMA(see KLEENE, 1952). A predicate is in A, if, and only if, it is recursive in predicates in El (or ITl). 14.2.3

14.2.4 LEMMA. The predicate T(n): “w, contains at least three distinct elements” is XI. PROOF.

T(n) 0( 3 ~ [(xh )

€On

& ( ~ 1 Ewn 1 & (x), Em, & ( i

*j , 0

Ii, j I2 => (X)i

* (x),)]

.

For any x (be it a sequence number or not) the predicate E(y, x ) : “ y extends x &Seq (y)” defined by E(y, x)*Seq(y)

& Zh(x) 5 Zh(y) & ( V i ) ( i < Zh(x)=>(~)~= (y),)

is (primitive) recursive. Hence there exists a recursive function t (n, x ) such that t(n, x ) > n a n d w , ( , , , = { y : y € o , & E ( y , x ) } . Main construction. We define a sequence of integers ai such that { (ao,...,a,, 1)},“= has limit A. (We add the 1at the end simply in order to avoid violating the definition of sequence number.) Stage 1. a,=O. Stage n+ 1. Suppose a,, ..., a , , , - , have been determined and t(n, (a09 ... a m ” - * ))=y (wherey>n). a) If T (y) then enumerate a,,until three distinct elements u< u < w 9

Ch. 141

171

QUASI-PINITENESS

+

have been obtained. Let m 1= max (lh (u), lh (u), lh ( w ) ) , (m is always defined since a, has been determined.) If u=(a,, ..., a,,- I , ..., ak) then a,, ..., ak are thereby. determined and if k 1<m then set

+

ak+,

= - a * =

a, = 0 and

m,, = m .

b) If 1 T(y) then set mn=m,,-, thereby determining no new ai. 14.2.5 LEMMA.For each n, a,, is eventually determined and no a,, is ever changed once determined. PROOF. a, is determined. If a,, ..., a,,,-, have been determined then since every set is repeated inllnitely many times in the enumeration oi we can always find 3 extensions of any given sequence number. Thus a,, ..., a,,,- has a proper extension. The first part of the lemma now follows by induction and the second assertion is trivial.

Letus defineA,={x:xA} to be thelowerandupper Dedekind sections determined by A. 14.2.6

LEMMA.A,, A,, are not r.e.

PROOF. If A, is r.e. then A,=o,, for some n and T(n) holds since A,

is infinite. Thus o,, contains three distinct sequence numbers u< u < w. Extend the sequences by 0’s to u’, u’, w’ so that lh (u’)= Ih (v), lh (w))= m + 1 and abbreviate m,,-l by r ; then U’=(ao,...tar, b r + 1 , * * * ,bm), u‘ = (a,, ...,a,, a,+,, ..., a,), w’ = (a,, ..., a,, c,+,, ..., c,),

say.

Now A lies strictly within the open interval (of real numbers) (u, u) since u’, u’, w’ all differ and (a,, ..., a,,) for n>m extends u’. Hence w$A, which is a contradiction. Similarly A,, is not r.e. We note that since A lies in the intersection of all the intervals (u, w ) the set {(a,, ..., a,, 1)},“=, has precisely one limit point. Now the procedure for constructing A is clearly recursive in El hence by lemma 14.2.3 A is A 2 , thus the predicate A(a):“A extends a” defined by (3n) (n 2 Ih(a) & E(
172

[Ch. 14

CONSTRIJCIIVE ORDER TYPES

is Z2 and so we have

A (4*(W (VY) R (a, x , y )

9

for some recursive R. Define the set B by a e B n o ( 3 x )(Vy) (y I n *R(a, x , y ) & Zh(a) 5 n ) , m

B = (J Bn, A=O

then B is clearly r.e. Since A,, A,, are not r,e. there are infinitely many ai += 0 hence C=(c:c~B&Seq(c)} is an infinite r.e. set containing infinitely many sequence numbers of the form (ao,..., a,, 1) and hence has A as alimit point. Finally, let C = [C ] then C is the required ordering. For if C had a recursive ascending chain then C would have a limit point distinct from A. This is impossible for let L be such a limit point then L, A are Merent limits of the sequence {(Uo,...,

a,

Q}2=0.

Finally, the order type of C is o + (a* +o)z+o*for some countable typez by theorem 14.2.1. But if 2 2 1 then C would have a limit point distinct from A which we have shown is impossible. This completes the proof. 14.2.7 THEOREM. (Recursive) Quasi-finite C.0.T.s are closed under addition and multiplication. PROOF.(Recursive) Quords are closed under addition and multiplication so the theorem follows from definition 14.1.1.

Summing up we get

THEOREM. There exist No recursive quasi-finite C.0.T.s of type ( o + o * ) * n for every n>O. PROOF. By theorem 14.2.2there exists a recursive quasi-finite C.O.T. A, say, of type o+o*.By theorem 14.2.7 A - n is a recursive quasi-finite C.O.T. for every n and by corollary 2.4.12, A.n=B-n&n>O implies A=B so it suffices to prove there exist KO recursive quasi-finite C.0.T.s of type w + a*. Let A be a r.e. quasi-finite linear ordering of type o+o*and let B be 14.2.8

Ch. 141

173

QUASI-FINITENESS

A [B where B is any infinite r.e. proper subset of C'A. Since C'A is infinite such a B exists. B is clearly a r.e. linear ordering. But if B contained a recursive ascending or descending chain then so would A hence 6 is r.e. and quasi-finite. But B A by theorem 14.1.4. Clearly, we can then repeat the process to obtain K O such B, such that A 3 B i 3 Bj for i< j , but A+ Bi+ B, if iij . Taking C.0.T.s yields the required result.

+

14.2.9

THEOREM. If A is quasi-finite then A

=B

+A +

C=O*B

=C=O.

PROOF. By theorem 3.2.7, A = B + A + C implies C=O. By theorems 3.2.7 and 2.4.2, A = B + A implies B=O.

14.3 We defined quasi-finite C.0.T.s as those C.0.T.s A such that both A and A* are quords, so most of our results for I obtained in previous chapters for quords also apply to I* for quasi-finite C.0.T.s with the obvious exceptions caused by the failure of closure conditions, etc. is We shall omit detailed mention of the results. But we note that I* reflexive, anti-symmetric and transitive on quasi-finite C.0.T.s and is a tree ordering. It is not a linear ordering or a partial well-ordering. The former follows from theorem 14.2.8 for let A be a quasi-finite C.O.T. of type ( 0 + 0 * ) * 2 then if AEA there exists B c A of type o * + w . Clearly A and COT(B) are incomparable under I* since they have final segments with incomparable (classical) order types. I* is not a partial well-ordering for if A is a quasi-finite C.O.T. as given by theorem 14.2.8, then A is expressible in the form n B,, for each n where B,, is quasi-finite and

+

B,<*B,,

if n < m ,

(where A <* B abbreviates A I* B & A =I= B ) i.e. {B,} is a descending chain of quasi-finite C.0.T.s under I*. We do remark, however, that although c is not anti-symmetric in general we have THEOREM. E is a partial ordering of the quasi-finite C.0.T.s. PROOF. If A E B and B E A then there exist AEA , BE B and recursive isomorphisms p , q such that 14.3.1

p : A - B, c B,q:B-A, E A .

174

CONSTRUCTWE ORDER TYPES

[Ch. 14

Hence qp: A

N

A, c A .

By theorem 14.1.4,qp extends the identity on A and similarlypq extends the identity on B, thus A- B and A = B. The other properties are obvious. By the proof of theorem 14.2.8, -c is not a partial well-ordering of the quasi-finite C.O.T.’s.

CHAPTER 1 5

LOSOLS

15.1 We saw in the previous chapter that quasi-finite C.0.T.s do not have very reasonable properties. We now turn to losols which we consider are nice and reasonable. This chapter is devoted to examining their basic properties: some of the proofs become very simple but with more complicated proofs we obtain some more far reaching cancellation laws. However, we leave over the general problem of the theory analogous to the Myhill-Nerode theory of combinatorial functions to another occasion. Aczel has shown that the obvious analogue is not fruitful, basically because there is no obvious correspondence between numbers and order types but Nerode and the author have now found methods of circumventing these difficulties. Since losols are special cases of C.0.T.s all the definitions given so far apply to losols and we shall not comment on any restrictions since these are obvious. We do, however, stress again that all linear orderings we consider are to be embeddablein R by means of one-one, partial recursive, order preserving maps.

15.1.1 THEOREM. If A N Band C'A is

1

finite then C'B is immune

5 finite

/ immune.

PROOF. Left to the reader. THEOREM. The following statements are equivalent for A = C'A. (i) A contains an infinite r.e. subset, (ii) A contains an infinite recursive subset, (iii) A is not isolated.

15.1.2

176

CONSTRUCIlVE ORDER TYPES

[Ch.15

PROOF. See theorem 13.1.4. 15.1.3 THEOREM. If C‘A is isolated then A is quasi-finite. The converse is false. PROOF.If A is not quasi-finite then A contains a recursive (ascending or descending) chain; u fortiori C‘A contains an infinite r.e. subset and hence is not isolated. In theorem 14.2.2it was shown that there exist r.e. quasi-finite linear orderings so the converse is false. 15.1.4 DEFINITION. A linear ordering A is said to be isolated if C‘A is isolated. A C.O.T. A is said to be a losol if it contains an isolated linear ordering. We write 9 = { A : A is a losol}.

COROLLARY. A is a losol*A is quasi-finite. PROOF. Immediate from theorem 15.1.3.

15.1.5

THEOREM.If some AEA is isolated then every AEA is isolated. PROOF. Left to the reader.

15.1.6

The term “losol” was invented by Joseph Rosenstein and the author after the author had considered several other words. It has the following motivations: it suggests linearly ordered isol, which is not strictly accurate but losols are C.0.T.s of linear orderings whose fields are isolated (and embeddable in R); further, it is palindromic and losols have a lot of symmetric properties as we shall show, and finally there is a paronomasial rapport with “isol”. 15.1.7

THEOREM. A recursive C.O.T. is a losol if, and only if, it is

finite.

In the next section we shall give Hamilton and Nerode’s proof that for each countably infinite order type z, there exist 2 ” O losols of type z. 15.2 In this section we commence discussion on the order relations on losols. We observe that if A is a losol then A is quasi-finite so both A and A* are quords and most of our results are applicable to <* too,

Ch. 151

177

LOSOLS

where we now re-define I and I *by

+

A 5 B o ( 3 C ) ( A c =B), AI*Be-(3C)(C+A=B).

We recall that we write A

c B for ( ~ A E A( )~ B E B (A ) E 6);

we shall also write A c B for A G B &A =kB.

A is said to be an initial (middle, final) segment of 15.2.1 DEFINITION. B if A IB ( ( 3 C ) ( 3 D )(C + A + D = B), A I *B ) . A is said to be a weak predecessor of B if A E B. The next theorem is inspired by theorem 41 of DEKKER and MYHILL (1960). 15.2.2 THEOREM. (i) A I B or A I * B implies A EB, (ii) Every infinite losol has KO initial (middle, finite) segments and 2'O weak predecessors, (iii) A E B does not imply A S B nor A s * B. PROOF. The first part is obvious and the third follows from the second. Since A is infinite any AEA contains countably infinitely many elements in its field. Let A be iixed and for UEC'Alet A,=COT(a)A) then A , IA .

By part (i) of the present theorem and theorem 14.1.5 A , + A and A,+=A, if u*b. Thus there are exactly KO distinct initial segments of A . The result follows for the other two types of segment since (i) initial segments are middle segments (put C=O) and (ii) A -A,, is a final segment of A. Since if A is a losol and for AEA, C'A is infinite, C'A has 2" infinite subsets. As we have argued many times before these subsets give rise to 2"O C.0.T.s all of which must be losols.

15.2.3 THEOREM. (i)-(iii) below are equivalent and so are (i)*-(iii)*. (i) A I B, (i)* A I * B , (ii) (3C) (A+C=B), (ii)* ( 3 C ) (C+A=B), (iii) A* I* B*, (iii)* A* IB*. PROOF.Left to the reader.

178

CONSTRUCTWEORDER TYPES

[Ch.15

15.2.4 THEOREM. If A is a quord and A
.

15.2.7 LEMMA(HAMILTON-NERODE). Let T be a countably infinite order type, then there exists a collection d ={ A i :~ E Iof } distinct subsets of Seq such that (i) cardinal of I =2'O, (ii) A j =[Ai] is of order type 7, (iii) A, is isolated (equivalently, immune), (iv) if i,jtzI, i + j , then there is no one-one partial recursive function mapping Ai into Aj.

Remark. Although the ordering < does not apparently enter into the statement of the theorem (or lemma) in any significant way, nevertheless we have not found it possible to prove the result directly from DEKKER and MYHILL'S (1960) theorem that there exist O 2' infinite isols.

PROOFof the lemma. Let Seq' denote Seq - {0) and let A, be a subset of Seq' with order type 7. (Such a set exists by the classical version of theorem 1.2.3.) Further, since A, is countable let a,, a,, ... be a (possibly non-effective) enumeration of A, without repetitions. Let cpo, pi, q2,... be a fixed (again non-effective) enumeration of all one-one partial recursive functions whose runges contain an infinite number of points of Seq' and let cpo be the (total) identity function. (We could restrict ourselves to order preserving cp to prove the theorem but this is clearly not necessary.) We now define a full binary tree whose nodes lie in Seq'. The A, will be the branchesof this treeand (i)willbe satisfied automatically.

Ch. 151

179

LOSOLS

We shall make extensive use of Kleene's indefinite description operator v which has the property that if % is a partial recursive predicate then

(3x1 %(x3 Y)* % ( V X W ,Y ) , Y ) and vx%(x, y ) is a partial recursive function of y uniformly in 2. We define our tree in stages. The nodes will be the points xsi where i ~ 2 "We . shall regard 2" as the set of all sequences of length s of zeros and ones, (well-)ordered lexicographically. Let TSibe the set of nodes of the tree which have been defined before xsi and let BSi be the set of nodes of the tree which have been defined before xsi and lie on the same branch asx,,, i.e. those oftheformxrjwherer<sandi=k ,... k,and j = k o...kr (ki=O or 1). We proceed to define our construction in stages. Stage s(sr0). For each i ~ 2 define " successively xsi to be an element t of Seq' such that (a) t stands in the same order relation (with respect to i) to the elements of BSi as a, does to the elements ao,..., as-1, (b) t E U Sp, but ( V j ) ( j l s & t € ~ p j ~ p j ( t ) B T s i ) , Osjss

(c> (Vj) (jls*(Vz) (ZETsi n6pj*t* pj (2))) (d) t B { pr ( Z r ) :Y I S } , where Zk = V,,,{(Pk(W)iS defined & p k ( w ) # { ( p j ( Z j ) : j < k } U { X j l : j < k & 1E2'} 7

U{xkt:

1
where I c i means I precedes i in the lexicographic ordering of 2". - Now writing A for the cardinal of the set A, Tsj,Bsi are finite for every s, i and therefore, since R is dense, pp, 2 Seq' and p q j n Seq' is infinite for a l l j we can always find a t satisfying (a)-(d) above. By condition (c) and the fact that pois the identity, no number chosen as t at some point can have been used previously. Now let A, for i ~ 2 O(i.e. i an infinite sequence of zeros and ones) be

{asis:is = ko ... k, and i = ko ...k, ..., s = 0, 1, ...}. Since Tm=2'O and i + j implies A i + A j by the remark above (and condition (c)), condition (i) of the lemma is fulfilled. Each A i= [A i] has order type z by (a) so condition (ii) is satisfied. (iii) is also satisfied since Ai contains no point of { p i ( z i ) : i = O ,1, ...} and therefore A, omits at least one point from every infinite r.e. set c Seq'. Suppose p is a one-one partial recursive function whose domain

180

CONSTRUCTIVE ORDER TYPES

[Ch. 15

includes Ai and which maps Ai into Aj. Then p is (Pk for some k. If Ai Aj then, by the definition of the Ai as branches in the tree, condition (c) implies that for sufficiently large s, s < t, u implies xri$.x u j where x t i , x u jare the elements of A,, A, defined at stage t , u, respectively. Hence for some r, t > k we have (Pk(&)=Xtj$.x,p Since q k ( x r i )is defined, if xri is defined before x r j we obtain a contradiction with condition (c) and if xri is defined after x t j we obtain a contradiction with condition (b). This completes the proof.

+

15.3.

THEOREM. If A , BE^, then (i) A + B e 2 , (ii) A . B E ~and , (iii) AB€$P. PROOF. (i) follows at once from theorem 13.3.1. (ii) Let A E A , BEB, then C‘A.B={j(a, b):a€C‘A,bEC‘B)=C, say. Suppose C contains an infinite r.e. set D then one of the two r.e. sets 15.3.1

A, = {.:(36)(j(a,

b)ED)),

or B, = { b : ( 3 a )( j ( u , ED)) is infinite, since otherwise C is contained inj(X, Y) where X, Y are finite. It follows that C‘A or C‘B is not$solated. (iii) In similar fashion to (ii) we show losols are closed under exponentiation. Let AEA, BEB and let C = CAB.

By our general blanket assumption we may suppose BE R. Let us define “b occursl in n” to be short for “n is of the form

(::: : : : ::)” .“.

and similarly define “a occursz in n” with a on the bottom line. Suppose C contains an infinite recursive subset D then let

A, = { a : ( 3 n ) ( n ~ &aoccurs,inn)), D

Ch. 151

LOSOLS

181

B, = { b : ( 3 n ) ED & b occurs, in n)}, A, = [A,] and B, = [B,]. Clearly A,, B, are r.e. sets contained in C'A, C'B, respectively. We shall obtain a contradiction to the assumption that A,, B, are both finite. Suppose B, is finite then B, may be enumerated as b,>b,>b3*.*> bk. By the definition of exponentiation it follows that each bracket symbol in (A,, B,) must contain Ik columns. HenceifA, has rn elements (A,, B,) contains at most 1+ mb bracket symbols and hence C'A,'' is finite. This is the required contradiction since C A Bcontains the infinite (recursive) subset D. It is, we hope, now clear why quords are not closed under exponentiation (cf. theorem 7.2.1, (ii)). Although A, B may have no recursive descending chains, nevertheless it is possible to find (and Parikh found) an A and a B such that A' contains a recursive descending chain where the length (equals number of columns) of corresponding bracket symbols increases as the chain descends. 15.3.2

THEOREM.If either (i) A + B ~ 9 o r (ii) O + A * B Eor~ (iii) A B ~ L ? & A + O , 1 &B+O, then

A, B E 9 . PROOF. If AEA, BEB then under the hypotheses of the theorem there exist p , , p z , p 3 such that p , :A E A

+ B, p z : A E A * B, p 3 :A E A'

and similarly for B there exist q,, q2, q3. For example, let

where we assume O=kmin(B). We leave other definitions to the reader. It now follows by theorem 15.2.5. that A , BEY.

CHAPTER 16

ARITHMETIC LAWS FOR LOSOLS

16.1 Now we are entirely concerned with losols unless otherwise stated.

In particular, with our new definition A 5 B is equivalent to ( 3 C ) ( A+ C = B). This definition does not agree with that given earlier for C.0.T.s since there are 2'O initial segments of any ordering of type but only KO separable segments of such an ordering. This chapter is concerned almost entirely with cancellation laws. The author has been inspired by DEKKER and MYHILL'Smonograph (1960) particularly the proof of theorem 116(a). However, it turned out that the proofs instead of being more complicated are in fact simpler than those proofs. This is mainly because of the trivial classical lemma below. 16.1.1 LEMMA, If A, B, C are finite linearly ordered sets such that C'B,C'C have the same number of elements and A+0 then there are unique minimal order isomorphisms p , q, r with domains C'B,G'AB, C'AC, respectively such that

p:B-C, q:AB AC, r : BA N

- CA

and d ( a , b) = j ( a , p ( b ) ) ,

r j ( b , a) = j ( p ( b ) , 0).

PROOF. Any linearly ordered, finite set is well-ordered. The existence of p , 4, r now follows from theorem 8.3.4. Since q, r are order isomorphisms the proof is complete.

Ch. 161

ARITHMETIC LAWS FOR LOSOLS

183

We leave the reader to extend this lemma to the cases of addition and exponentiation. THEOREM. (i) If A+O and AB=AC, then B=C, (ii) If A P 0 and BA = CA, then B = C. PROOF. (i) Let AEA, BEB, CEC, A=C‘A, B=C‘B, C=C‘C, B’= C A B and C’=C‘AC. We may assume without loss of generality that A, B, C c R and ( 1 ) = 2 ~ A . By hypothesis there is a recursive isomorphism p such that p:AB-AC. 16.1.2

We consider a formal system (not in standard formalization) whose formulae are of the forms X E A ,XEB,X E C ,XEB’,XEC’,

where x is a non-negative integer. (We shall refer to these as A-formulae, B-formulae, etc ...). There is just one axiom namely ~ E and A there are three rules of inference: R1. To inferj(u, ~ ) E Bfrom ’ UEAand ~ E and B conversely. R2. To infer j ( u , c)EC’ from UEAand C E Cand conversely. R3. To infer CEC‘from ~ E B if’ p ( b ) is defined and equals c and similarly to infer ~ E Bfrom ’ CEC’if p-’ (c) is defined and equals b. If 5 is such a formula we write Cons (5) for the set of consequences of 3 i.e. the set of formulae obtained from 5 (and the axiom) by a finite number of applications of R1-R3. We leave the reader to verify that Rl-R3 always lead from true formulae to true formulae. In particular we note that R3 can always be legitimately applied to a formula deduced only from true formulae since Sp 2 B’, p p 2 C’. Sometimes Cons (5) is finite and these will give the important cases we shall use to define the recursive isomorphism from B to C. We observe first that if Cons(%) is finite then we can decide when the whole of Cons(5) has been generated (by applications of Rl-R3) for this is so when any further application of R3 is (a) legitimate and (b) yields only a formula ~ E Bor ’ CEC’which has already been generated. Since only a finite number of formulae have been generated at any stage this can be checked in finitely many steps. In particular, if 5 is true then Cons(%) is finite since otherwise we should generate an infinite number of formulae and therefore an infinite

184

CONSlXUClWE ORDER TYPES

[Ch.16

number of A-formulae or of B-formulae or etc. This is impossible since A, B, C, B', C' are isolated. If Cons(5) is finite, let r, s, t, y , z be the number of A-, B-, C-, B'-, C'-formulae in Cons(%). Then rs=y since Cons(%) is closed under R1, rt =z since Cons (5)is closed under R2, y =z since Cons (8)is closed under R3 and finally s = t since r+O as Cons@) contains the axiom ~ E ACorresponding . to the A-formulae we can enumerate the numbers such that a e A is in Cons@) as a,

4-.*< a, A* = { u I , ..., a,},

(1)

since 4 is recursive. Similarly we obtain

b,

b,

B* = { b , , ..., b,},

(2)

<.*-
c * = { c l , ..., c , } .

(3)

-<em-<

and c1

Now if Cons@) is finite, since p is order preserving everywhere it is defined, we have p j ( a , b j ) = j ( a , , ci) for

1 I i 5 r, 1 5 j 5 s

(4)

(s may be zero, though r 2 1, but this makes no differenceto the argument).

If Cons (5)is finite, then p is an order isomorphism between finite sets and hence, by the lemma, unique. Finally, we define 4 (x) = b j (2, 4 if Cons(xeB) is finite, undefined otherwise. Now, if Cons(xEB) is finite, Cons(xeB) consists of XEB,~ E ACEC, , j(2, X)E B' and j(2, C)E C' for some (unique) c, hence q is one-one. It is clearly order-preserving by (4). qmaps B onto C for i f y d then lp-'j(2,y) is defined and equals x , say. But then Cons(xeB) is finite and q ( x ) = y . This completes the proof of (i). (ii) follows at once by permuting AB, AC where necessary with the appropriate corresponding notational changes. By theorem 14.1.4 we know that if A, ByC are linearly ordered isolated sets then there is a minimal p such that p' :AB N AC implies p' extends p so, on C'B, the q we obtained is uniquely determined. However, in determining (a suitable) q we only use finite sub-functions of p . We shall use this fact to establish other cancellation laws. We observe that if A, B do

Ch. 161

ARITHMETIC LAWS FOR LOSOLS

185

not have isolated fields then there may be many recursive isomorphisms from LA onto LBeven if L is finite. For, classically, we know that q + l + q = q and 2 - q = 2 * ( q + l + q ) = q and we can map any recursively linearly ordered set of type q onto itself in KO distinct ways.

16.2.

THEOREM. (i) If A+O and A B s A C then BcC. (ii) If A+O and B A G C A then BGC. PROOF. Exactly as for theorem 16.1.2 except we omit the part showing that the q constructed is onto. 16.2.1

THEOREM. (i) If A +O and AB< AC, then B I C, (ii) If A += 0 and BA ICAYthen B I C. PROOF. Let A, B, C be as in the proof of theorem 16.1.2. Construct q as in that proof then it is sufficient to show that (a) q maps B onto an initial segment of C and (b)q(B) is separable from its complement C"=c'C - C'q( B). (a) Clearly it is sufficient to show that if C E Cand there exist C'ECand ~'EB such that c c C c ' and q ( b ' ) = c', 16.2.2

then there exists ~ E with B q(b)=c. Let c', b' be as above. Then j(2, c) < d ( 2 , c') = pj(2, b'), s o j ( 2 , c)=pj(u, b) for some UEA,some beB with b
cd

implies a; =al so j(2, c) = pj(2, b) and it now follows that q ( b ) = M(2, b) = c , as required. (b) By our hypothesis A B I ACit follows that if we set AC[(j(A, C)j(A, B))=D then AB) (D. But x ~ B o j ( 2x, ) ~ j ( AB) , and x ~ C " e j ( 2 x, ) ~ j ( AC'D). , (*)

186

Finally, xEB-q(x)EC

CONSTRUCTIVE ORDER TYPES

[Ch. 16

so we obtain

which with (*) gives q(B)) (C". This completes the proof of (i) and the proof of (ii) follows at once by permuting AB, AC etc. as in the proof of theorem 16.1.2,(ii). 16.2.3 THEOREM. (i) If A 9 0 and A B S * A C , then B S * C . (ii) If A 9 0 and BAs*CA then B I * C . PROOF. Exactly as for theorem 16.2.2. reading "<" for ">" in the ordering relations. 16.2.4 COROLLARY. (i) AB< AC=-B< C, (ii) BA < CA+ B < C, (iii) AB< * AC*B< * C, (iv) BA<*CA=>B<*C. PROOF.Immediate from the fact that A < B e A I B & A + B and similarly for <*.

These results are in marked contrast to those for isols. The principal reason for this is that here we have our classical lemma 16.1.1 whereas its cardinal analogue is false: there are lots of isomorphisms in general in the cardinal case.

16.3.

+

16.3.1 THEOREM. B I C implies neither B A I C + A nor BA 5 CA nor ( A + o = > B ~ cA). I PROOF. Let A =COT (A) where A is an immune set with order type o (such an A exists by theorem 15.2.6 or by taking the natural ordering by magnitude of integers on an immune set). Since A=!=Wand n + A S A e W S A it follows that m + A $ n + A if m
Ch. 161

ARITHMETIC LAWS FOR LOSOLS

187

A. We conclude that if B A < C A then A=2A whence by lemma 11.2.3 we obtain A = W which is false. The converse implications have, of course, been established earlier since all losols are quords. We also established laws like A B s A C for B I C and A T 1.

16.3.2 THEOREM. (i) AB= AC and A 2 2 imply B= C, (ii) A +O & (1+ B)A= (1+ C)" imply B= C, (iii) AB
+

from aOEA',..., a,EA', boEB, ..., b,EB and conversely. R2. To infer

from a, EA', ...,a,,E A', co EC, ...,c,,E C and conversely. R3. To infer CEC'from ~ E B if' p(b) is defined and equals c and similarly to infer bEB' from CEC' if p - l (c) is defined and equals b. We take ~ E A~, E as A axioms. Cons@) is now defined to be the set of consequences of 5 under our new rules (and axioms). If 5 is true then, as before, Cons@) contains only true formulae. Again, as previously, Cons(5) is finite if 5 is true and we can decide when Cons@) has been completely enumerated. If Cons(5) is finite, let r, s, t, y , z be defined as before. Then r " = y + l , r'=z+l,y=zandhence, sincer>l,s=t. (Thereasonwehave to take y + 1, z+ 1 rather than y, z is that the minimum element of EF

188

CONSTRUCTIVE ORDER TYPES

[Ch. 16

when E, F are finite exists and is 0 and we do not have OEB', OEC' in Cons (5)). As before we obtain (l),(2), (3) and since p is order preserving everywhere it is defined, we also have

and

biu and ci, > c c . . . > ~ c i u

bi,

for all possible bracket symbols obtainable from A*, B*, C*. If Cons (5) is finite then p is an order isomorphism between finite sets (here we do not need to bother about adding 0 as minimum element) and hence, by lemma 16.1.1, unique. Finally we define

if Cons(xeB) is finite, undefined otherwise. The rest of the proof now follows analogously to that of theorems 16.1.2 et seq. and we are through.

16.4 This section is devoted to the proof of a cancellation metatheorem for losols. The idea of the proof is just like the ones for previous cancellation theorems, namely, given P(X)G P(Y) we look at all possible elements (effectively)"generated" from a single element X E C'X and then, since C'X is isolated this set is finite so we can use the corresponding cancellation theorem for finite sets (or numbers) and thence obtain a recursive isomorphism from X into Y. Notation: u;(xl,

...)x")=xi.

16.4.1 LEMMA. If p is a (number-theoretic) non-constant one-place function obtained by finite compositions of the functions

u;, x + y, x - y ,(1 + x)'

-1

and (finite) parameters, then p(x) I p(y)

if, and only if, x 5 Y

PROOF. We proceed by induction on the complexity of the function p .

Ch. 161

189

ARITHMETIC LAWS FOR LOSOLS

Clearly Ul has the required property when all arguments except the ith are held constant. All the other functions listed are strictly monotone increasing in each argument or constant so they satisfy the condition. Now suppose h (x) = Uf(fi (x, i i ) , ...)f n ( x , i n ) ) 7

where the f i are functions obtained by composition from the given functions and the di are sequences ail,...,aikrof finite parameters. Then h (x)=L( x , di) so this case reduces to a simpler one. Finally suppose h ( X I = 9 (fl(x, i l ) , f z (x, 4))3 where at most one offl, f z is constant and the rest ofS,, fz satisfy the condition. Then h ( x ) is strictly monotonic increasing since it is increasing in both arguments and thefi are increasing too. Hence x I y-h(x)

Ih(y).

We note that these are the only cases we have to consider since we do not need to distinguish between constant functions and parameters as far as their values are concerned. 16.4.2 THEOREM (CANCELLATION METATHEOREM FOR LOSOLS). If P is a non-constant one-place function (from losols to losols) obtained by finite compositions of the functions

ui”,x + Y,x - Y ,(1 + X)‘

- 1, x*

and (losol) parameters, then (i) P(X)G P(Y)if, and only if, XG Y and (ii) P(X)=P(Y)if, and only if, X= Y. PROOF. Suppose the losol parameters involved in P are Al, ..., A,. Then, since P is non-constant we must have certain of the Ai >O. We shall assume that in this case if A,EA,>O then min(A,)=O. We shall also assume that all the linear orderings we consider are E R but that min (X), min(Y)*O. We associate with P a set of canonical embeddings of X in P(X). P is a function from linear orderings to linear orderings inducing P . We shall assume without loss of generality that P is obtained by composition analogously to P where the basic functions correspond as

190

CONSTRUCllVE ORDER TYPES

[Ch. 16

below.

u; ,

U/ X+Y x.Y (1 + X)' - 1 X*

X +Y rn (X, Y) e(X, Y) r (XI

(see theorem 2.2.2), (see theorem 6.1.6), (see below), (see below).

Let R denote R[(Seq-(0)) then R, R'* are recursive dense linear orderings without first or last elements and so by theorem 1.2.4, for some r, r : R'*E R'. We take r to be fmd. This is the r of the table above. Since we are assuming min(X)+O we can take as representative of l + X , X+={
9

+

+

Notes. 1. Since P is non-constant, in the cases of multiplication and

Ch. 161

ARITHMEnC LAWS FOR LOSOLS

191

exponentiation we take a to be a fixed element of a fixed AEA and observe that such an a exists whenever that function is essentially involved in the definition of P since we saw above that the appropriate A, were non-zero losols. 2. Since it is clear that parameters can be put in one at a time in any order it follows that C (P) is defined uniquely, independent of the order of the parameters A,. 3. I f p e C ( P ) then O$X implies O$p(X). We leave the reader to prove that p:X-p(X)_c P(X) and p , p - ' are one-one and order preserving wherever they are defined. It is also trivial that if X c Y thenp(X)Ep(Y). The implication (in the theorem) Xc Y implies P ( X ) c P ( Y) now follows easily by considering all the embeddings where the constants a in the definition of C(F) above are allowed to range over the whole of the (appropriate) Ai. We leave this trivial verification to the reader. Now suppose P ( X ) c P(Y) where P is as above and the A, are the fixed representatives of the losols A,. By hypothesis there is a one-one partial recursive functionfwhich is order preserving wherever it is defined and such that f :P(X) N Z c P(Y) c R , where P(X), X, Y c R . We make one further assumption on the Ai, namely, that we can choose a = 2 = ( 1 ) 9 0 in the construction of C(P). This is merely to avoid excessive complication in the notation and clearly causes no loss of generality. Now given any finite subsets X' EX, A: E A,, P (X', A')is defined and finite where A: = [A:], the linear ordering generated by A: c Seq. This is because all the given functions lead from finite sets to finite sets. Moreover, if ~ E C ' P ( Y , A )then y€C'P(Y', A') for some minimum finite Y', A; ( i = l , ..., n) which are effectively computable from the description of P and y. We leave the reader to verify this by induction. For example if y ~ C ' ( 1C A)', then (uniquely) Y l **. Y m = (al ... ,

..>

192

[Ch. 16

CONSTRUCTIVE ORDER TYPES

where y l > y z > ~ ~ - > yand m the aiEA; so let Y'=[{yl, ..., y,}], A'=[{a,, ..., a,}]. We observe that this process is uniform since min (1.7. A) is independent of A. We write

Y' = c ( y ) , Af = PX,(y)

and P;(y) = C'P;(y),

etc.

Now if a necessary and sufficient condition that P ( X , A ) be nonconstant is that A i l,...,Ai,>O and A j ,,..., A j p > l then we assume min (Aik)=O,k=l,..., m ; min(Ajk[Ajk-{0})=(1) ( = 2 ) . Clearly this can be done without loss of generality. Now we set up our formal system. We have X-, Y-, P(X)-, P(Y)- and A,-formulae ( i = l , ..., n). The axioms are all formulae of the form Q E A ~ ,2 € A j which are true. So there is a finite number of axioms. For brevity we write x,, ...,X,EX or even Z'EX instead of x1 EX, ..., X,EX. The rules are as follows: Rla. To infer XEP(X)from x , , ,.., x,EX, a,,, a,, ,..., anjnE A, if x E P (X', A')where

...,a l j 1 ~ A l...; ;

X' = {x1, ..., x,}, Af = {uiIy..., aij,} ( i = 1, ..., n).

Rlb. To infer x,, ..., x,EX; a,,,..., a l j , ~ A 1...; ; a,,, ..., a n j n ~ Afrom , xEP(X)if{x ,,..., x m } c P ~ ( x ){q,,..., , a i j , } s P ~ , ( x()i = l , . . . ,n). R2. As for R1 (a and b) with x, X replaced by y, Y. R3. To infer p P ( Y ) from XEP(X)ifp(x) is defined and equals y and similarly to infer XEP(X) from YEP(Y) if p e l ( y ) is defined and equals x. As usual Cons (5)is the set of consequences of 5 and the axioms (if any) under Rl-R3. As in previous proofs of this type the rules preserve truth and if 8 is true then because all the A, X and Y are isolated we can generate Cons (5)and decide (recursively) when the whole of Cons (5) has been generated. If Cons (5)is finite then let the numbers such that UEA,is in Cons(%) be enumerated as ail

aij,

<me.<

A; = {ail, ..., aij,} i = 1, ..., IZ.

Similarly we have enumerations x1

<..a<

x,

X' = {x,, ..., x,}

forX-formulae,

Ch. 161

AIUTHM)?TIC LAWS FOR LOSOLS

193

Y‘= { y 1, ..., y,} for Y-formulae , Y , <-..i ys P’ = { p 1, ..., p,,} for P(X)-formulae , p1 <...iP” q1 <...i q, Q‘ = {ql, ..., q”} for P(Y)-formulae. Now by R1, R2, respectively,

P’ = P(X’,A’), Q’ = P(Y’,A‘) and by R3 f:P‘ N Q‘ (or more precisely, a finite restriction off: P’ N a’).It follows at once that u=u. Now let c(r,j) be the cardinality of P(X, A) where X, Ai have cardinalities r, ji,respectively. Then c is a number-theoretic function which satisfies the hypotheses of lemma 16.4.1 (proof by induction on the complexity of P) since O€Ai, 2eAj are always in Cons@) whenever they are required to ensure P is non-constant. Hence, if s is the cardinality of Q’, c (r, j )= u = u = c (s, j ) implies x = y by the lemma and hence r =s. We now show that if p is any canonical embedding (though we need only consider a particular one) then f:p (X’) N p (Y’). We then define g (x) =p-lfp (x) from which it follows at once that g:X-Y. Since (as we have just showed)f:P(X’)= P(Y’) and P(X’), P(Y’) are finite it suffices to prove the above assertion for finite X , Y. But this is clear since given any canonical embedding p then we can explicitly compute from the description of p and the number of elements in X and the Ai which element (in the ordering < of P ( X ) ) p ( x ) is, where XEX, say, the nJh. Since we only used the number of elements of X in this calculation (and the numbers of elements in X and the A, are computable) it follows that if x,y are the mth elements of X, Y, respectively, then p (x),p (y) are the nJh elements of P (X), P (Y), respectively. Butf: P (X)P (Y) is an order-preserving map between finite sets so fp (x) = p (y). This completes the proof. We shall develop this technique further in our forthcoming work with Nerode.

APPENDIX A

A MIXED CANCELLATION THEOREM by P. H. G . ACZEL and J. N. CROSSLEY

A.1 DEFINITION. If F :B?-+ 9,F is one-one and A 5 B o F ( A )5 F(B),

then F is said to be an order embedding where I is used in the strong sense that A S B o ( 3 C ) (A+C=B).A1 Let & ( A ) = C + A , P,(A)=C.A. By corollary 3.2.9, if C is a quord then S, is an order embedding. In this appendix we show that if C is a quord such that C 2 1 then Pc is also an order embedding. The proof of theorem A.4 below was inspired by SIERPINSKI (1948).

A.2 LEMMA.Iff: C- X c C where C’ is a proper initial segment of C then C is not a quasi-well-ordering. PROOF. The lemma is merely a restatement of lemma 3.2.6.

+

A.3 LEMMA.If A B = A’ +B’ is a quord, then B < B’ implies A’ < A . PROOF. Assume the hypothesis. By the directed refinement theorem (2.3.2) either A I A’ and B‘ <* B , or A ‘ I A and B I * B ’ .

If B < B‘, then B $ * B by theorem 2.4.9. Hence A’ I A and BI * B’. But if A’ = A , then by corollary 3.2.9, B= B‘ so we conclude A’ < A .

AP.A1

A MIXED CANCELLATION THEOREM

195

A.4 THEOREM. If C 2 1 is a quord and C,, C2
+ C2

implies A = B .

PROOF. Let AEA, BEB, CECand let Ci ( i = l , 2) be initial segments of C. We may assume A, B, C c R and min(C)=O. As usual we set A=C'A, etc. Let A'=A+{(O, 0)}, B'=B+{(O, O)}. By hypothesis there is a recursive isomorphism

p':m(C*A)+ m(Cl*{(O,O)})- m ( C * B ) + m(C2.{(0,0)}) (cf. the proof of theorem 6.1.6). In order to simplify the notation however we shall consider instead a recursive isomorphism p:C*A

+ C,*{(O, 0)}

N

+

C*B C2*{(0,0))

and leave the reader to observe that from our construction of the function q :A N B it readily follows that we can construct q': A- B such that q' is order preserving wherever it is defined. This is because the natures of A, B do not enter into the arguments insofar as their recursive properties are concerned. Let q l ( a ) = M ( 0 , a ) , q 2 ( b )= b - ' j ( O , b ) . We shall eventually define q:A+B such that either q is a sub-function of q, or q-' is a subfunction of q2. The statement of the theorem is clearly symmetric so we consider the pairs A, B; Ci, C2; P, P-'; C, C ; 41, q z ; etc. as dual (not in the sense of $2.4). The dual of a statement is obtained by replacing any term by its dual. We shall make frequent use of the fact that each lemma or definition has a dual. By lemma A S below it will be clear that q2 is one-one on B and maps B into A preserving order. Dually, q1 is one-one on A and maps A into B preserving order. The major part of the proof is devoted to showing that q1 maps A onto B and hence (dually) that q2 maps B onto A. The remainder of the proof shows that either q, or q2 has a one-one partial recursive sub-function defined on A or B respectively. The following notions are useful. We say beB is generated if b = q1 (u) for some aEA'. Dually we define the generated elements of A.

196

P. H. G. ACZEL AND J. N. CROSSLEY

Define the functions fa,b by

It should be clear that these functions have the following properties : 1) fa,b is one-one, partial recursive, 2) aEA, ~ E imply B fa,, is order-preserving (in particular on C), 3) aEA', b=O imply fa,, is order-preserving (in particular on C,), 4) a = 0, ~ E Bimply ' fa;: is order-preserving (in particular on C,). We shall make use of these facts without referring to them. A.5 LEMMA.Let aEA', c ' c C c such that if a=O then CEC,. Then p j ( c , a ) = j ( c o , b,) and pj(c', a ) = j ( O , b,)

imply bo=b, and 0 < c,. PROOF. Assume the hypothesis. Then j ( 0 , b,) < ,,,j(c,, b, I Bsb,. Suppose bl <,, b,, then fa, b ,

: C N X E C [c

b,),

so

for some X ,

which by lemma A.2 contradicts the fact that c is quasi-well-ordered. Hence b, =b, and then trivially we must have 0 < c, (since 0 =min (C)). If X is a linear ordering set x-i x y if, and only if, x<,y and there is no z such that x<,z<,y i.e. if, and only if, y is the immediate successor of x in X. A.6 LEMMA.Let aEA', 0 < c such that if a = 0 then

C EC,.

Then

pi (c, a ) = j ( 0 , b,) and p j ( 0 , a ) = j ( c 1 9b,)

imply O
,,

But

AP. -41

197

A MIXED CANCELLATION THEOREM

Hence by lemma A S b =b, giving a contradiction. Therefore b1i B, bo. The following will be used throughout the rest of the proof. A.7 DEFINITION. Suppose b,€B such that if b,=O then OEC,. Define recursively a", a,, b"+', b,+ for all n 2 0 by p j (a", an) = j (0, bn) p j ( 0 , a,) = j ( b " + l ,b,+,). 9

(*I Also define

A, = COT (C [a"), Bn+l = COT(C[b"+'),

(**I for each n 2 0.

A.8 LEMMA. 1) If bo=O, OEC,, ao+O, then a 0 ~ , , Oand aO+O, 2) if b0eB', (bo=O implies OEC,) then for all n20

+ 0, b"+' 9 0,

a"" an + 1-1

A' an, bn + 1-1 8'

bn

and

B,+* + A , = c , A,+1+ B,+1 = c. PROOF. 1) Assume the hypothesis. Then u , < ~ O . Suppose u,,<~,u<,.O, i.e. ~ z , < ~ uThenfa:J:C-XsCC, . for some giving a contradiction. So a,+ ArO. Now suppose a' =O. Then&,: :C N X E C, for some X, again giving a contradiction. So a0+O. 2) Assume the hypothesis. By lemma A.6, if d+O, then b"+'=t=Oand bn+l-IBjbn. By the dual of lemma A.6 if b"+'iO then a"+'+O and But uo+O. Hence by induction we have a,++

x

a""

+ 0,

+ 0,

b""

an+ 1-1 A' am bn+ 1-1

8'

bn

9

for all n 2 0. By these results we must have .fa,+,,

b , + i **

and fan2

b,+l'

C [b""

- b"+')C

N

N

a"+') C

C [a".

But b"+l)C, C[b"+l are separated by the disjoint r.e. sets sfan+,,

b,,+ly sfan,

b,+l-

198

P. H. 0.ACZEL AND J. N. CROSSLEY

Hence An+' +Bn+1 = C *

Also a") C, C [a"are separated by the disjoint r.e. sets Pfa,,, b , + l ? Pfa,,, b, so that

Bn+l

+ An

=

c-

+

A.9 LEMMA.If b,€B is not generated then A , C = C. PROOF. Suppose b,eB is not generated, then ao+O. From (i), (ii) below it will follow that fao,bo: Ccia') c giving the conclusion of the lemma as we have already observed the r.e. separability of C [a', a') C. (i) If O<, c andf,,,,(c)=c' for a e A then a=a, and ~ ' < ~ c ' . Clearly a , ~ ~ If a U. , < ~ U thenpj(0, a)=j(c", b,) for some c"EC, i.e. bo is generated contradicting the hypothesis of the lemma. So a=ao and then trivially aoc< c'. (ii) If ao<,c' andf,,,,(c)=c' then b=b, and O
A.10 LEMMA.Every b, E B is generated. PROOF. Suppose b,eB is not generated. Then a0+O and by lemmata A.8 and A.9

for all n TO,where we have temporarily put B, = C. But COT (C[b') = B , < B , = C as C i s a quord. By lemma A.3, (R,) and (R,,,), B,+,
A.ll a,+

LEMMA.If b,EB and ao+O then there is an a-,eA' such that and a bo such that

(***I

p j ( 0 , a-

= j ( b o , bo).

PROOF. By lemma A.lO, bo is generated, i.e. there is an U - ~ E A ' , ~ ' E Csuch that (***) holds. Then by lemma A.6, a,+ Ata-land bo + O .

AP. A1

A MIXED CANCELLATION THEOREM

199

Call UEAa tail element of A if there is a finite sequence u = a0 +A

+A***+A

0,

u,+K

Dually we define the tail elements of B. A.12 LEMMA.Let boEB and ao+O be such that if bo is a tail element of B then p : C A - CB. Then there are a,,, b, for n
and a,,+,+Aa,, b,+,+ Bb,for all n ( n 2 O or n
But pj(0, O)=j(bo,b,)EC'CB andj(0, O)$C'CA. Thus a _ , 90,i.e. a-,EA. By the dual of lemma A.11, there are a-'=l=Oand b - l such that p j ( a - ' , a-,) = j ( O , b - l ) and

bo+ B t b - l

and as above we may show that b - , 1 0 so that b-,EB. Repeating the above procedure we construct a",a,, b"", b, for n
bn+ 1 + B

bn *

Define A,, B,+, for n
+Bn+1 = C , Bn+1 +An =C,

4 + 1

for n n giving a contra-

200

P. H. G. ACZEL A N D J. N. CROSSLEY

[AP.A

diction. If d" <,a" then A,+1 < A , and as in lemma A.10 A,+1 < A , for all m 2 n again giving a contradiction. Hence d'+'=a". So A,,=Anfl which by R,,, R,+, and corollary 3.2.9 gives B,= B,,,,. Hence b"=b"+l as B,, is a quord, concluding the proof of the lemma. From lemma A.12 it is immediately clear that if bo is a tail element of B andp:CA-CB then ao=O. In order to complete the proof of the theorem we consider the following exclusive and exhaustive cases : 1 p:CA-CB, 2 pj(c, u)=j(O, 0) for some CEC,UEA, 3 pj(O,O)=j(c, b) for some CEC,b E B . Case 1. p:CA-CB. Let q(a)=b if, and only if pi(0, a) = j ( b l , b ) , p j ( a O ,a,) = j ( o , b ) , Pj(0, ao) = j ( b l , bl),

for some a', a,, b', b,. Clearly q is a one-one partial recursive sub-function of 4,. By lemma A.10 and the assumption thatp: CANCB it follows that q1 maps A onto B. Hence to show that q : A- B we need only prove that q is defined on A. Let a ~ A , p j ( Oa, ) = j ( b l , b). If bl=O, then clearly q(a)=b with a'= b'=O and ao=a, b,=b. If bl+O, letpj(aO,a,)=j(O, b), then ao+O and the hypothesis of lemma A.12 holds with bo=b. Hence by lemma A.12, pj(0, a,)=j(b', b,) so that q(a)=b. Case 2. pj(c, a)=j(O, 0) for CEC,U E A . Clearly we must have OEC,. Hence the hypotheses of lemma A.8 1) and 2) hold with b, = 0, a, =a, a' = c. Define the recursive functions rl, r, by r,(n)=a,, rZ(n)=bnf1 for all n>O. Then by lemma A.8 r l ( n + l ) + A r , ( n ) , rl (O)+A,O, r2(n + l)+Brz(n), r2(O)--4Bto and 41 ( r l ( 4 ) = rz ( n ) * Let q (a) =b if, and only if, p j (0, a ) = j (bl, b) , p j ( a O ,a,) = j ( 0 , b), p j ( 0 , a,) = j ( b l , b,) for some a', a,, b', b , ,

AP.A1

A MIXED

201

CANCELLATION THEOREM

or there is an n such that a=r,(n) and b=r2(n). Clearly q is a one-one partial recursive sub-function of q1 and q1 maps A onto B. It remains to show that q maps A onto B. Let UEA. If a is a tail element of A then a = r l ( n ) for some n and 4 (4 = rz (4. If u is not a tail element of A andpj(0, a ) = j ( b l , b) then b'=O implies q(u)=b. If b'+O then ifpj(aO,uo)=j(O, b) holds so does the hypothesis of lemma A.12 with bo=b since b cannot be a tail element of B. Hence by lemma A.12pj(OYu O ) = j ( b l ,b,) so that q(a)=b. Case 3. pj(O,O)=j(c,b) for ceC, ~ E B Using . the dual argument to case 2 we have a sub-function q of q2 such that q: B -A.

A.13 THEOREM. If C 2 1 is a quord, then Pc is an order embedding, i.e. 1) CA = CB if, and only if A = B, 2 ) (30) ( C A + D = C B ) if, and only if (30)( A + D = B ) . PROOF. 1) follows from theorem A.4 with C, = C, =O. 2) The implication from right to left follows from the proof of theorem 6.2.1. Now suppose C A + D = C B for some D, then by lemma A.14 below, CA = CB'+ Cz for some B', C, where (3E) (B'+ E = B) and ( 3 F ) (C2+F=C&F+O). Hence by theorem A.4, A=B' and the required result follows. A.14 LEMMA.If C 2 1 is a quord, then ( 3 E ) ( D + E = C B ) implies D = C B + C2 for some B', C , such that (W) (B'+F=B) and (3G)(C, + G = C&G+O). PROOF (generalizing lemma 9.2.2). Let CEC, BE By and D E= CB. Choose D E D to be an initial segment of CB. Let C=c'C, B=C'B, D =C'D, and let

+

B'= B[{~:(VC)(CEC*~(C,ED)}.

Let B' = C'B'. As D is recursively separable from C'CB-D we must have that B' is recursively separable from B-B'. Hence, as B' is an initial segment of B, B' =COT (B) is such that B' +F= B for some C.O.T. F. If D+CB' then there is a (unique) ~ E B - B ' such thatj(c,b)ED for some c e C . Let C, = C [ ( c : j ( c , b ) € D } .Then C, is a proper initial segment

202

P. H. G. ACZEL AND 1. N. CROSSLEY

[AP.A

of C and C‘C, is recursively separable from C-C‘C, so that C, =COT(C,) is such that C, + G = C for some non-zero G, since C is a quord. Also D = C.B’$ C , . { ( b , b ) } ,

+

hence D = CB’ C, .

APPENDIX B

INFINITE PRODUCTS AND PRINCIPAL NUMBERS FOR MULTIPLICATION by A. G. HAMILTON

We first of all define infinite products of sequence orderings in a manner closely related to the definition of exponentiation and then proceed to show that there exist 2'O principal numbers for multiplication of classical ordinal oWo which are not of the form W A .

B.l DEFINITION.Let A = (Ai: C) be a standard sequence ordering (see following definition 12.2.3) such that every Ai has a minimum element. Let

&(Vrn)(O 5 rn 5 n=>a,EC'A,, & a , =k min A,,)}. ( K may be the empty bracket symbol.) Now define

II(A) = {(e(K), e(K')):K , K ' E E ( A )

B.2 LEMMA.If (A,:C) is a standard sequence ordering with C'C r.e. and C has a minimum element, co, say, then II(Ai: C) II AC;II(Ai:

C[(CC- {cO})),

204

[AP. B

A. 0. HAMILTON

PROOF. By our definition of standard sequence ordering every A i R ~ C E R). Let f be the function defined by

(and

'

(: ... ""-'>)if c, co each c i ~ C ' C , j(min(Aco), "'i:)) if c, co ... j (a", e

=

*"

u"-l

&

=i=

e(i:

& each ci E C'C , undefined if any ci 4 C'C or if the argument o f f is not of the above form.

I

It is easily verified that f is one-one and maps CII(Ai:C) onto C'A,;II(Ai: C [(CC- {co})), Since, by hypothesis C C is r.e. and Seq = C'R is r.e. it follows that 6f is r.e. Finally we leave the reader to check that f is order preserving on the whole of its domain. B.3 COROLLARY. With the hypotheses of lemma B.2, if D E C and min (D) = co then

II(Ai: D)

N

Aco.II(A,:D [(C'D

- (~0))).

PROOF. The same function f as in the proof of the lemma is the required recursive isomorphism. Notation: If C has a first element, we shall denote C[(C'C-{min(C)}) by C'. If C has an initial segment of type n (finite) and n > 1, then C(") denotes (C'"- I))'.

B.4 COROLLARY. Let C be a well-ordering of type o (cR) with r.e. field, and let D be a sub-ordering of C with an initial segment do,..., d,,-l (i.e. the first n elements are d o . . . in that order). Then

n ( A i : D)NA,j;**-.Ad,- ;II(Ai: D'"'). PROOF.If n = 1 , let C , = C [ ( X : X E C C & ( ~x)EC}. ~, Since do can have at most a finite number of predecessors in C, C, has an r.e. field and we can apply corollary B.3 to get

II(Ai: D)

N

Ad;n(Ai: D').

Now assume n > 1. Suppose, as induction hypothesis, that

II(Ai: D)

N

Ado*..*.Ad,-2.n(Ai: D("-')>.

AP. BI

INFINITE PRODUCTS

205

We have to show that

But this follows immediately from the first part of the proof. So we conclude n ( A i : D) N A,,.*.*.A,,-,*II(Ai: D'"'). We now define a collection 42 of sub-orderings U of W' as follows: By theorem B.5 below, there exists a collection of 2'O isolated sets none of which is mapped into any other by a finite-to-one partial recursive function. We can suppose that none of these sets contains 0, 1 or 2. Now order each set by magnitude, and we get 2'O well-orderings of type o.For each such well-ordering U' let U = {((rn), (n)) :(rn, n ) ~ u ' } . Then the collection 42 of all such U has 2'O members, each of which is of type o,r W but not recursively isomorphic to W and moreover no two members of 42 are recursively isomorphic and no finite-to-onepartial recursive function maps the field of one into the field of any other. Every U E 42 satisfies the hypotheses of corollary B.4 which apply to D (with W' as the well-ordering C). B.5 THEOREM. There exists a collection d = { A i : i E I }of sets of (nonnegative) integers with the following properties: (i) the cardinal of I is 2'O, (ii) for all i , j ~ with I i + j there is no finite-to-one partial recursive function mapping Ai into A j , (iii) each Ai ( ~ E I is ) isolated. PROOF. Let cpo, cpl, ... be an enumeration (non-effective) of all finiteto-one partial recursive functions with infinite range. "20/

We construct a tree

which branches into two at each node, each node being an integer. The sets Ai will be the branches of the completed tree.

206

LAP. B

A. G. HAMILTON

Stage 0. Let ro be any integer not equal to cpo(zo)where zo = v, { q o ( w ) is defined).

We describe stage s for s 2 1. Stage s. We define successively Yo, Y1, ..., Yz'-'. Define 2' for 0 5 i l 2 ' - 1 as follows : Let Tsi=the set of all elements of the tree which have been defined before xsi, B,,=the set of all elements defined before Y i which lie on the same branch as x", TL=the set of all elements of the tree defined when stage s- 1 has been completed. Choose 9' to be some (say the least) integer t satisfying (a) t#TSi, (b) either t$ U d q j or t E U 6 q j and (Vj'jrs) (t&pj=.qj(t)$Tsi), jss

jss

*

(c) ( V j s8 ) ( V Z ) (zeTsi nh j * q j ( z ) t ) , (d) t $ { q j ( z j ) : O S j s s +l), where .zj=vw(qj(w) is defined and (Pj(w)#(qk(zk)

Ik<j) uT(i3.

At every stage we have infinitely many numbers to choose from; each of conditions (a)-(d) disqualifies only a finite number of integers since each j is finite-to-one with inlinite range. Thus such a t always exists. This describes the construction of the tree. The branches of this tree are the sets Ai. (i) The cardinal of I is 2N0since all the branches are distinct, (ii) Suppose q k : A i s A where j i i j . Then Ai*Aj by the construction, and we may find x€Ai-Aj such that x appears in the construction of the tree after the kth stage. Then y = q k ( x ) ~ Aandy+x. j Ify appears in the tree before x, then (b) is violated when x is put in the tree. If y appears after x then (c) is violated when y is put in. Thus we have a contradiction so condition (ii) of the theorem is satisfied. (iii) Suppose Ai is not isolated, then Ai contains an infinite r.e. set, say p ( p k (it must be the range of one of the 9;s). At stage k we kept Pk ( z k ) out of every branch of the tree, and at dl later stages. But z k was chosen so that qk(zk)did not occur in the tree before stage k. So qk(zk)$AI which is a contradiction. Thus condition (iii) of the theorem is met and the theorem is proved. We now write ui for the ith element of U in the obvious way.

AP.BI

207

INFINITE PRODUCTS

B.6 LEMMA.If UEQ, {Pi},,, is a strictly increasing sequence ofprincipal numbers for multiplication and PiEPi for all i, then COTJI( Pi: U) is a principal number for multiplication. PROOF. We have to show that if B is recursively isomorphic to an initial segment of II( Pi: U) then B*II(Pi: U)

N

II(Pi: U ) .

Suppose

B N B' < II(Pi: U> and suppose

U = {(ui, uj): i I j } . We must have

B1
forsomej,

i.e.

B' < lI(P,: U [u,) = Pu,""'P,,-,

= pui-i

9

since Pui-, is a principal number for multiplication > P,, for i < j - 1. Now, using corollary B.4 and properties of principal numbers we have:

B*II(Pi: U ) N B'.II(P,: U) N B' .P , .P,, * * P-, ,* II (Pi :U"') N B'*P,,-,*II(Pi: U"') N Pu,-,.n(Pi:U"') N p,,..... PUj-;II(Pi: U"') N II(P,: U). Hence COTII( Pi: U ) is a principal number for multiplication.

B.7 LEMMA.If B N WAfor some A, then the function h a 2 is recursively representable on B. PROOF. If B=WAthen C'B consists of elements of the form

and the ni (01i s m ) are non-zero integers. The function f defined on pe as follows recursively represents the

208

[AP.B

A. G . HAMILTON

function Iaa2 on B: fe

("no ... n,

am)

=

(2n0n, .*.... n,

am)

a0 '1

'

i.e.

If

(x)lB= (X(B.2 whenever XEC'B

and f(x)$C'B

if x$C'B.

This function f is obviously one-one and partial recursive mapping C'B into C'B. That it represents laa2 on B can be seen by mapping the set i x : (e

(an: ::::),

x)

EB

'"'))

(

& (x, e 2n0n, a '.'.*. n,

in a one-one order preserving way onto the set

( ('

y : y , e no *... ** n,

E

EB

1,

1

B ,

as follows

Now if g : B ' z B=WAthen g-'fg

recursively represents lam2 on B'.

B.8 LEMMA.If UE@and is a strictly increasing sequence of principal numbers for multiplication, then Pi€Pi can be chosen so that the function h a 2 is not recursively representable on II( Pi: U). PROOF. Let U = {(ui,u j ) : i s j } .Since all the Piare principal numbers, we must have Pi > 3 for all i, so we can choose the P,EP,with the following properties: min(Pi) = 0, min(PI) = (1) = 2, rnin(P1')) = u, . That we can choose orderings thus and still have them included in R is n easily seen (e.g. if QiePi, take ui+2Qi and replace the first three elements by 0,2, ui. The order must remain correct since ui >2 for all i.) With this choice of { Pi)i<w,consider

AP. Bl

209

INFINITE PRODUCTS

Suppose a is the ordinal represented by e Then the element representing a2 is e e(

i)

can be mapped exactly twice in a one-one order preserving way into

the set of predecessors of e (co a,

(3

by the identity function and the function

... cn) ... an

--t

(j c o ...... 2a,

Cn)

an

Now if 1aa2 were recursively representable on lT(P,:U), function defined on numbers of the form e

(i),

then the

taking e(i) to e

(L)

would be partial recursive. However, this would imply that Xui was a (partial) recursive one-one function. But U is not recursively isomorphic to W, so liuiis not a recursive function. Therefore h a 2 is not recursively representable on ll( Pi: U).

B.9 COROLLARY. COTII( Pi:U) is a principal number for multiplication not of the form W A . PROOF. By lemma B.6, COTII( Pi: U) is a principal number for multiplication. Suppose it were of the form W A ,then

n(Pi:U)=WA

forsomeA.

(*I

By lemma B.8 1aa2 is not recursively representable on 11(Pi: U). By lemma B.7, 1aa2 is recursively representable on WA.This contradicts (*) and the corollary is proved.

+ COT II ( Pi : U) + COT II (Pi:

B.10 LEMMA. If U, V E 42 and U V then

V).

PROOF. Suppose the contrary, then there is a partial recursive one-one function f such that f:II(P,: U) N II(P,:V). Now

fe

(9) ( ... ") = e co'*' a, a,

for some c,, ..., c , ~ c ' v and aiECLP,,

210

[AP. B

A. G . HAMILTON

so and therefore

is a finite-to-one partial recursive function mapping C'U into C'V. For suppose there were a v, E C'V which was the image of infinitely many u~EC'U.It would then follow that all of II( P,: U> is mapped onto a proper initial segment of II( P i : V) by f, namely that initial segment determined by e

pi1)

(since U has type w and the set of ui mapped

to v, is cofinal with U). This is a contradiction, so AxZ((fe ($,)

is

finite-to-one. That it is partial recursive is obvious. But the existence of such a function contradicts the definition of 4. Hence there is no such f and therefore

COTII(P,:U>+COTII(Pi:V). B.ll THEOREM. There exist 2" principal numbers for multiplication of classical ordinal ow"which are not of the form WA. PROOF. By corollary B.9 COTII( P i : U) is a principal number for multiplication not of the form W A ,for each UE 4. By lemma B.10 all the co-ordinals COT n

product) = maw. This completes the proof. Theorem 10.2.10 shows that the result in theorem B.ll is best possible.

NOTATION AND TERMINOLOGY

1 Logical notation. We use logical symbolism freely for convenience in our informal exposition. We write &, v ,1, *,-, 3, V, E!, p x for and, or, not, implies, if and only if, there exists, for all, there is a unique, the least x such that, respectively. In this last we always mean the least natural number. We occasionally write (Vx < y )

or (Vx),,

for (Vx)(x < y*)

and analogously for 3. We also use the A-notation for functions (see e.g. KLEENE, 1952, p. 34) and we sometimes use dots for bracketing purposes in the usual way but we do not need rules for association since we do not use dots in such complicated ways. 2 Set-theoretic notions and notations. A’ denotes the set of natural numbers otherwise called non-negative integers. We use Ifor the usual ordering of natural numbers by magnitude. p i denotes the ith prime (p0=2) and ( x ) denotes ~ the exponent of the highest power of p i which divides x. If p i is the largest prime which divides x non-trivially then we set Zh(x) = i + 1. N.B. This is a deviation from KLEENE (1952). As usual the membership relation is denoted by E and E is used to denote an ordinal. We write x$A for 1 x E A as usual and similarly for . A set generally means a set of natural numbers unless we specifically say e.g. “a set of ordinals”. We use class and collection for talking

+

212

NOTATION

about aggregates of sets or classes. The empty set (of anything) is denoted by 0. We use Roman letters A, B, C, ... for sets. { x : ' p ( x ) } denotes the set of all elements satisfying 'p. (The context will make clear what kind of elements is intended.) { x } denotes the set whose only element is x but {a(n)}, {a,,}, {a(n)};=,, { u , , } ~ = all~ denote sequences whose nth member is a(.) or a,,. A-B={x:xEA&x~B},A=J~~-A, when A is a set of natural numbers except in the proofs of lemma 16.4.1 and theorem 16.4.2. A G B means XEA implies XEB. N.B. A c B always means A E B &A B. (x, y) is the ordered pair of the (natural) numbers x , y. A x B= {(x, ~ ) : ~ E A & ~ EAB2}=;A x A , etc. u, n denote union, intersection, respectively, as usual. Two sets are said to be disjoint if their intersection is the empty set. a, B, ... denote ordinals and an ordinal is construed as its set of predecessors. With one exception, which is explicitly noted, ordinal always means countable ordinal. If A is a set of ordinals then supA=limA=uA=the least ordinal greater than or equal to any ordinal in A. The order type of the rationals is denoted by q as usual. q is the type of any countable dense linearly ordered set without fist or last element. KO is the cardinal of N and 2'O=c is the cardinal of the continuum. A relation is a set of ordered pairs of numbers, i.e. a subset of M'. We use Gothic letters A, B, ... for relations. By the converse, A*, of a relation A we mean ( ( x , y ) : ( y , X)EA}. If A, B are relations such that A s B then we say A is cofinal in the B if for some x ( x , Y)EB then ( y , Z)EA for some z. The restriction of A to B, A[B=AnB2.

+

3 Functions. A map of a (possibly proper) subset of Nninto Jlr is called a (partial) function of n arguments. A partial function is total if it is d e k e d o n all of J". We use lower case italic letters (f,g, h, ...) for functions and we write either f ( x ) or f, for the value of the (one-place) function f at x. The domain of a (one-place) function f is the set of numbers for which f is defined. We write Sf for this set and we write pf for the range off, i.e. the set of values off. Byf(A) we mean {f(x):xEA} and similarly, if A is a relation we write

f(A)= {(f(x), f(J9): (x, Y> € 4*

213

NOTATION

Iff is one-one then we definef -'(x) = y if, and only if,f(y) is defined and = x. We sometimes talk of functions of ordinals and here we mean functions from (n-tuples of) countable ordinals into the countable ordinals. The classical set-theoretic definitions of addition, etc. for ordinals may be found in SIERPINSKI (1958). 4 Recursion theory.

We assume familiarity with the basic notions of partial recursive and (general) recursive functions and recursive and recursively enumerable (r.e.) sets. We sometimes use Turing machine methods for convenience (details can be found in KLEENE, 1952). The following two facts are basic: (i) A set A is recursive if, and only if, A and A are r.e. (ii) An (infinite) set A is recursive if, and only if, it can be enumerated in (strict) order of magnitude by a recursive function. We recall that a set containing no infinite r.e. subset is said to be immune and that such sets (indeed c of them) exist by D E K K E R ( ~ ~ ~ ~ ) . We use the well-known (primitive) recursive map j:M2+M which is 1 - 1 and onto, defined by j ( x , Y ) = +(.

+ Y ) ( X + Y + 1) + x

and also the (primitive) recursive k, 1 such that j ( k ( x ) , l(x)) = x .

We write j(A, n) = { j ( u , n):aEA}, j(A, B) = {j(a, b):aEA, b E B ) .

We use the usual (primitive) recursive function defined by a*b=c,

if a = 2ao.3a1..... and c

pE*, b = 2bo....-p>

....- p ~ m - p ~ + P + l - . . -where - p ~ + ,a,,,,

b, =k 0.

We make a lot of use of Kleene's indefinite description operator v, which from a partial recursive function f yields a partial recursive function 9 such that ( 3 Y ) ( f ( Y ) = 0)=. 9 (V,(f(Y) = 0)) = 0 *

214

NOTATION

We define x ~ asy usual by x-y=x-y

=O but we also define x

7

if x k y , otherwise ,

y = pz { y + z

= x}.

So x - y is a partial function in general.

Other unexplained notations may be found in KLEENE (1952) especially p. 538.

REFERENCES

P. H. G. ACZEL,1966,D. Phil. Thesis, Oxford. 1966a,Paths in Kleene’s 0, Archiv Math. Logik Grundlagenforschung 10, 8-12. - & J. N. CROSSLEY, 1966,Constructive Order Types, 111, Archiv Math. Logik Grundlagenforschung 9, 112-1 16. H. BACHMANN, 1955, Transfinite Zahlen (Berlin). G. CANTOR,1915, Contributions to the Founding of the Theory of Transfinite Numbers, (Dover Reprint). 1936,Formal Definitions in the Theory of Ordinal Numbers, A. CHURCH& S. C. KLEENE, Fundamenta Mathematicae 28, 11-21. J. N.CROSSLEY, 1963,D. Phil. Thesis, Oxford. -, 1965, Constructive Order Types, I , in Formal Systems and Recursive Functions, Eds J. N. Crossley&M. A. E. Dummett (Amsterdam) 189-264. -, 1966, Constructive Order Types, 11, JSL 31, 525-538. - &R. J. PARIKH, 1963, On Isomorphisms of Recursive Wefl-orderings (Abstract), JSL 28, 308171. - &K. SCHUTTE,1966,Non-uniquenessat w2 in Kleene’sO, Archiv Math. Logik Grundlagenforschung 9,95-101. J. C. E. DEKKER& J. MYHILL,1960, Recursive Equivalence Types, Un. of California Publications in Mathematics, n.s. 3, 67-214. A. EHRENFEUCHT, 1957, Applications of games to some problems of mathematical logic, Bull. Acad. Polon. Sci. 5 pp. 35-37. E. ELLENTUCK, 1963,Solution of aproblem of R. Friedberg, Mathematische Zeitschrift, 82, 101-103. S. FEFERMAN, 1968,Systems of Predicative Analysis, II; JSL 33, 193-220. R. FRIEDBERG, 1961,The Uniqueness of Finite Division for Recursive Equivalence Types, Math. Zeitschrift 75,3-7. R. 0.GANDY, 1960,Proofof Mostowski’s Conjecture,Bull. Polon. Acad. Sci. 8,571-575. A. G. HAMILTON, 1968, An unsolvedproblem in the theory of constructive order types, JSL. 33, 565-567 S. C. KLEENE,1952, Introduction to Metamathematics (Amsterdam). -, 1955, On the Forms of Predicates in the Theory of Constructive Ordinals (Second Paper), Am. J. Math. 77,405-428.

216

REFERENW

G. KREISEL,1960, Non-uniqueness Results for Transfinite Progressions, Bull. Polon. Acad. Sci. 8,287-290. J. MCCARTHY, 1956, The Inversion of Functions defined by Turing Machines, Automata Studies, Annals of Mathematics Study no. 34, (Princeton) 177-181. A. NERODE,1961, Extensions to Isols, Annals of Mathematics 73, 362403. R. J. PARIKH,1962, Some Generalisations of the Notion of Well-ordering (Abstract), Notices Am. Math. SOC.9, 412. -, 1966, Some Generalisaiionsof the Notion of Weil-ordering,Zeitschrift Math. Logik Grundlagen der Mathematik 12,333-340. H. G. RICE,1956, Recursive and recursively enumerable orders, Transactions Am. Math. SOC. 83,277-300. K.ScHUme, 1965, Predicattve Well-Orderings,in Formal Systems and Recursive Functions. Eds J. N. Crossley and M.A. E. Dummett (Amsterdam), 280-303. W.SIERPINSKI, 1948, Sur la Division des Types Ordinaux, Fundamenta Mathematicae, 35,l-12. -, 1958, Cardinal and Ordinal Numbers, (Warsaw). R. I. SOARE, 1969, Constructive order types on cuts (to appear in JSL). C. SPECTOR, 1955, Recursive Well-Orderings,JSL 20, 151-163. A. TARSKI,1949, Cardinal Algebras, (New York). -, 1956, Ordinal Algebras, (Amsterdam). J. S. ULLIAN,1960, Splinters of Recursive Functions, JSL 25, 33-38. 0. VEBLEN,1908, Continuous Increasing Functions of Finite and Transfinite Ordinals, Transactions Am. Math. SOC.9, 280-292. A. N. WHITEHEAD & B. RUSSELL, 1927, Principia Mathematica, Vol. I1 (Cambridge).

NOTES

INTRODUCTION 01 This is closely related to the results of CROSSLEY AND PARIKH (1963). 02 The term “recursive isotonism” which has been used instead of “recursive iso(1963, 1965), ACZELAND CROSSLEY (1966) and some morphism’’ in CROSSLEY other places has a slightly different meaning. 03 This will be exploited more in work by NERODE and the author which is in preparation. 04 We abbreviate “recursively enumerable” by “r.e.”.

CHAPTER 1

11 The author has discovered (July 1966) that this theorem was essentially proved by NCE (1956, theorem 20) and an essentially identical one by SPECTOR (1955). 12 There will be no confusion between sequence numbers of sequences of two elements and ordered pairs since the context will always make clear which is intended. is defined on p. 213. CHAPTER

2

21 Note that this notation will causeno confusion for if A is set-theoreticallyincluded inBthenA=B. 22 We use the word “step” here in the sense of a whole phase in the calculation rather than moving just one square on the Turing machine tape. CHAPTER 3 31 The quantifier (Vf) ranges over all one-place functions and the quantifier (VB) ranges over all sets of natural numbers. 32 This use of the word “splinter” is derived but differs from that in ULLIAN (1960).

218

NOrnS

CHAPTER 4 41 Such exists by theorem 3.1.5.

CHAPTER 5 51 Although these are not quite the same orderings as in CROSSLEY (1965), the

co-ordinals are the same. 52 Since we get most of our counterexamples from V. 53 This simple group-theoretic way of presenting the proof is due to Alex Rosenberg.

CHAPTER 6 61 By “minimal” we mean minimal with respect to domain and range. 62 This argument is basically due to TARSKI(1956). 63 This argument and those to the end of this chapter are due to TARSKI (1956)

though the proofs of the supporting theorems are very different from his. CHAPTER

7

71 Since we are assuming min A =0 we shall never misidentify two sequences since no sequence can end in 0. 72 This is a strengthened version of theorem VIII.2.2 of CROSSLEY (1965).

CHAPTER 8 81 This is inspired by TARSKI(1956). Compare also theorem 2.4.11. 82 We are here using an extension procedure similar to that in the proof of theorem 8.3.5. 83 We write e(A, B) for { e ( A ) : AE (A, B)}. 84 Recall ( x ) ~ =exponent of 2 in prime factorization of x. CHAPTER 9

91 In fact we show that many co-ordinals may be expressed as polynomials in W

with large exponents. 92 This theorem was first conjectured by A. L. Tritter. 93 We are assuming min(A) =0 as usual. CHAPTER 10 101 This definition is adapted from KREISEL (1960). 102 The problem of whether there exist principal numbers for multiplication not of the form W Ais solved affumatively in the appendix B by A. Hamilton.

CHAPTER11 111 The results in this chapter were obtained by P. H. G. Aczel and the author and

NOTES

219

(1966). The name “Eappeared in their original form in ACZELand CROSSLEY number” is intended simply to convey that these co-ordinals are closely related to (classical) .+numbers. It should not be confused with the E(I), etc. of VEBLEN (1908). 112 ea =PO (09 = / 3 & / b cy for y< a} or, equivalently, the a-th (classical) principal number for exponentiation greater than w. 113 Here XI 2 E . . , 2 is an abbreviation for (XS, X I > E

E&.

..&<xr, X~-I>E E .

CHAPTER 12 121 7 i s defined on p. 14. CHAPTER 13 131 The reader who is familiar with R.E.T.s and isols is advised to omit this chapter. 132 There will be no confusion although we use the same type founts for (e.g.) R.E.T.s and C.0.T.s since the context will make clear which is intended (generally, R.E.T.s in this chapter). 133 Here again we are dealing with unordered sets so N is not ambiguous.

CHAPTER 14 141 The construction given here is due to C. G. Jockusch. APPENDIX A1 The question whether the theorem holds with the weak sense of 5 is open. REFERENCES 171 See the correction in the author’s abstract JSL 31,292-3.

INDEX OF SYMBOLS

8,

0 W1

8L V

1

5 5 6 211 211 211 211 211 211 211 211 211 211 211 211 211 211 21I 211 212 212 212 212 212

212 212 212 212 190,212 26, 32, 189 212 212 212 212 212 212 212 212 212 212 212 212 212 212 212 212 212 212

221

INDEX OF SYMBOLS

212 212 213 213 213 213 213 213 214 214 21 22 22 22 22 23 23 25 26 26 26 26 27 29 30 30 31 32 32 32 34 34 34 38 38 38 29 43

46 46 46 46 212 52 54

54 54 54 54 56 56 56 56

(b...'bn),(A, B) ao...an

56 56 60 62 62 63 75 75 76 76 76 76 77 79 80 82 93 93 97 97 103 103

222

As,B A = (Ai :C) S.O. s.W.O. CST(A)

z %4 0, (for R.E.T.s) R (R.E.T.) A + B (for R.E.T.s) I (for R.E.T.s) E for C.0.T.s N

on

xi, ni, Ai T(4 E(Y, x) A, A, 9 I ,I *, redefined

Seq'

INDEX OF SYMBOLS

133 142 1 42 142 143 144 144 145 145 159 160 161 162 168 170 170 170 170 171 171 176 177 178

INDEX OF TERMS

Aczel, P. H. G., 5, 6, 12, 16, 17, 22, 131, 142,153, 175, 194 addition of C.0.T.s 12, 34 - of C.S.T.s 146 - of R.E.T.s 161 antisymmetry 21, 68 ascending chain 167 Bachmann, H. 11,92, 130 bound, upper 16, 150 - -, existence of 152 bracket symbol 75 cancellation laws for C.0.T.s 92 et seq, 194 et seq - - for isols 165, 166 - - for losols 182 et seq - - metatheorem for losols 189 Cantor, G. 12, 24 Cantor normal form 15, 110 et seq, 120 cardinality of Q, etc. 47 Church, A. 11, 122 class 211 classical results 11 cofinal212 collection 212 commuting co-ordinals 57 constructive means 6 - order type 12, 23 - - _ , standard 29 - sequence type, 16, 143

- - -, standard 145 converse 37, 167 co-ordinals 13, 46 -, commuting 57 -, incomparable 55 -, natural 54 -, recursive 46 -, sequence 143 counterexample, basic 54 Dedekind section 169 Dekker, J. C. E. 5, 12, 16, 23, 159 et seq, 177, 178 descending chain 41 directed refinement theorem, 36 - - -, generalized 148 disjoint sets 212 -, strictly 30 dots 21 1 dual 38, 195 Ehrenfeucht, A. 16 Ellentuck, E. 166 E-number 16, 131 et seq. exponentiation of C.0.T.s 15, 75 - of R.E.T.s 165 Feferman, S . 5 field of a relation 21 final segment 38

224

INDEX OF TERMS

function, partial 212 -, total 212 -, domain of 212 -, range of 212 Gandy, R.0. 13. 15 generates 29, 195 Hamilton, A. G. 17, 18, 176, 178,203 immune set 213 infinite product 203 et seq - sum 144 et seq initial segment 38 isol, 16, 164 et seq isolated set 160 - linear ordering 176 isomorphic 21 isomorphism, classical 15 -, of sequence orderings 142 -, recursive 11, 22 -, recursive, pair 143 - with classical ordinals 129, 141 Jockusch, C. G. 17, 169 k-attained 79 Kleene, S. C. 5, 25, 26, 43, 78, 122, 143, 170, 179,211 Kreisel, G. 5, 6, 122, 167 A-notation 21 1 limit number, 52 Lindenbaum, A. 13,39 losol 16, 175 et seq McCarthy, J. 23 minimum element 25 Morley, M. 12, 13, 39, 40, 94 multiplication of C.0.T.s 14, 60 - of R.E.T.s 162 Myhill, J. 5, 7, 12, 16, 17, 23, 159 et seq, 175, 177, 178 natural well-ordering 5, 54 - co-ordinal 54 Nerode, A. 7, 17, 29, 166, 167, 175, 176, 178, 193 notation 21 1 -, unexplained 214

order embedding 194 - type 22 _ _ of quords 13 ordering, dense 25 -, linear 21 -, natural well- 5 -, partial 21 -, partial well- 49 -, rooted tree 49 -, sequence 142 -, sequence well- 142 -, tree 49 -, well- 41 ordinal algebra 5 -, classical 46 -, first non-recursive 6 - number 11 - recursive 11 - sum29 Parikh, R. J. 5, 6, 13, 15, 17, 42, 43, 78, 122, 167 partial ordering 21 - recursive functional for addition 34 _ - _ _ multiplication 62 - - - - exponentiation 77 - _ _ _ infinite sums 144 path 16, 56 -, long 153 pathological features 12, 28, 51 predecessor, weak 177 principal number 15 - _ for addition 92 _ _ _ multiplication 97, 203 et seq - - _ exponentiation 103, 135 quasi-finite, 17, 167 -well-ordering 13,41 _ _ -, recursive 42 quord, 13,43 -, recursive 43 -

recursive equivalence 159 - equivalence type 16, 159 et seq - isomorphism 11, 22 - isomorphism pair 143 - permutation 23 - sequence type 16 refinement theorem, directed 36

INDEX OF TERMS

- -,for R.E.T.s 161

- -,

generalized directed 148 relation 21, 212 -, field of 21 -, recursive 22 -, r.e. 22 -, reflexive 21 -, restriction of 212 Rice, H. G. 12, 25 Rosenberg, A. 58 Russell, B. 30 Schiitte, K. 6, 7 segment, closed 56 -, final 38 -, initial 38 -, open 56 separable 30 separation lemma 12, 35 - _ , generalized 148 sequence co-ordinal 143 - number 26 - well-ordering 142 sets 211 -, disjoint 212 -, empty 212 -, finite 14 -, immune 160, 213 -, isolated 160 Shepherdson, J. C. 5 Sierpinski, W. 11, 37, 39, 44, 57, 60,92, 98, 102, 194 Soare, R. I. 23

225

Spector, C. 13,43 splinter 44 standard C.O.T. 29 - C.S.T. 145 strictly disjoint 30 - a-unique 122 successor 53 - number 52 Tarski, A. 5, 13, 14, 15, 39, 68, 72, 94, 102, 147, 149 Tennenbaum, S. 17, 169 transitivity 21 tree ordering 49 - property 14 trichotomy law 21 Tritter. A. L. 111 Ullian, J. S. 44 unique factorization 130 -, a- 122 -,E W , co-ordinals 141 -, am-, co-ordinals 125 -, wWm-,co-ordinals 128 upper bounds 16,150 Veblen, 0. 5, 131 well-ordering 41

-. natural 5

-, partial 49 -, sequence 142 Whitehead, A. N. 30