Constructive Order Types, I

CONSTRUCTIVE ORDER TYPES, II) JOHN N. CROSSLEy2) St. Catherine's College, Oxford, UK

Introduction 1. The theory of constructive order types constitutes a new approach to the problem of providing a constructive analogue of ordinal number theory. Recently, Dekker, Myhillet al. (e.g. [8]) have considered a generalization of the notion of cardinal number which may be regarded as a constructive analogue of Cantor's theory. Ordinal number theory may be approached in two ways. (1) Ordinals may be considered as being generated in a certain way (v. [1] § 3, p. 19; [2] p. 87). (2) Ordinals may be regarded as the equivalence classes of well-ordered sets under (arbitrary) one-one order-preserving maps (isotonisms). Church and Kleene ([3]) considered a constructive analogue of (1). In the present work we embark on a constructive analogue of (2). We define constructive order types as equivalence classes of (linear) orderings under effective one-one order-preserving maps (recursive isotonisms). (We are only concerned with denumerable orderings.) In particular, co-ordinals are the equivalence classes of well-orderings obtained under recursive isotonisms. Since there are only denumerably infinitely many recursive isotonisms, co-ordinals are, in general, proper sub-classes of the corresponding classical ordinals. In establishing our results we use classical set theory together with 1) The author is deeply indebted to Prof. G. Kreisel for his valuable suggestions and comments on the problems discussed here. 2) The work presented here was done whilst the author was a Junior Research Fellow at Merton College, Oxford.

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JOHN N. CROSSLEY

recursive function theory. We define addition, multiplication and exponentiation in a way which agrees (with respect to the orderings) with the classical versions of these functions. Most of our basic results in the additive and multiplicative theory hold not only for co-ordinals but also for a collection of constructive order types we call quords. Quords are the constructive order types of those linear orderings which contain no effective infinite descending chains. As there were close similarities between some aspects of Tarski's Cardinal algebras [16] and Dekker and Myhill's Recursive equivalence types [8], it is not surprising that there should be analogous connections between Tarski's Ordinal algebras [17] and the present work. However, neither in the theory of recursive equivalence types nor in the theory of constructive order types is there a natural analogue of the infinite sum which Tarski introduced (v. [7], p. 197). This may be regarded as the main reason why constructive order types do not naturally form an ordinal algebra. 2. In section I we introduce various kinds of isotonisms, i.e. one-one, order-preserving maps, and show that two recursive linear orderings are recursively isomorphic if and only if they are recursively isotonic. Thus the theory of recursively isomorphic well-orderings is part of the theory of constructive order types. Addition is defined in § II. For this we require the notion of (r.e.) separable relations (cf. [15]) which is studied in § II. 1. The rest of § II is devoted to establishing elementary properties of addition and of an ordering by initial segments (:0:;). In particular, we prove the Separation Lemma (II. 5 . 1) and the Directed Refinement Theorem (II. 5 .2) which are fundamental for much of the later work. The Directed Refinement Theorem also shows that :0:; is a tree ordering, i.e. A :0:; C and B :0:; C imply A :0:; B or B :0:; A, as well as being a quasi-ordering of all constructive order types. We restrict our attention exclusively to linearly ordered sets and their constructive order types from § III. There we consider an analogue of the descending chain condition. We caIl linearly ordered sets which contain no infinite recursive descending chain quasi-well-orderings, and we call their constructive order types quords. For quords we also have the cancellation law A + B = A + C implies B = C. However, quords are not partially well-ordered by :0:;.

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Co-ordinals, which we discuss in § IV, are the constructive order types of well-orderings. Although ~ is a partial well-ordering") of C(? (the collection of all co-ordinals) it is not a well-ordering and for every (classical) limit number there exist uncountably many co-ordinals which are subclasses of that classical ordinal. It follows that these co-ordinals are incomparable. Because of this [an analogue of] a classical law for addition fails for co-ordinals. However, when we introduce the notion of a principal number for addition then the [analogue of the] law holds for predecessors of any given principal number. For such co-ordinals we also obtain a unique additive decomposition. A negative result is that a co-ordinal A may be such that there is a (classical) ordinal T, less than the (classical) ordinal of A, for which there is no co-ordinal C which is both < A and of classical ordinal T (example IV.5. I). But we do have the following Representation Theorem (IV. 5 .4): For every (classical) ordinal r, there is a co-ordinal C (or ordinal T) which is such that, for every ordinal Ll < r, there is a co-ordinal D of ordinal A such that D < C. If T is infinite there are uncountably many such co-ordinals (corollary IV. 5 .5). In § V we prove that a collection of quords (a fortiori of co-ordinals) has a least upper bound if, and only if, it has a maximum, i.e. there are no non-trivial least upper bounds. Multiplication is introduced in § VI and some of its basic properties derived. Most of the fundamental classical laws [have analogues which] go through. In order to show that AB = AC implies B = C (if A '# 0) we prove that the (classical) isotonism between representatives of Band C can be extended to a recursive isotonism. Analogously to the situation for addition, the law A < B implies AC ~ BC does not hold in general; but we show that it does hold for predecessors of a given principal number for multiplication. Exponentiation is defined in § VII and the development is very similar to that of § VI. In § VIII we prove that the collections of principal numbers for addiWw tion and multiplication of ordinal less than W W and w , respectively, lie in a single branch of the tree of co-ordinals. This result is best possible in the sense that there exist incomparable principal numbers for addition Ww and multiplication of ordinal co" and w , respectively. 1) :::; is a tree ordering, also, by theorem II. 5.3.

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Because predecessors of a principal number for (e.g.) addition obey all the classical laws for addition, the well-orderings belonging to co-ordinals lying in the branch of the tree just mentioned may be called natural well-orderings (with respect to addition). By increasing the number of functions considered and ensuring that the collections of principal numbers form a nested sequence, we can get characterizations of natural well-orderings up to larger segments of the Cantor second number class in terms of their co-ordinals. 1) Terminology and notation

The development of the theory of constructive order types will be informal but we shall use logical symbolism freely for brevity. We write "&", "v", "I", "---+", "+--).", "3", "V", "E!", "flx" for "and", "or", "not", "implies", "if and only if", "there exists", "for all", "there is a unique", "the least x such that", respectively, and we also use the Anotation (cf. [9], p. 34). We sometimes use dots for bracketing purposes in the usual way. A number means a natural number (0, I, 2, ... ) unless otherwise stated. A set is a collection of numbers and a class is a collection of sets. We denote the set of all natural numbers by.J' and the empty set by 0. We use lower case Greek letters for sets. {x : P(x)} is the set of all elements satisfying the predicateP. &('J.- 13 = {x : x s ('J. & x ¢ f3}. ii = .J' -('J.. e('J. £: 13 means x s ('J. -+ x e 13 and ('J. c: 13 means ('J. £: 13 & ('J. i= 13.
=

('J.X('J..

A relation is a set of ordered pairs of natural numbers, i.e. a subset of

.J'2. We use upper case bold face letters (A, B,... ) for relations. A relation A is said to be reflexive if

(x, y) e A

-+


The converse of a relation A is {
CONSTRUCTIVE ORDER TYPES, I

193

{x : (3y) «x, y) B A v 1) arguments is the set of all (n-tuples of) numbers for which the function is defined. The range offis the set of values off By f(rx) we mean {I(x) : x B a} and by leA), wherefis a function of one argument, we mean {
:
B

A}.

We assume familiarity with classical ordinal number theory (as in e.g.

[1] or [14]). We also assume familiarity with the notions of recursive

and partial recursive functions and recursive and recursively enumerable (r.e.) sets. We sometimes use Turing machine methods for convenience (for details see e.g. [5] or [9]). We make heavy use in the sequel of the facts: (i) If a is a r.e, set and a ~ of, thenf(a) is r.e., (ii) If a is an infinite r.e. set, then a is recursive if, and only if, there is a recursive function which enumerates a in order of magnitude ([13], p. 291). We recall that a set containing no infinite r.e. subset is said to be immune and that there exist immune sets and r.e. non-recursive sets ([6], p. 89, [13], p. 291). We use the well-known (primitive) recursive functions defined by

= !(x+Y) (x+y+ l)+x, j(k(x), lex)) = x j(x, y)

(v. [5], p. 43). j maps .?2 one-one onto f

{j(x, n) : x

and (A ; n) for

B

We write j(a, n) for «}

{
B A}.

Unexplained notations may be found in [9], p. 538.

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JOHN N. CROSSLEY

I. Recursive isotonism

1.1. All relations are assumed to be reflexive unless otherwise stated.

A function f from the field of a relation A to the field of a relation B is said to be relation preserving (between A and B) if

(x, y) s A +-+
In the above definition and in definitions I. I .2 and I. I .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 cases f- 1 is well-defined on pi (Under definition 1.1.2 we may haveff-t :1: 1.) Clearly isotonism is an equivalence relation. We write RT(A) = {B : B ~ A} and if A = RT(A), then A is said to be a relation type. DEFINITION I. 1.2: Suppose A and B are relations. Then a map p(x) is said to be a recursive isotonism from A to B if (i) p is a partial recursive function, (ii) p is one-one, (iii) {)p ;2 C'A and p( C'A) = C'B, (iv) p is relation preserving between A and B. A is recursively isotonic to B if there is a map p which is a recursive isotonism from A to B. We write p : A ~ B if p is a recursive isotonism from A to B and A ~ B if there is a recursive isotonism from A to B.

We claim that recursive isotonism is an equivalence relation. The identity map is recursive, hence recursive isotonism is reflexive. If p is a one-one partial recursive function, then p - 1 (defined, of course, only on pp) is also partial recursive (see [11], p. 177). It follows that if p : A ~ B, then p-l : B ~ A. It is clear that recursive isotonism is a transitive relation. We can now introduce our next definition. 1.1.3: If A = {B : B ~ A}, then A is said to be a constructive relation type. We write A = CRT(A). DEFINITION

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DEFINITION 1.1.4: A function f is said to be a recursive permutation iff is recursive and maps f one-one onto itself. DEFINITION 1.1.5: A relation A is said to be totally recursively isotonic to a relation B if there is a recursive permutation f which is a recursive isotonism from A to B. We write A ~ B if A is totally recursively isotonic to B.

Again, totally recursive isotonism, is an equivalence relation; we write TRRT(A) = {B : B ~ A} and if A = TRRT(A) for some relation A, then A is said to be a total recursive relation type.

1.2. From now on we use upper case Roman letters for constructive relation types (C.R.T.s). The collection ofall C.R.T.s will be denoted by (jI. I. 2. I : (i) A ~ B -. A ~ B -. A ~ B, (ii) There exist relations A, B such that A ~ B but A ~ B, liii) There exist relations C, D such that C ~ D but C D. THEOREM

PROOF. (i)

(ii) Let

(X

*

Clear from definitions 1.1.1, 2 and 5. be a r.e. non-recursive set. Then (X is infinite with an infinite

non-r.e. complement ii. Let

A = {(a, a') : a, a' s (X & a

B = {(b, b') : b, b' s ii & b

~

a'},

~

b'}.

Then A '" B, since A, B both represent well-orderings of type t». But A ~ B implies ii = f«(X) for some partial recursive function! This implies ii is r.e., contradicting the choice of (x. (iii) Let C = {(c, c') : 0 ~ c ~ c'}, and D = {(d, d') : I s d s d'}.

*

Then if f(n) = n+ 1, f: C ~ D. But C D, for if C ~ D by g, then g-l(O) is undefined, which is in contradiction with g being a recursive permutation.

Corollary 1.2.2. (i) RT(A);2 CRT(A);2 TRRT(A),(ii) There is a relation C such that RT(C) ::::> CRT(C) ::::> TRRT(C).

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JOHN N. CROSSLEY

PROOF. (ii) Let C, D be as above in the proof of theorem 1.2.1, and let E = {
1.3. We observe that A ~ B if, and only if, A* ~ B* (and similarly for '" and ~). DEFINITION 1. 3.1: A* is said to be the converse of (the C.R.T.). A if A = CRT(A) and A* = {B : B ~ A*}. THEOREM 1. 3 . 2: (i) A * = {B* : B (ii) {A* : A e.?Jl} = .?Jl, (iii) A** = A.

~

A}

where

A

= CRT(A),

THEOREM 1.3.3: Let A ~ B, (f. = C'A and f3 = C'B. Then (i) (f. is r.e. +-4 f3 is r.e., (ii) (f. is immune +-4 f3 is immune, (iii) There exist relations A', B' such that C'A' is recursive, C'B' is not recursive and A' ~ B'. PROOF. (i), (ii) Left to the reader. (iii) Let f3 be a r.e, non-recursive set enumerated without repetitions by the recursive function ben). Let

B'

=

{
:i

s

j} and A' = {
s

j}.

Then h : A' ~ B'.

1.4. DEFINITION 1.4.1: A relation A is said to be recursive (r.e.) if there is a recursive function f(a, b) (f(a, b, c) such that
(= {x : (3z)j(x, x, z) = O} if A is Le.). DEFINITION 1.4.3: A relation A is said to be recursively isomorphic to a relation B if there is a recursive predicate L(X, y) such that, for some

CONSTRUCTIVE ORDER TYPES,

197

I

isotonism, f, between A and B, I(X, y) ~ f(x) = y. In this case we write I : A == B and if there is such an I we write A == B. If A = {B : B == A} then A is said to be a recursive isomorphism type and we write A = RIT(A). Recursive isomorphism is an equivalence relation. Since (i) I : A == A if I(X, y) ~ X = y, (ii) if I : A == B, then 1* : B == A where I*(X, y) ~ I(Y, x), (iii) if I : A == Band K : B == C, then A : A == C where

A(X, y)

~

(3z) (I(X, z) & K(Z, y))

~

(V'z) (I(X, z)

--t

K(Z, y)),

since A is recursive by [9], theorem VI (p. 284). THEOREM 1.4.4: If A, B are recursive relations, then A == B if; and only if, A ~ B. PROOF. Suppose A, B are recursive relations and I : A == B. Then I(x,y) &1 (x, z) --t y = zby definitionI.4.3, and hence that thefunctionf, defined by f(x) = flyl(X, y), is partial recursive. C1early,jis an isotonism, Thus f: A ~ B. Conversely, suppose f : A ~ B. Then by theorem 1.4.2 IX = C'A and f3 = C'B are recursive. If IX or f3 = 0, then the assertion is trivial. Hence we may assume there exist numbers a e IX and b s f3 such that b = j(a). Set

I(X, y)

~ f(a{l...:...

cix)} + xcix)) = (b+ 1) (1...:... cpCy)) + ycp(y)

where cix) = 1 if x belongs to the recursive set y, = 0 otherwise. It is easily verified that I : A == B. This completes the proof. This theorem allows us, when discussing recursive relations, to work with partial recursive functions rather than with predicates. THEOREM 1. 4.5: There exist r.e. relations which are not recursively isotonic to any recursive relation. PROOF. Let

IX

be a r.e. non-recursive set and let

A = {
IX

Hence


~

(3z) (I x- y 1{ If(z)-x 1+ 1y-(x+ 1) I} = 0),

JOHN· N. CROSSLEY

198

and it follows that A is a r.e. relation. Suppose p : A ~ B for some recursive relation B = {
= 0).

Let ex = C'A. If ex is finite there is nothing to prove since then A is finite and hence recursive. Otherwise ex is infinite r.e. and there is a one-one recursive function, g, such that ex = g(J) (cf. [5], p. 73). Since A is linear, (Vx) (Vy) (3z) (f(g(x), g(y), z)

= 0 v f(g(y), g(x), z) = 0).

This is equivalent to (Vx) (Vy) (3z) (j(g(x), g(y), z)· f(g(y), g(x), z) = 0).

(1)

Since A is anti-symmetric, f(g(x), g(y), zo)

= 0 &f(g(y), g(x), Z1) = 0

-+

g(x) = g(y),

and hence x = y and f(g(y) , g(x), zo) = O. Now let B be the relation defined by
+-+

f(g(x), g(y), liz {[(g(x), g(y), z) . f(g(y), g(x), z) = O})

Then B is recursive, by (1), and clearly g : B

~

A.

=

O.

CONSTRUCTIVE ORDER TYPES, I

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II. Addition 11.1. If A and B are arbitrary relations, then the ordinal sum of A and

B is defined by

A

+-

B = A u B u (CA x CB).

If the ordinal sum of two relation types is defined as the relation type of the ordinal sum of arbitrary representatives of the given relation types, then this definition is not, in general, unique. This is because the fields of the representative relations may have non-empty intersection in some cases, depending on the choice of representatives. But if we define the relation type of the sum in terms of representatives which do have disjoint fields, then the definition is unique (cf. [19], pp. 341, 345 * 160.48). Two relations are said to be strictly disjoint if their fields are disjoint. We observe that if A, B are reflexive relations, then A n B= 0

~

C'A n C'B = 0.

Now, in order to define a constructive version of ordinal sum we require "constructive disjointness" i.e. CA and CB must be contained in sets which are "effectively disjoint". If this is not the case, then the following situation arises: let IX be a r.e. non-recursive set and let fJ be a r.e, set containing a. Then there is no (partial) recursive function, defined on Ji which agrees with f(x) = x on IX and g(x) = x + 1 on fJ. Hence there can be no (partial) recursive function defined on C(A +- B) where A = IX2 and B = aZ, although A and B are strictly disjoint. DEFINITION 11.1.1: A is r.e. separable from B if there are disjoint r.e. relations Ai' Bi such that A ~ Ai and B ~ Bi. If A is r.e. separable from B we write A)( B. Note. In general we shall be concerned only with r.e. separability and shall omit the qualification "r.e.".

DEFINITION 11.1.2: Ais recursively separable from Bif there are disjoint recursive relations Ai' B, such that A ~ Ai and B ~ Bi. If A is recursively separable from B we write A

>
THEOREM 11.1.3: (i) A)( B ~ A*)( B*, (ii) A B ~A* B*,

><

><

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JOHN N. CROSSLEY

(iii) A ) ( B -. A)( B, (iv) There exist relations A, B such that A )( B but not A ) ( B. PROOF. (i) ((ii)) follows from the fact that the converse of a r.e. (recursive) relation is r.e. (recursive). (iii) Every recursive relation is r.e, (iv) We call two sets o; P r.e. (recursively) separable if the relations a 2 , p2 are r.e. (recursively) separable. Let !!Z be a consistent incomplete formal system containing formal arithmetic and let

T = {(x, y) : x is (a Godel number of) a proof of the sentence (with GOdel number) y}, To = {x: (3y) (y, x)

8

R)},

R = {
Ro

= {x : (3y) (y, x) 8 R)}.

Then T, R are both (primitive) recursive relations (though not reflexive relations [9], p. 252-5) and To and R o are r.e. sets. Let To = T5 and Ro = R~. Then To and Ro are r.e. and disjoint (since !!Z is consistent), i.e. T o ) ( Ro. By [15], theorem 22 (p. 59), To and R o are not recursively separable. Hence To and Ro are not recursively separable. Let A = To and B = Ro and (iv) is established. II. 1.4: (i) A is r.e. (recursively) separable from B if, and only if, there are r.e. (recursive) sets (Xl' Pl such that C'A S (Xi> C'B S Pl and THEOREM

(Xl

n

Pl

= 0·

(ii) If C'A or C'B is finite, then A)


PROOF. (i) If A is r.e. (recursively) separable from B, then there are r.e. (recursive) relations A l , Bl such that A S Ai> B S B. and A l n B, = 0. Let (Xl = C'A l , Pl = C'B l , then (Xu Pl are r.e, (recursive) by theorem 1.4.2. Since A l , Bl are reflexive, x 8 (Xl +-+
CONSTRUCTIVE ORDER TYPES, I

201

and the fact that every finite set is recursive and so is the complement of a finite set. The second version of part (i) of this theorem is false if the relations are not assumed to be reflexive. For let T, R be the relations defined in the proof of theorem II. I .3. (iv); and suppose that there exist recursive sets " p such that CT s; , and CR s; p where, n p = 0. Then the sets,' = {x : x s C'T and x is (a Godel number of) a single formula} and p' = {x : x e C'R and x is (a Godel number of) a single formula} are recursively separable by r and p, But r = To and p' = R o which contradicts [15] theorem 22. The converse assertion, namely, that if there exist disjoint recursive sets containing the fields of A and B, then A and B are recursively separable, still holds, of course. THEOREM ILl.5: Let (X = CA, /3 = CB; then A)( B if, and only if, there is a partial recursive function, p, such that

f>P:2

(X

U

/3,

pp s; {O, I} (S)

and x e (X u /3 implies x s

(X -

p(x) = 0 .&. x s /3 - p(x) = l.

PROOF. If A )( B, then by the preceding theorem, there are r.e. sets

0(1 :2 0( and /31:2 /3 such that 0(1 n /31

= 0. For arbitrary r.e. set y let

c;(x) be the partial recursive function defined only on y such that c;(x) = 1 for x e y. Then x e 0(1 implies 1 ~ C~,(x) = 0 and x e /31 implies cp,(x) = 1. Let T, be a Turing machine which calculates 1..:... C~, and let

Tp be a Turing machine which calculates Cp,' Further, let T(m, n) = the number (represented) on the tape of the Turing machine T at the m-th step") in the calculation for argument n. Now let a new machine To be defined such that To(m, n) is as follows: (i) If Ta , Tp have not halted before the m-th step for argument n, then To(2m + l , n) = Ta(m + l , n), (ii) If T; has not halted before the (m + 1)-st step and Tp has not halted before the m-th step, then To(2m + 2, n) = TpCm + 1, n), (iii) If 1;. halts at the m-tll step and T p has not halted before the m-th step, then To halts at the (2m + 1)-st step, 1) "Step" does not mean here just one operation of the Turing machine, but a whole phase in the calculation. We assume m ~ 1.

JOHN N. CROSSLEY

202

(iv) If Tp halts at the m-th step and T~ has not halted before the + 1)-st step, then To halts at the (2m + 2)-nd step. 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 P, T~ halts if, and only if, Tp does not. Conversely, let a l = {x: p(x) = O} and Pl = {x: p(x) = I}. Then a l and Pl are r.e. and disjoint; the required result follows from the preceding theorem. (m

THEOREM 11.1.6: If A, Bare r.e. (recursive) relations, then A )( B +--+ A ('\ B

=0

(A ) ( B +--+ A ('\ B

= 0).

THEOREM II.I .7: Any two C.R.T.s have recursively separable representatives. PROOF (v. [8], theorem 9(a)). Let A s A and Be B and let

C

= {(2x, 2y): (x, y) e

A},

D = {(2x+ 1, 2y+ 1): (x, y) s B}. Then C

~

A, D

~

Band C ) ( D.

11.2. THEOREM 11.2.1: Let A l +- B1 ~ A l +- Bz·

~

Al

Al ,

s,

~

a; A l )( s, and A l

)(

Bl , then

PROOF. Let a i = C' Ai' Pi = C'B i (i = 1,2). By hypothesis there exist recursive isotonisms p, q such that p: Al ~ Az and q: B1 ~ Bz. P; (i = 1,2) such that P; = 0. Let Further, there are r.e. sets Pl be the partial recursive function with domain ~P ('\ which is equal to P on ~Pl and let Pl be the partial recursive function with range PPI ('\ which is equal to PI on ~Pl' Let ql be the partial recursive function whose definition is obtained by replacing P by q and a by P in the preceding sentence. Then ~Pz ('\ ~ql = 0 and ppz ('\ pqz = 0. Hence r: A l +- Bl ~ A l +- Bl where r is the partial recursive function which is equal to Pz on its domain and equal to qz on its (disjoint) domain r is one-one since PPl ('\ pqz = 0 and Pz- ql are one-one. The other requirements are obviously satisfied. By virtue of this theorem we can now define addition of C.R.T.s uniquely as follows:

a;,

a;

a; ('\ a;

CONSTRUCTIVE ORDER TYPES, I

203

DEFINITION II .2.2: A +B = CRT(A-tB) whereAe A, Be Band A)( B. Notation. 0

= CRT(0).

We write "A+B" for "A-tB" when A)( B.

THEOREM 11.2.3: (i) A+O = O+A (ii) A + B = 0 +-+ A = 0 = B, (iii) (A+B)* = B*+A*.

= A,

PROOF of (ii). Let A s A, B e B where A )( B. Then A + B = A u B u CAC'B) = 0. Hence A = B = 0.

0

implies

THEOREM II. 2.4: + is associative, viz.for all A, B, C e:Jl, A + (B+ C)

= (A+B)+C.

PROOF. By definition II . 2 . 2 there exist A s A, B s Band C e C such that B )( C and A)( {B+C}. Now the latter implies A )( B and A )( C, hence A+B is defined, {A+B})(C and (A+B)+C is well-defined. We leave the reader to verify that A+(B+C) = (A+B)+C. As in the classical case addition is not commutative in general. 11.3. We can now introduce two relations on the collection :Jl of all C.R.T.s. These relations are reflexive and transitive, i.e. are quasiorderings. Later (§§ III, IV) we shall show that the former of these two quasi-orderings is anti-symmetric on a sub-collection of :Jl and is a partial well-ordering of C.R.T.s of well-orderings. DEFINITION 11.3.1: A :-:;; B if there is a C.R.T. C such that A + C = B. A < B if there is a C.R.T. C i= 0 such that A + C = B. A < B A = {
is not, in general, equivalent to A :-:;; B & A i= B. For let y) : y :-:;; x}, B = A [(J - {O}), A = CRT(A) and B = CRT(B). B are both of classical order type w* and clearly B + 11 = A, = CRT({ <0, O)}). But A ~ B under the map x -+ x+ 1, hence

DEFINITION 1I.3.2:A:-:;;* BifthereisaC.R.T.CsuchthatC+A

=

B.

We shall refer to ":-:;;" as "the ordering by initial segments" and":-:;; *" as "the ordering by final segments".

JOHN N. CROSSLEY

204

THEOREM II.3.3: (i) A ::s:; A, (i)* A ::s:;* A, (ii) o s A, (ii)* 0 ::s:; * A, (iii) A ::s:; 0 ~ A = 0, (iii)* A ::s:; * 0 ~ A = 0, (iv) A::s:; B & B ::s:; C --+ A ::s:; C, (iv)* A ::s:; * B & B ::s:; * C --+ A ::s:; * C, (v) s « C --+ A+B ::s:; A+C, (v)* A ::s:;* B --+ A+C::s:;* B+C. PROOF. (i)-(iii)* follow from theorem II. 2.3, (iv)-(v)* follow from theorem II. 2 . 4.

Corollary II.3.4. ::s:; and ::s:;* are quasi-orderings of9f!. THEOREM II.3.5: There exist C.R.T.s of well-orderings, A, B, say, such that A ::s:; B but not A ::s:; * B. PROOF (as in the classical case). Let A be the natural ordering (by magnitude) on J - {O} and let A = CRT(A). Let 11 be as in § II. 3, then, setting B = 11 , clearly A = B+A and B::s:; A. But we cannot have B ::s:; * A, since A has no last element and C + B has last element 0 for every (separable) C.

11.4. We introduce some notation. o

n

I

i~O

ceO =

Ai = A o,

+

L

1

i~O

n

Ai =

L

i~O

A i+A n + 1 •

0, IY..n = {j(a,m): m < n Sc a e cq,

o .o: = {j(a,n) :neJ&aer:t.}; A.O =

0, A.n = {
A.w = {(j(a, m), j(a', m'» : m < m' & a, a' s C'A .V.m = m'&
A.O = 0, A.(n

+

1) = A.n+A, A.w = CRT(A.w) for A e A.

Part (ii) of the following theorem shows that it is immaterial which element A of A we use to define A. w.

CONSTRUCTIVE ORDER TYPES, THEOREM 11.4.1: (i) A ~ B --+ A.n ~ B .n, (ii) A ~ B --+ A.w ~ B.w, (iii) A . n eA. n, (iv) O.n = 0, (v) A.(m+n) = A.m+A.n, (vi) A.(mn) = (A.m).n, (vii) A. w = A + A .w, (viii) (A.n).w = A.w, (ix) if n > 0, then A .n = 0 +-+ A . to = 0 +-+ A (x) m ~ n --+ A.m ~ A.n, (xi) m

s

m

n

--+

i

I

205

= 0,

n

I

Ai

=0

s I i

Ai'

=0

PROOF. The proofs of the various parts of this theorem are elementary and we only prove parts (ii), (viii) as examples, and leave the other parts to the reader. (ii) Suppose p : A ~ B, then q : A. t» ~ B. w where q(z) = j(pk(z), fez)). (viii) Let A e A and A.w then belongs to A.w by part (ii). Let q(x), rex) be the (primitive) recursive functions such that

x

= nq(x)+r(x) and 0

~

rex) < n

and letp(x) = j(j(k(x), r(l(x))), q(l(x))). Thenp is one-one and (primitive) recursive. We assert that p is relation preserving between A.m and (A.n).m, for

(A.n)m

and where

= {(j(j(a, s), u),j(j(b, t), v) : (u < v .v. u = v & s < t) & a, b e C'A .v. u = v & s = t &
(1)


o~

r, r' < n & nq-s-r < nq' +r' & a, a' e C'A or nq+r

(2)

= nq' +r' &
Condition (2) is equivalent to:

(q < q' .v. q

= q'

& r < r') & a, a' e C'A

.v. q = q' & r = r' &
(3)

206

JOHN N. CROSSLEY

Comparison of (I) and (3) immediately shows thatp is relation preserving. This completes the proof. THEOREM II.4. 2: (i) A ) ( B ~ A . w )( B. w, (ii) (A+B).n+A = A+(B+A).n, (iii) (A+B).(n+ 1) = A+(B+A).n+B, (iv) (A+B).w = A+(B+A).w.

PROOF. (i) Let p be a partial recursive function satisfying the requirements (S) in theorem II. 1.5 for A and B. Now x e C'A. w ~ k(x) e C'A and similarly for B. w. Hence if x s C'A. w u C'B. w, then x s cA. w ~ pk(x) = 0.&. x e C'B.w ~ pk(x) = 1. (ii), (iii). Proof by induction on n using the associativity of addition (theorem 11.2.4). (iv) Let A s A, Be B, a = C'A and 13 = C'B where A)( B. It is easily verified that (A+B).w = (A; O)+C where C

= ~

m<"

m=O

{(B; m) u (A; m+ 1) u U(f3, m) xj(a, n)]

u U(a, m + 1) xj(f3, n + I)]}.

We construct a recursive isotonism p such that p: C ~ (B+A) .w. Suppose ai' 131 are r.e. sets separating A and B (v. theorem II .1.4.(i)). Let p be the partial recursive function defined only on the r.e. set (al'oo u f31.(0)-j(a1, 0) by p(z)=z

ifzef31'00

=j(k(z),I(z)~I) if

ZBa 1.00-j(a1,0).

By part (i) p is well-defined. It is then readily verified that p has the required properties. 11.5.

11.5.1: (SEPARATION LEMMA.) If A = B+ C and A e A, then there are relations B e Band C e C such that B )(C and B+C = A. LEMMA

PROOF. Suppose A e A and A = B+ C. Then there are relations B' s Band C s C such that B' )( C and for some f, f: B' +C ~ A. Let B = J(B') and C = J(C). Clearly, B e Band C s C. By theorem II. 1 .5,

207

CONSTRUCTIVE ORDER TYPES, I

there is a partial recursive function, p, such that Jp;2 C'B' pp ~ {O, I} and if x s C'B' u C'C' then x s C'B' ...... p(x) =

°.&. x s C'C' ...... p(x)

U

C'C',

= 1.

If y s C'B u C'C then, sincefis one-one, there is a unique x s C'B' u C'C' such that y = f(x). Thus pF 1 is partial recursive, J(pf-l) ;2 C'B u C'C, p(pf- l) ~ {O, I} and if x e C'B u C'C then x e C'B ...... pf-l(X) = 0.&. x e C'C ...... pf-I(X) = 1.

Hence by theorem II. 1.5, B)( e. Clearly B+C complete.

= A,

thus the proof is

11.5.2: (DIRECTED REFINEMENT THEOREM.) If A +C = B+D then there is an E such that either A = B+E and E+C = D THEOREM

or

A+E = Band C = E+D.

PROOF. If D = 0, let E = C. Otherwise we may assume D =F 0. Let A 6 A and C 6 C where A )( C. Then by the Separation Lemma (11.5.1) there are relations B 6 Band D 6 D such that B)( D and A +C = B+ D. Let rx = C'A, {3 = C'B, y = C'C and J = C'D. Then rx u y = P u J. Case 1. If J n rx =F 0, let " = rx - {3 and E = D [ n, By construction, rx = {3 and n s: J. Therefore B)( E and E)(e. Now, B ~ A+C and E ~ A+C, further x 6 {3 and y 6" imply y 6 J and
u"

6"

208 THEOREM

E such that

JOHN N. CROSSLEY

II. 5.3: If B. w

=

A + C and C

=1=

0, then there exist n, D,

A = B.n+D, D+E = Band E+B.w = C. PROOF. We consider only the non-trivial case where A, B, C are all non-zero. Let Be B, then by the Separation Lemma (II. 5.1) there exist A e A and C s C such that A)( C and B. w = A + C. By assumption C =1= 0 =1= A, hence there is ace c-c where c = j(b, n') for some b e CB and some n' ~ 0. Therefore 11 = {lea) : a e CA} is a set of natural numbers bounded by n', Let n be the maximum number in 11 and let D = A [j(!3, n), E = C [j(!3, n). Then, as in the proof of the Directed Refinement Theorem it is easily verified that D)( E and A = B.n+D, D+ E = Band E+B.w [{x: lex) > n} = C. (1) We observe that B.w [{x: lex) > n} ~ B.w under the map p : x - j(k(x), lex) ~ (n+ 1» defined only on {x ; lex) > n}. Taking C.R.T.s of both sides of the equations in (1) completes the proof.

Note. There is no obvious link between theorems II.5.2 and II.5.3 for the following reason. We know that B.n+B.w = B.w for all n (by theorem II. 1.4. (vii) and induction). Suppose B. w = A + C, then by theorem II. 5.2 it easily follows that for each n, either A ::;; B. n or B.n s A. If A ::;; B.n for some n, then we are through, but A ~ B.n for all n does not l ) imply A ~ B.w nor is ::;; anti-symmetric on~.

We sum up the properties of ::;; in the following theorem. THEOREM II. 5.4: The relation ::;; is a quasi-ordering on ~ and satisfies the following tree condition

A ::;; C and B ::;; C imply A ::;; B or B

s

A.

By definition II. 3 . 1, if A ::;; C and B ::;; C then there exist D, E such that A+D = C and B+E = C. Hence by theorem II.5.2 there is an F such that either PROOF.

or

A+F

A

= Band

= B + F and

D

=

F+E

(1)

=E

(2)

F+ D

1) For it follows from theorems IV.4. 5 that if B for all n, but A :2: B. w (= W).

=

1 and A

=

V, then A > B. n

CONSTRUCTIVE ORDER TYPES, I

209

In case (I), A s B and in case (2), B ~ A. ~ is a quasi-ordering of fJi by corollary II. 3 .4. III. Quords 111.1. We now commence our study of proper subsets of fJi. DEFINITION 111.1. I : A C.R.T. A is said to be a constructive order type (C.O.T.) if there is an A e A which is a linear ordering. Since C.R.T.s are subsets of the corresponding (classical) relation types, if A is a constructive order type then every relation A e A is a linear ordering. DEFINITION III. 1. 2: A sequence {a i};~ 0 in the field of a linear ordering relation A is said to be an infinite recursive descending chain if the function Aiai is recursive and for all i,
if, it contains no splinter.

PROOF. Let A be a linear ordering and suppose that {gJ;"= 0 is an infinite recursive descending chain in A. Define f, a as follows: a = go. f(n) = g(l+ll y{g(y) = tI})

Then {l(a)};"=

0

(v. [18], p.33).2)

is a splinter in A.

1) This use of the word "splinter" is derived from that in [18).

2) By our convention (v. Terminology and notation) we write gi for the value of

gat t.

JOHN N. CROSSLEY

210

Conversely. suppose that {l(a)}(= 0 is a splinter in the linear ordering A. Let g be the function defined by g(O)

=

a, g(n + 1)

= f(g(n».

Then g is totally defined and computable, hence recursive. I.e. {giL"'= 0 is an infinite recursive descending chain in A. THEOREM III. 1.6: If A is a quasi-well-ordering and f is a one-one partial recursive function such that f: A ~ A then f is the identity map on C'A. PROOF. (This proof is essentially that in [14J. p. 264.) If f =F 1 on C' A then there is an a e C' A such that f(a) =F a. Since A is a linear ordering, either (f(a), a) e A or
DEFINITION III.2.2: A C.O.T. A is said to be a quord if there is an A s A which is a quasi-well-ordering. We write .fl for the collection of all quords. It follows at once from example 111.2.1, that there are some quords which are not e.R.T.s of well-orderings, though we shall show later (§ IV) that the cardinal numbers of quords and of e.O.T.s of wellorderings are the same (namely c, theorem IV. 3.3). We shall see that quords possess many additive and multiplicative properties analogous to those of classical ordinals. THEOREM 111.2.3: A is a quord if, and only if, every A s A is a quasiwell-ordering.

211

CONSTRUCTIVE ORDER TYPES, I

PROOF. By definition III. 2.2 there is a B.s A which is a quasi-wellordering. Let A be any other relation e A, then there is an f such that f: A ~ B. Suppose that A is not a quasi-well-ordering, then there is an infinite recursive descending chain {aJ;"= 0 in A. But then {f(a;)};"= 0 is an infinite recursive descending chain in B since lif(a i) is totally defined. This contradicts our assumption and we conclude that A is a quasi-wellordering. The converse is trivial. THEOREM III. 2.4: (i) 0 e .2, (ii) A = B+C implies A e.2 ...... B, C e.2. (iii) A e.2 ...... (3n) (n "# 0 & A.n e..@) ...... (Vn) (A.n e..@) ...... A.w e.2.

PROOF.

Left to the reader.

Let .91 be a collection of C.R.T.s, then if we define A :s; .RIB to mean (3 C) (C e.91 & A + C = B) then :s; is absolute for quords in a certain sense by the following corollary. Corollary III. 2 .5. If A, B e PROOF.

s. then A s

B

+-+ A

:s; fiB.

Immediate from theorem 1I1.2.4.(ii).

THEOREM

111.2.6:

If A

is a quord, then A

= B+A+C -+ C = O. = B+A+C where C"#

PROOF. Suppose A is a quord and A O. Then there exist quasi-well-orderings A e A, Be Band C s C and a recursive isotonism f such that B+A+C is well-defined and f: B+A+ C ~ A. Since C "# 0, C "# 0 and hence there is an element c e C'C and for this element, f(c) e C'A. Now A)( C, hence C'A () C'C = 0 and c "# f(c). But
ir

Corollary III.2.7. If A is a quord, then A+B

=

A +-+B

= O.

Corollary III.2.8. If A is a quord, then B < A ...... B:s; A & B"# A. Corollary Ill. 2 .9. If A or B is a quord, then A :s; B & B

~

A

-+

A

= B.

PROOF. By hypothesis there exist C, D such that A + C = Band B+D = A. Hence A = (A+C)+D = A+(C+D) and B = B+(D+C) and if A is a quord, then C, Dare quords by theorem III. 2.4. (ii) and C = D = 0 by corollary 111.2.7 and theorem 11.2.3; similarly if B is a quord.

212

JOHN N. CROSSLEY

Corollary III. 2 .10. If A or B is a quord, then A :::;; B & B :::;;* A

-+

A

= B.

PROOF. By hypothesis, there exist C, D such that A + C = Band D+B = A. If A is a quord, then A = D+A+C and by theorem III.2.6, C = 0; hence A = B. Similarly if B is a quord.

Corollary III.2.ll. If A is a quord, then A+B PROOF. Suppose Theorem (II. 5 .2) E + B = C or A + E III. 2. 7, E = 0 and

=

A+C+-+B

=

C.

A + B = A + C, then by the Directed Refinement there is an E such that either A = A + E and = A and B = E + C. In either case, by corollary B = C. The converse is trivial.

Corollary III. 2 .12. If A is a quord, then B< C-+A+B
Corollary III.2.9 establishes that z; is a partial ordering of !2. We shall show later (Theorem IV. 2.6) that :::;; is a partial well-ordering of the collection of all C.R.T.s of well-orderings. :::;; is not a partial wellordering of !2 as is shown by the example below. Example III. 2. 13. Let A be as in example III.2.1 and suppose A = { n} and An = A [tin' Then for all n, An is a quasi-well-ordering. Let An = CRT(An), then In < n -+ Am > An since C' Am contains only finitely many more elements than does C'A n (using theorem II.1.4.(ii» and Am ¥- An by corollary III.2.8. Hence the sequence {AJr'= 0 is a strictly descending infinite sequence of quords, thus !2 is not partially well-ordered by :::;;.

n.

Corollary III. 2. 10 may be regarded as a constructive analogue of the following theorem attributed to Lindenbaum (given in [14], 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." By corollaries III. 2.9 and 10, A = B is equivalent (in !2) both to A ::;; B & B :::;; A and to A :::;; B & B :::;; * A. But it follows from the existence of quords incomparable with respect to :::;; (see below § IV. 4)

CONSTRUCTIVE ORDER TYPES,

I

213

that :-s; * is not anti-symmetric on .£U) For let A, B be two incomparable quords and let C = (A+B).w and D = B+(A+B).w. Then clearly C:-s;* D and D:-S;* C. If C = D, then by theorems II.4.I.(vii) and II 5 . 4 it easily follows that A and B are comparable, contradicting our assumption. 0

IV. Co-ordinals IV.l. In this section we establish some properties of co-ordinals which are the C. R. T.s of well-orderings. We regard classical ordinals as relation types (v. § 1.1) of (denumerable) well-orderings. Hence co-ordinals are sub-classes of the corresponding (classical) ordinals. We use upper case Greek letters for classical ordinals (and variables over the denumerable classical ordinals). Notation. 10 = 0, 11 = CRT { 1, In = Il.n. THEOREM IV .1.1: (i) {In: n s J} £ .2, (ii) In = (i ii) 1m + In = 1m + m (iv) Imon = I mn, (v) 1m + 1m W = I.; t», (vi) if m =I=I- n, Im·w = In·w, (vii) In+A = In+B ~ A = B, (viii) A+ln = B+ln ~ A = B.

I:, 0

°

0

PROOF. We prove only part (viii) leaving the other parts to the reader. (viii) A+ln = B+ln ~ (A+l n)* = (B+l n)* ~ I:+A* = I:+B* (by theorem n.203) ~ In+A* = In+B* (by part (ii)) ~ A* = B* (by (vi)) ~A=B.

DEFINITION IV. I .2: A C. O.T. is said to be finite (or a finite co-ordinal) if it is In for some n. We remark that this definition corresponds to the classical definition of finite sets as sets which are inductive. A search for an analogue of Dekker and Myhill's Isols (v. [8]) proved abortive. 1) This example is based on that in [17] p. 25.

214

JOHN N. CROSSLEY

THEOREM IV. 1.3: Any two linear orderings with fields of the same finite cardinal are totally recursively isotonic. PROOF. Since any two finite sets of the same cardinal can be mapped onto each other in a one-one manner by a recursive permutation (any permutation of the natural numbers which interchanges only a finite number of numbers is recursive), and since every finite linearly ordered set is well-ordered and so is its converse, it follows that any two linearly ordered sets of the same finite cardinal are totally recursively isotonic. IV.2. DEFINITION IV. 2.1: If A is the C.R.T. of a well-ordering, then A is said to be a co-ordinal. We let 'fJ denote the collection of all co-ordinals. If E is a classical ordinal such that E :2 A, then E is said to be the (classical) ordinal of A and we write E = 1 A I. Corollary IV.2.2. If A is a co-ordinal then As; then A = I A I.

1

A

I

and if A is finite

THEOREM IV. 2.3: (i) 0 e -e, (ii) A = B+ C implies A e'fJ ...... B, C s 'fJ, (iii) A s 'fJ ...... (3n) (n "# 0 & A.n s 'fJ) +-+ (Vn) (A.n s 'fJ) ...... A.w s 'fJ. Corollary IV.2. 4. If A, Be 'fJ, then A s B ...... A s 'lB. Thus ~ is absolute for co-ordinals as well as for quords (cf. corollary III.2.5).

LEMMA IV.2.5:

I A+B I = I AI+I

B

I·

THEOREM IV. 2.6: ~ is a partial well-ordering of'fJ and satisfies the tree condition (v. theorem II.5.4). PROOF. As we remarked earlier (§ 111.2) every co-ordinal is a quord, hence ~ is a partial order of'fJ by corollary 111.2.9. Further, ~ satisfies the tree condition by theorem 11.5.4 and corollary IV. 2.4. Now suppose {AJ?= 0 is an infinite descending chain of co-ordinals and let E j = I A j I for each i. Then, by lemma IV. 2.5, {E j }?= 0 is an infinite descending chain of (classical) ordinals under the natural ordering of ordinals. This is impossible, hence ~ is a partial well-ordering of 'fJ.

CONSTRUCTIVE ORDER TYPES,

I

215

Corollary IV.2. 7. If A is a co-ordinal, B:::; A and C:::; A and B = C.

1B I = I C I, then

PROOF. Since :::; is a tree ordering, either B :::; C or C :::; B. Suppose B < C, then there is an E #= 0 such that B + E = C. Therefore, by lemma IV. 2.5, I B I + I E I = I C I and I E I #= 0; thus I B I < I C I where < denotes the classical ordering of ordinals. Similarly we cannot have I C I < I B I· Corollary IV.2.8. If A is a co-ordinal, then &(A)

= {B : B < A} and &+(A) = {B : B :::; A}

are well-ordered by :::;. IV. 3. DEFINITION IV. 3 . 1: A co-ordinal A is said to be infinite if, for some A e A, C'A is infinite. It follows at once that a co-ordinal is infinite if, and only if, it is not finite. THEOREM IV. 3 .2: A co-ordinal A is infinite ff, and only if, In < A for all n. PROOF. Let A e A and suppose that In < A for all n. Then by the Separation Lemma (II. 5 . I), for each n there exist Bn such that An + Bn = A, where

An = {
:

i :::; j

< n} and A = {
It follows at once that C'A 2 {a i

: i s $} and that C'A is infinite. Conversely, if A is infinite, then using theorem II. I .4. (ii) one easily shows that In < A for all n. We leave the details to the reader.

Notation. By virtue of theorems IV. 1. I and IV. 3.2 we now write "n" for "In" and "$" for "{In: n is a natural number}" where there is no danger of confusion.

THEOREM IV.3.3: (i)$ c Cfl c fl c!3£, (ii) The cardinalities ofCfl, fl,!3£ are all c (the cardinal of the continuum). PROOF. (i) Every finite linearly ordered set is well-ordered, hence

JOHN N. CROSSLEY

216

J! s;

C{}. There exist infinite well-ordered sets, hence J! #- rr5. By example III.2. 1 there exist quasi-well-orderings which are not well-orderings but every well-ordering is a quasi-well-ordering, hence C6' c fl. Finally, let W* be the converse of the natural ordering of the natural numbers and let W* = CRT(W*). Clearly, W* is not a quord. Hence fl c !Jf. (ii) The cardinality of !Jf is :0;: 2N~ since any C.R.T. is an equivalence class of subsets of J!2. But 2N~ = c; thus in order to prove (ii) it suffices to prove that the cardinality of rr5 ~ c. Now every equivalence class of well-orderings contains at most ~o well-orderings since there are only No recursive isotonisms, Further, there are at least c distinct well-orderings of subsets of J!. Hence, if x is the number ofelements ofrr5, then ~o. x ~ c and it follows, using the axiom of choice that x ~ c. This completes the proof.

As in the classical case subtraction does not playa major role, but we introduce the notion now for notational convenience. By corollary III. 2. 11, if A = B + C then C is uniquely determined by A and B; it follows by theorem IV. 3 . 3 that the same is true for co-ordinals. Hence the following definition gives a unique value for A - B (which by Theorem IV. 2. 3(ii) is a co-ordinal if B, A are co-ordinals). DEFINITION IV. 3 .4: If A ~ B and (A and) Bare quords, then A - B is the unique C such that A = B + C. THEOREM IV. 3 . 5: If A, B, Care quords, then (i) A-A = 0,

(ii) (A+B)-A = B, (iii) if B :0;: A, then B+(A-B) = A, (iv) if A+B :0;: C, then C-(A+B) = (C-A)-B. PROOF OF (iv). A + B

:0;:

C

-+

(ElD) (C = A + B + D), hence

C-(A+B) Also, C-A

= B+D and

=

D.

therefore

(C-A)-B = D. THEOREM IV. 3 .6: If I A I is a successor number A + m, where A is a limit number, then for each n there is a unique BII which is comparable with A and of classical ordinal A+n; further, B; = A ± I m-n I (where

CONSTRUCTIVE ORDER TYPES,

I m - n I is the modulus of m - n and the as En < A or En ;:::: A).

+ or

217

I

- sign is taken according

This theorem follows at once from the fact that if A e A then A has a final segment of type m which is finite, and hence, by theorem II. 1 .4. (ii), separable from its complement in A. We leave the details to the reader. It follows from this theorem that every co-ordinal has a unique successor. We shall show later (§ V) that limits of strictly increasing sequences of co-ordinals are never uniquely determined by such sequences without other conditions. IVA. THEOREM IV .4.1: For each limit number A there exist ceo-ordinals

of ordinal A.

PROOF. There are c distinct infinite subsets of .Y. Let each of these be well-ordered with ordinal A (this is possible since A is denumerable). Then these c subsets are spread among, say, x equivalence classes containing at most No members each since there are only No recursive isotonisms. Hence No.X = c and therefore (using the axiom of choice)

x = c.

Corollary IVA. 2. There are c co-ordinals

V~

such that

I V~ I = w.

Notation. W = {
v = {
=

CRT(V).

DEFINITION IV. 4.3: W is said to be the standard well-ordering of type w, W is the standard w-co-ordinal; V is called the generic counterexample.i) THEOREM IV .4.4: (i) 1 + W = W, (ii) 1 + V"# V. PROOF. (i) It is easily verified that W = I. w, hence by theorem 1I.4.1.(vii) with A = 1, 1+ W = W. (ii) Suppose 1 + V = V. Since p is r.e. non-recursive, p is non-empty, 1) Since most of our counterexamples are based on V.

218

JOHN N. CROSSLEY

say ao s p. Therefore there is a recursive isotonism

f such that

»:m

f: {(ao, ao)}+V ~ V. Hence V = {(r(ao),f"(a o

~

n}

and g : W ~ V where g(n) = I"(ao). But then g enumerates p in order of magnitude, and it follows by Post's lemma ([13], p. 291) that p is recursive, contradicting our choice of p. Corollary IVA.5. V and Ware incomparable.

PROOF. By the theorem V '" W. But V < Wor W < V implies Vor W (respectively) is finite. This corollary shows that there exist incomparable co-ordinals, and hence, that there are incomparable quords. It therefore completes the demonstration (end of § 111.2) that ~ * is not anti-symmetric (even on ..2). THEOREM IV.4.6: There exist co-ordinals A, B, C such that A < B but A+C $ B+C. PROOF. Let A = I, B = V and C = W. Then by theorem IV.3.2, A < B, and by theorem IV.4.4.(i)A+C = C. If A+C ~ B+C, then C ~ B + C and B ~ B + C. Hence by theorem IV. 2. 6, Band Care comparable which contradicts corollary IV.4. 5. We now consider important classes of co-ordinals for which the law (+) A < B - A + C ~ B+ C does hold. In fact, if A, B, C are predecessors of the same principal number for addition then (+) holds. Classically, a principal number for addition, otherwise called a y-number ([I], p. 67) or a prime component ([14], p. 279), may be defined as an ordinal II '" 0 satisfying one of the three (equivalent) conditions (l C)-(3C) below. r
r,

+ L1 = II L1

-+

L1

=0

or L1

< II - r + L1 < II

= II

We consider constructive analogues of these conditions, viz.: B < A - B+A = A B + C = A -+ C = 0 or C B, C < A _ B + C < A

=

(I),

A

(2), (3).

CONSTRUCTIVE ORDER TYPES, I

219

THEOREM IV. 4.7: If A is a co-ordinal #- 0, then (i) (1) +-+ (2), (ii) (1) ~ (3), (2) ~ (3), (iii) (3) -+-+ (1), (3) -+-+ (2). PROOF. (i) Suppose B+C = A and (1) holds. Then if C #- 0, B < A, and by (1), B + A = A. Hence by corollary III . 2 . 11, A = C. Conversely suppose (2) holds and B < A. Then there is a C#-O such that B+ C = A and by (2) we have C = A, hence B + A = A. (ii) Suppose (1) holds and B, C < A. Then B+A = A and C+A = A, hence (B+C)+A = B+(C+A) = B+A = A and since A #- 0, B+C< A. That (2) ~ (3) follows from (i). (iii) It suffices to prove (3) +} (1). Let A = V, then B < A implies B is finite, hence (3) holds for V. Since 1 < V, if (1) held we would have 1 + V = V contradicting theorem IV. 4.4. (ii). This completes the proof.

°

DEFINITION IV. 4.8: A co-ordinal A is said to be a principal number for addition if A =1= and B < A ~ B+A = A. If A = 1, then A is called an improper principal number for addition and if A#-1 then A is called a proper principal number for addition. We write£'( +) for the collection of all principal numbers for addition.

THEOREM IV. 4 .9: Every proper principal number is a co-ordinal whose classical ordinal is a limit number. PROOF. Clearly, no finite co-ordinal is a proper principal number. Suppose A is a proper principal number for addition and I A I = A + m, where A is a limit number and m is finite. Then by theorem IV. 3 .6 there is a co-ordinal u; < A such that I Bo I = A. Hence I B o + A I = A.2+m> I A I and consequently Bo+A #- A. THEOREM IV.4.lO: If P e£'( +), then P.w e£'( +) and P < P.w. PROOF. Suppose P is a principal number for addition and A < P. w; then, by theorem 11.5.3, there is an n and a D such that A = P.n+D, where D < P. Hence A+P.w = A+(P+P.w) [by theorem 11.4.1. (vii)] = (A+P)+P.w = (P.n+D+P)+P.w = P.(n+l)+P.w [sinceP

220

JOHN N. CROSSLEY

is a principal number for addition and D < P] = P. co [by (n + I) applications of theorem II.4.I.(vii)]. Hence P.we£(+). Clearly, P < P.w since P "* O. THEOREM IV 04.11: (i) If P is a principal number for addition and A, B, C < P, then A < B--+ A+C s B+C. (ii) Similarly under the hypothesis that A, B, C

s

P.

PROOF. (i) By theorem IV.4.7.(ii), A < B--+ A+C < P&B+C < P. Therefore, by theorem II. 5 . 4, A + C and B + C are comparable. But, classically, ep < 'JI --+ ep + r ~ 'JI+ r, hence by lemma IV. 2.5, A + C s B + C. (ii) follows at once from (i) using P. co instead of P and theorem IVA.IO. Corollary IVA .12. If A, B s P and P is a principal number for addition, then B s A + B. Now we prove that any (non-zero) predecessor of a principal number for addition is uniquely expressible as a finite sum of non-increasing principal numbers for addition.')

THEOREM IV. 4. 13: If 0 < A < P e£ ( +), then there exist principal numbers for addition C l' . . . , Cn such that P > Cn 2: Cn _ 1 •.. 2: C 1 and A = Cn + ... +Cl • Further, if A = Cn + ... +C1 and A = D m + ... +D l are two decompositions such that P 2: Cn 2: Cn - 1 . . . 2: C 1 and P > D m 2: D m _ 1 . . . 2: D l and all the C, and D, are principal numbers for addition, then n = m and for all r S n, Cr = Dr' Conversely, if A is expressible as Cn + ... +C l where C; 2: Cn _ 1 2: ... C l and all the Cr are principal numbers for addition, then there is a principal number, namely,

c..«

2: A.

PROOF by transfinite induction with respect to the partial well-ordering S. We assume 0 < A < P e£( +) and take as induction hypothesis: If 0 < B < A, then B is uniquely expressible as a finite sum of principal numbers < P. If A is a principal number for addition, then there is nothing to prove. 1)

This theorem was conjectured by A. L. Tritter.

CONSTRUCTIVE ORDER TYPES, I

221

Now suppose A is not a principal number, then there exist B, C such that B+ C = A, where C "# 0, A (and hence B "# 0).

(4)

By corollary IV.4.l2, C < A. Let C 1 be the least C satisfying (4) (i.e. under the ordering by initial segments). We now show that C 1 is a principal number for addition. Suppose C 1 = D + E, then by corollary IV.4 .12, E < P and hence by theorem 11.5.4, C 1 and E are comparable. But 1E 1 ~I C 1 I, hence E s C 1 and by the minimality of C b E = C 1 • Thus C 1 is a principal number by theorem lV.4.7.(i). Now let B 1 be the least B such that B+C 1 = A. Then if B 1 = 0 we only have to prove uniqueness, and otherwise by the hypothesis of the induction, B 1 has a (unique) decomposition B = Cn + + ... +C z where P> C, ~ ... ~ C z and all the Cr(r = 2, ... , n) are principal numbers. Hence A = Cn+ ... +C 1 and, since C 1 < P, all the C, (r = 1, ... , n) are comparable. Suppose C z < C 1 , then by the definition of a principal number for addition, Cz + C 1 = C l' hence A = (C n+ ... +C Z)+C 1

= (C n+··· +C 3)+C 1 •

Now C z "# 0 -+ Cn+ ... +C 3 < Cn+ ... +C z = B 1 • But B 1 was chosen as the least B such that B+C 1 = A. We therefore cannot have C z < C 1 and must have Cz ~ C L: Thus A = C, + ... + C 1 is a decomposition of the required type. As regards uniqueness, letA = Cn+ ... + C 1 and A = Dm+ ... +D 1 be two decompositions of A as a sum of non-increasing principal numbers. By theorem 11.5.4, C, and D m are comparable. Suppose Cn > D m , then D m + C n = C, since C n is a principal number for addition. Therefore A = D m + C n + ... + C 1 and by substituting D m + C, for C, m times more we obtain A = Dm.(m+ 1)+A which implies Dm.(m+ 1) < A. Now if i ~ m, then D;+D m = Dm or Dm.2 according as Di < Dm or D i = Dm. Therefore Dm.(m+l)+A = A < A+Dm.m ~ Dm.2m and hence, by corollary JIL2.11, A ~ Dm.m. This contradicts Dm.(m+l) < A and we therefore cannot have D m < Cn" Similarly, Cn -{: D m and we conclude C; = Dm . Now by corollary 111.2.11 it follows that

222

JOHN N. CROSSLEY

Repeating this argument the minimum of m and n times and letting s be this minimum, we obtain C, r = Dm _, (r = 0, ... , s) and hence either Ct+ ... +C t = OorDt+ ... +D t = 0 where t = 1m - Ill. By theorem 11.2.3. (i i) it follows that t = 0 and hence that n = m and C, = D, for every r. Conversely, if A = Cn+ ... +C t , then as for Dm above, A+Cn.n :::;; :::;; Cn . 2n and hence by theorems IV.4 .10 and 11.4.1. (vii) it easily follows that A < Cn • w which is a principal number for addition. This completes the proof. -r

This theorem is not an immediate corollary of theorem 2, p. 280 in [14] for the following reasons: (i) it may be the case that I P I is a classical principal number while P is not a principal number, e.g. V, (ii) P may be a principal number but I P I may not be a classical principal number (see § VIII .1) and (iii) comparability conditions have to be established.

IV.5. By theorem IV .4. 1 above there are c co-ordinals corresponding to each limit number (and these co-ordinals are therefore incomparable with each other) but there are some limit number co-ordinals which have no predecessors of some smaller ordinal. More formally: Let A, B, C range over co-ordinals, over classical ordinals, then

e

(3A) (3B) (3 e)

(/

A

& (V C)

I = r & IB I = A & A < B & r < e < A (I C 1"1= e v C 1: B v A 1: C».

This is shown by example IV. 5.1 below. If, however, we restrict ourselves to recursive co-ordinals then this situation does not arise. We hope to present the results for recursive co-ordinals in [4]. Example IV. 5 .J. P is as given in § IV.4. Let T be the well-ordering of type w. 2 defined by x e p & yep v xc y e p &x:::;; y v x, yep & x :::;; y.

Let T = CRT(T). Suppose T = V + V' where I V I = I V' I = w, then by the Separation Lemma (II. 5 .1) there exist relations U, U' such that U )( U' and T = U + U'. Hence C'U and C'U' are contained in disjoint

CONSTRUCTIVE ORDER TYPES, r.e. sets (x, p. But this implies trary to the choice of p.

(X

223

I

= p & P = P and that p is recursive, con-

e

We observe that if the condition on above is satisfied for some successor number l ' then by theorem IV. 3 . 6 it is satisfied for some limit number 2 • On the other hand we do have c co-ordinals which have predecessors representing all ordinals less than that of the given co-ordinal. This is the content of the representation theorem below. We shall use the following classical theorems in proving the representation theorem.

e

e

THEOREM IV.5.2: ([14], p. 379, theorem 1.) Every denumerable ordinal which is a limit number is the limit ofa strictly increasing sequence, of type ill, of ordinals less than the given number. i

THEOREM IV. 5.3: ([14], p. 264, corol/ary 3.) If A and B are isotonic weI/orderings then there is an isotonism f such that every isotonism between A and B is an extension off. THEOREM IV. 5.4: (REPRESENTATION THEOREM.) Let F, ..1 range over (denumerable) classical ordinals, C, D over co-ordinals, then

('IT) (3C)

(I C 1=

r &('1..1) (..1


->

(E!D)

(I

D

1=

L1 & D < C))).

PROOF BY TRANSFINITE INDUCTION. The assertion is trivial if T = O. We assume the assertion holds for all ordinals less than T, If r = e + 1, then by the hypothesis of the induction there is a co-ordinal T such that

I T I = e & ('1..1) (..1 < e

->

(ElD)(1 D

I = ..1 & D <

T)).

Then by theorem IV.3.6, T < T+ I and D < T+ 1 -> D s T. Let C = T + 1, then by corollary IV. 2.7, it easily follows that C has the required properties. If r is a limit number, then by theorem IV. 5.2, F is the limit of a strictly increasing sequence {4>;} j < w of ordinals. We may assume 4>0 #- O. Put II 0 = 4>0' Il, + 1 = 4>j + 1- 4>j (by [14], p. 275, Il, is well-defined). Then

By the hypothesis of the induction, for each i there is a P, such that

224

JOHN N. CROSSLEY

I Pi I = IIi & ('v'A)(A < IIi

(E! D)(I D I = A & D < Pi»'

-+

(5)

Using the axiom of choice, choose a fixed Pi in Pi (such that 0 a CP i for each i 1 Now define

».

C

= {
< n

.v. m = n & a Pm}

and C = CRT(C). Clearly,

L

IC I = Now suppose A

<

i
r, then for

IIi =

r.

some n,

A <

n

L IIi i; 0

and we may assume that n is minimal. Therefore A

where e and T <

n- 1

= L u.s o i; 0

< II n- From (5) it follows that there is a T such that I T I =

r; Let D =

n - 1

e

n

L Pi + T, then I D I = A and D::;; L0 i; 0 i;

Pi'

Since ::;; is a tree-ordering, in order to complete the proof it suffices to prove that, for all n, n L r;« C. i; 0

Let Pen) = C[{x: lex) ::;; n}, then it is easily verified that

Pen) a

n

L Pi' i; 0

Further, let pen) = C[{x: lex) > n}. Then p(n»( pen) since if x a CP(n) U cp(n) (= CC), then xaCP(n)+-->l(x)

s:

U

n .&. xaCp(n)+-->l(x) > n.

Hence, if p is the partial recursive function sg (l(x)-=- n), then p satisfies 1) We shall use this auxiliary condition in the proof of corollary IV. 5.5.

225

CONSTRUCTIVE ORDER TYPES, I

the conditions in theorem II .1. 5. Hence p(n) + p
i

L= 0 Pi <

C

and the proof is complete. Corollary IV.5 .5. There are ceo-ordinals C A for each ordinal T such that I C A I = rand

('ILl) (Ll < PROOF.

r -+ (E! D)( I D I =

Ll & D < C A ) ) .

~

w

(6)

Case 1. If F is a limit number.

Let VA be a co-ordinal such that I v~ I = W & VA i= W, by corollary IV.4.2 there are c such co-ordinals. Let YA e VA and suppose YA = {
C' = {G(p,

» : P e eP

vm ) , j(q, vn

m

& q s eP n & m <

. v. m = n &
/I

(7)

and C' = CRT(C'). As before, I C'I = rand (6) holds with C' replacing CA' Clearly, C '" C' under the map f: x -+ j(k(x), v/(x» and hence, by theorem IV. 5.3, every isotonism between C and C' is an extension off. Therefore, if C = C', then g : C ~ C' for some partial recursive extension of f. In particular, gj(O, m) = j(O, vm ) for every m and hence the map m -+ Vm is partial recursive. This contradicts our choice of VA' and we conclude C i= C'. Similarly, if C" is obtained from Vn then C i= C" and C' i= C" since the former implies that the map m -+ U m is partial recursive and the latter that the map Vm -+ u.; is partial recursive, where Yn = {
Suppose T = e + n where e is a limit number. Then by case 1, there exist c co-ordinals LA such that I LA I = e and (6) holds with LA re-

226

JOHN N. CROSSLEY

placing CA' Let C A = LA follows for this case also.

+ n, then

by theorem IV. 3 . 6 the conclusion

We observe that the limit of a sequence of recursive co-ordinals (coordinals containing a recursive well-ordering) is not uniquely defined either, since by theorem 1. 4. 6 the generic counterexample is a recursive co-ordinal and using this Vas the VA of the corollary proof we obtain a C' "# C. In the case of recursive co-ordinals, however, there are only ~o distinct "limits".') V. Bounds

V.I. Since the ordering by initial segments is a partial order on f2 and on l(/ we can define upper and lower bounds in the usual way. We use the techniques developed in the previous section in order to show that there are no non-trivial upper bounds for collections of quords or co-ordinals. DEFINITION V .1.1: A quord B is said to be a lower (upper) bound for a collection of quords, d, if Qed implies B s Q (B ~ Q).

V. I .2: A quord B is said to be a greatest lower bound (least upper bound) for a collection of quords, d, if B is a lower bound (upper bound) for d and every lower bound (upper bound) for d is ~ B (is ~ B). DEFINITION

By the anti-symmetry of ~, least upper bounds and greatest lower bounds are unique if they exist at all. (The proof of the following lemma is based on the idea in the proof of theorem 4lb in [8].) LEMMA

decessors.

V. I .3: A quord has at most denumerably infinitely many pre-

Let A be a quord and A s A. For fixed A we show that every r.e. set determines at most one predecessor of A and that every predecessor of A determines at least one r.e. set. The lemma follows at once from these results. Let 13 be a r.e. set, then 13 determines at most one predecessor of A as follows: Let B = A [13, then B = CRT(B) ~ A only if there is a r.e. set PROOF.

1) Since there are only ~o recursive co-ordinals.

227

CONSTRUCTIVE ORDER TYPES, I

Y such that B)( A [y and B+A [y = A (using the Separation Lemma II. 5 .1). On the other hand, if B ~ A, then there is a B such that B e Band B is contained in some r.e. set p separating B from A [ (C'A - C'B). This completes the proof of the lemma. LEMMA V.I. 4: A denumerable collection of quords has an upper bound if, and only if, every two members of the collection are comparable. PROOF. Let d = {Ai: i e J'} be a collection of quords. If there is an upper bound, U, for d, then A; ~ U for all i and by theorem II. 5.4, for all i, j, either A; ~ A j or A j ~ Ai' Conversely, suppose A; and A j are comparable for all pairs i, j. We

may assume that there is no maximum A; since the assertion is trivial in that case. We now set Bo = A" B; + 1 = A /(;), where r = Jls{ As =F O}, t(i) = Jls{A s > B i } · Clearly, i < j --+ B, < B j • Hence the C i , defined by Co = B o, C i + 1 B; + 1 -B;, are all non-zero. For each i, let C; be a fixed representative of Ci such that 0 e C'C;. Now let

U

=

{(j(e, m),j(d, n»: e s C'C m & d e C'Cn & m < n . v. m = n & (c, d) e Cm}.

Further, let U = CRT(U), Yn = {x: lex) ~ n}, Urn) = U [Y(n) and urn) = U [~n' We shall prove:

~n

=

{x: lex) > n};

1) urn) =F 0, 2) Urn) )( urn),

3) for all n, Urn) s e; 1) urn) =F 0 since m > n --+ (j(0, m), j(O, m» e urn) by construction and the choice of the C; 2) For each n, x s C'U implies x e C'U(n) ~ sg (l(x) -=-n) = 0 & x e C'u(n) ~ sg (l(x) -=-n) = 1. Hence by theorem II .1.5, 2) holds. 3) U(o) = (Co; 0) ~ Co s Co = B o. Now we assume Urn) e B; and prove Urn + 1) e B n + i -

228

JOHN N. CROSSLEY

By construction, (C n+ 1 ; n+1) <:; Urn»)( Urn)' Hence Urn)+ ( C n+ 1 ; n+1) = T, say, is well-defined and T s Bn+Cn = B n + r- But T = Urn +1) by construction of U and 1'n + r- Hence, for all n, Urn) e Bn" This proves 3)1). Using 1) it follows at once that B; < V for all n and it only remains to prove that V is a quord. Suppose U(Xi' ninr'= 0 is an infinite recursive decending chain in U. Then by the definition of U, {ni: i <; J} has a maximum; let this be n. Then the given chain is also an infinite recursive descending chain in U(n) which is impossible, since U(n) s B; and B; is a quord. Thus the proof is complete.

v.2. An analysis of the proof of corollary IV. 5. 5 shows that there exist c incomparable limits to certain increasing sequences of co-ordinals. We now prove the stronger result that any increasing sequence of coordinals without a maximum has c incomparable co-ordinals whose classical ordinals are all equal to the limit of the classical ordinals of the given sequence. This will be a corollary of the next theorem. THEOREM V. 2. l: A collection of quords has a least upper bound if, and only if, it has a maximum.

The "if" part is trivial. Now suppose that sf is a collection of quords without a maximum, but with a least upper bound L. By lemma V. 1.3, L has at most denumerably infinitely many predecessors, hence sf is at most denumerably infinite. And sf is not finite since sf has no maximum. In order to prove the theorem we construct two incomparable upper bounds V and U' which are, in a certain sense, minimal upper bounds and obtain a contradiction. Since sf is denumerable we construct U and V exactly as in the proof of lemma V. 1.4. Let V be a well-ordering in the generic counterexample, say, V = {
1) We are here using the fact that, for any finite set of one-one partial recursive functions with mutually disjoint domains and ranges, there is a one-one partial recursive function which agrees with each member of the given set on its respective domain.

CONSTRUCTIVE ORDER TYPES, I

229

V.l.4 easily yields that V' is also an upper bound for d (see especially footnote.') p. 224), the details of this verification we leave to the reader. We now prove A < V

-+

(3n) (A < En)'

(I)

Suppose A < V, then there exist relations A, D, such that A)( D, A + D = U and D :1= 0 (by the Separation Lemma 11.5. 1 and corollary 111.2.8). Hence there is a number j(x, m) e CD for some m. Clearly, CA ~ {x : l(x) ~ m}, hence, using the same notation as in the proof of lemma V.l.4,

U = A+D [{x: l(x)

s

m}+um.

Taking C.R.T.s we obtain A s Em and if n = m+ 1, then A < En since c, :1= 0. This proves (1). Similarly one proves (I) with U' replacing V. It follows at once that L ~ V and L ~ U', Thus in order to complete the proof we only need to establish V:I= V But V = U' implies there is a recursive isotonism g such that g : U ~ U'. But U' = l(U) and hence U = g[(U). Hence by theorem III .1. 6, gf = 1 on CU. But this implies the map h : j(O, n) -+ j(O, vn) defined only on numbers of the formj(O, n) has a partial recursive inverse, namely g, which implies that h is partial recursive contrary to our choice of V. Thus the proof is complete. f

•

Corollary V. 2 .2. A collection of co-ordinals has a least upper bound if, and only if, it has a maximum. Corollary V.2. 3. Let d be a collection of co-ordinals with no maximum, but such that all its members are comparable. Further, let lim

A.JII

IA I =

A,

then there exist c incomparable upper bounds V4> such that PROOF.

I V4> I =

A.

Clearly,

IV I=

lim

A.JII

IA I

by the construction of V. We leave the reader to verify that c upper bounds

230

JOHN N. CROSSLEY

U~ can be constructed from the c incomparable co-ordinals corollary IV. 4.2 (cf. proof of corollary IV. 5.5).

V~

given by

V.3. Since 0 ~ A for every quord A, every collection of quords has a lower bound. There exist collections of quords with a greatest lower bound but no minimum as the following example shows. Example V. 3 . 1. Let U be a co-ordinal such that I U I = wand there is a U e U such that C'U is immune. Then, clearly U =F W. U - n is well defined for all n and if m =F n, then U- m =F U- n since otherwise U = r+ U for some r. We shall show later (Lemma VIII.l.4) that this last equation implies U =F W. However, U'
PROOF.

Immediate from theorem 11.5.4.

The converse of this theorem is false. For example, let ir be the collection of all co-ordinals of classical ordinal ro, then every finite co-ordinal is a lower bound for ir but there is no greatest lower bound for the ordinal of any greatest lower bound would be w. VI. Multiplication VI. 1. From now on we shall be principally concerned with co-ordinals and unless otherwise stated all C.R.T.s mentioned will be assumed to be co-ordinals. We give a natural definition of multiplication of C.O.T.s in this section and show that most of the [analogues of the] basic classical laws hold for co-ordinals. There is one striking breakdown, namely in the case of the law (1) A < B -+ AC s BC

CONSTRUCTIVE ORDER TYPES, I

231

which we shall show fails for some co-ordinals. If, however, A, B, C are all predecessors of a principal number for multiplication, then (I) does hold. Notation. A.B = {
THEOREM VI. 1. 1: If A, B are reflexive relations (linear orderings, quasi-well-orderings, well-orderings) then A. B is a reflexive relation (linear ordering, quasi-well-ordering, well-ordering). PROOF. All except the case of quasi-well-orderings follow at once from the classical definition of multiplication of relations (cf. [14J, p.229). Suppose A and Bare quasi-well-orderings and that A. B is not. Then there is an infinite recursive descending chain in A. B. Since every element ofthe field of A. B is of the formj(a, b), this chain must be ofthe form {jean> bnn:,= 0 where an e C' A and b; s COB. Let IX = {an} and f3 = {b n } , then there are four cases to consider: and f3 are both finite, infinite and f3 is finite, (iii) IX is finite and f3 is infinite, (iv) IX and f3 are both infinite. (i) (i i)

IX

IX is

(i) is impossible since then {jean> bn)} would only contain a finite number of elements. (ii) Since f3 is finite, there is at least one number be f3 for which {jean, b): an e IX} is infinite. Let the distinct an in this set be a(n;) (i = 0,1, ... ) where i < j ~ n, < nj • Then {j(a(n), b)}?= ° is an infinite recursive descending chain in A. B since

a(no) ain,

+

= a o,

1) = a(Jls{r < s

--+

a, # as})'

It follows at once that {a(ni)}~= 0 is an infinite recursive descending

chain in A which is a contradiction. (iii) This case is dealt with a manner very similar to (ii). We omit the details.

JOHN N. CROSSLEY

232

(iv) Let {b(n)};x;"o be the set of distinct b., where i <} ~ n, < nj' Then every ben) occurs in {b n };:'= 0 at most finitely many times for the following two reasons: 1. If b, = ben) for some fixed i and all} greater than some }o, then there are only finitely many distinct bn> namely, those occurring in b o, ... , b(jo)· 2. If i <} and
This is impossible since B is a quasi-well-ordering. This completes the proof of the theorem. THEOREM VI. 1.2: If Al ~ A2 and B1 ~ B2 , then Ai' B1 ~ A2 · B2 • PROOF. Suppose p : A 1 ~ A2 and q : B1 ~ B2 , then r: Ai' Bi ~ A2 . B2 where rex) = j(pk(x), ql(x». THEOREM VI. 1.3: (A 1.A2).A3

~

A1.(A2.A 3 ) .

PROOF.
j(j(a~,

a;), a;)

& ai' a; e C' Ai (i = 1, 2, 3)

.&:
x = j(a l,j(a2, a 3» & y = j(a~,j(a;, a;» & a.; e C' Ai (i = 1,2, 3)

a;

.&: e A 2 & a 2 i: a; · v. j(a 2, a3) = j(a;, a;) &
I

(2)

I

(3)

CONSTRUCTIVE ORDER TYPES, I

233

Since j is one-one, conditions (2) and (3) are equivalent and the proof is completed by using the recursive isotonism x

-+

DEFINITION VI. 1. 4: A. B

j(kk(x), j(kl(x), lex))).

=

CRT(A. B) where A s A and Be B.

Theorem VI. 1. 1 guarantees the uniqueness of this definition. (We often write "AB" for "A .B".) Corollary VI.I.5. Multiplication is associative, i.e. (A.B).C A.(B.C).

=

By virtue of this corollary we may omit brackets in a product of several C.O.T.s. THEOREM VI. 1.6: A. B = 0 +-+ A = 0 v B = O. PROOF. j(a, /3)

=0

+-+ a

=0

v f3 =

0.

THEOREM VI. 1. 7: A(B + C) = AB + A C. PROOF. It is sufficient to establish that separability conditions are satisfied, since the proof that the order type is the same on both sides of the equation is proved exactly as in the classical case. Let A e A, B e Band C e C, then x s C'A. B +-+ k(x) s C'A & lex) 8 C'B

and

x e C'A.C

+-+ k(x)

e C'A & lex) e c-c.

Hence by theorem II .1.4.(i) A.B)( A.C

+-+ A =

0

v B)( C.

THEOREM VI.I.8: (i) For all n, A.In = A.n, (ii) A. W

= A.

PROOF. (i) If n = 0, then A.In = 0 = A.n. If n = 1, then A.I = A (by definition § 11.4) and A.I1 = CRT {(j(x, O),j(y, 0) : x, y e C'A & 0

= 0 & (x, y) s A}

where CRT(A) = A. If n > 1, then A.In = A.{Il.n) = (A.I1).n = A.n by the first part of the proof and corollary VI. 1.5. Hence, for all n, A . In = A. n. (ii) A. W = A.{Il'OJ) = (A.I1).OJ = A.OJ.

234

JOHN N. CROSSLEY

VI. 2. By analogy to principal numbers for addition, we now introduce principal numbers for multiplication (v. [1], p. 66). DEFINITION VI. 2. 1: A co-ordinal A is said to be a principal number for multiplication if A i= 0, 1 and

°<

B < A

-+

BA

= A.

If A = 2, then A is called an improper principal number for multiplication, and if A i= 2, then A is called a proper principal number for multiplication. We write £(.) for the collection of all principal numbers for multiplication. THEOREM VI. 2.2: Every proper principal number for multiplication is a co-ordinal whose classical ordinal is a limit number. PROOF. Left to the reader (cf. theorem IV.4.9). As in the classical case, B < A -+ BA = A is a stronger condition than BC = A -+ B = A v C = A. But also, the former condition is stronger than B, C < A -+ BC < A for co-ordinals. For the generic counterexample V satisfies this last condition but is not a principal number for multiplication since 2. V i= Vas we shall show later (lemma VIII.2.4); in fact we show that if n.A = A for any n, then A = W. It follows at once that W is a principal number for multiplication. (Alternatively, that W is a principal number for multiplication follows immediately from theorem II.4.I.(viii).) We now establish analogues of classical laws for multiplication (of co-ordinals) and show that these all go through if the co-ordinals concerned are all predecessors of the same principal number for multiplication. THEOREM VI. 2.3: (i) If B i= 0, then A ~ AB, (it) If B > 1, then A < AB whenever A i= 0. PROOF. We prove only (it) leaving (i) to the reader. (it) B > 1 -+ (E!C) (B = I+C & C i= 0). Hence AB = A (I+C) = A+AC where AC i= 0 if A i= O. Thus A < AB. THEOREM VI.2.4: If A i= 0 and A, B, C are co-ordinals, then AB = AC -+ B = C.

CONSTRUCTIVE ORDER TYPES, I

235

PROOF. Let A e A, Be Band C e C and suppose p : AB ~ AC. Then AB ,.., AC and since AB and AC are well-orderings, it follows that p is an extension of the unique minimal isotonism, Pe, between AB and AC (theorem IV. 5.3). Now, classically, :F 0& = eJ -+ T = J. Therefore there is an isotonism qe (not necessarily partial recursive) such that qe : B ,.., C. Now the map Te :j(a, b) -+ j(a, qe(b» defined only on C'AB is an isotonism between AB and AC. Hence by theorem IV. 5 . 3 p is an extension of r.: Since A :F 0, there is an element, say a o, in C'A. Let p' be the map p with domain and range restricted to {j(ao, n) : n e J}, then p' is partial recursive. Further, if p'(j(ao, x) is defined then its value is j(ao, y) for some y. Now let q' be the map x -+ l(p'(j(ao, x»), then clearly q' is partial recursive and q' agrees with qe on C'B (again by theorem IV. 5.3). q' is one-one, since

e

= q'(y)

q'(x)

er

-+ l(p'(j(ao, x») -+ p'(j(ao,

x)

= l(p' (j(a o, y»)

= j(ao, c) & p'(j(ao, y» = j(ao, c)

(since pp' £ {j(ao, n) : n s J} by construction) -+ j(ao, x) = j(ao, y) -+

x

= y.

(since p is one-one)

Thus q' is partial recursive, agrees with qe on C'B and is one-one and order-preserving, i.e. q' : B ~ C, from which the theorem follows. LEMMA

VI. 2.5: If M is a principal number for multiplication, and

~C:F~~fflOC
Suppose BC < M, then B < M or C = I by theorem VI.2.3. (ii). In the former case BM = M and in the latter trivially, C < M. Now BC < M -+ BCM = M, and therefore, by theorem VI. 2.4, CM = M. Using theorem VI.2.3.(ii) it follows that C < M. Conversely, C < M -+ CM = M and B < M -+ BM = M. Hence (BC)M = B(CM) = BM = M and by theorem VI.2.3.(ii), BC < M. PROOF.

VI. 3

VI.3.1: (i) If A :F 0, then B < C C -+ AB s AC.

THEOREM

(ii) B

s

-+

AB < AC,

236

JOHN N. CROSSLEY

PROOF. (i) B < C -+ (E tD) (B + D = C & D "# 0). By theorem VI. I. 7, AC= A(B+D) = AB+AD. AD"#O by theorem VI. I. 6, hence AB < Ae. (ii) follows at once from (i). THEOREM VI. 3 .2: There exist co-ordinals A, B, C ("# 0) such that A < B but AC $ Be. PROOF. Let A = 1, B = Vand C = W, then AC = Wand BC = Vw. By theorem VI. 2 .3 . (i), V:s; Vw. Hence if W:s; VW, Wand V are comparable by theorem II. 5 .4 which contradicts corollary IV. 4.5. THEOREM VI. 3.3: If there is a principal number for multiplication such that B, C < M (or equivalently BC < M) then A < B -+ AC :s; Be. PROOF. If B or C = 0 there is nothing to prove. Similarly if A = O. Otherwise, by lemma VI. 2.5, A C < M and BC < M. Hence, by theorem II. 5.4, AC and BC are comparable. Now, classically, 4'> < lJI -+ 4'>r ..s lJIr, hence AC :s; BC. THEOREM VI. 3.4: If A, B, C are co-ordinals, then A C < BC

-+

A < B.

PROOF. If C = 0, then the assertion is trivial. If C "# 0, then by theorem VI. 2.3. (i), A :s; A C and B :s; Be. Hence by the transitivity of :s; and theorem II. 5 . 4, A and B are comparable. By the classical theorem 4'>r < lJIr -+ 4'> < lJI, we have I A I < I B I and hence A < B. THEOREM VI. 3.5: There exist co-ordinals A, B, C such thatA C :s; BC but A :$ B. PROOF. (As in the classical case.) Let A

=

2, B

=

1, C

=

THEOREM VI . 3 .6: If B, C are comparable, then AB < A C

W.

-+

B <

e.

PROOF. Immediate from theorem VI. 3 . 1. (i). THEOREM VI. 3.7: If there is a principal number for multiplication, M such that (AB <) AC < M, then AB < AC -+ B < e. PROOF. By lemma VI.2.5, AB < AC < M -+ B < M & C < M. Hence by theorem II. 5.4, Band C are comparable. The theorem now follows at once from theorem VI. 3.6. We leave the question of prime numbers and unique factorization of certain co-ordinals to a later paper [4].

CONSTRUCTIVE ORDER TYPES, I

237

VII. Exponentiation VII.t. In this section we define exponentiation but restrict our attention to co-ordinals, since although the collection of all C.O.T.s is closed under exponentiation, the collection of quords is not. This result is implicit in [12]. Since some properties of exponentiation depend on multiplicative properties (e.g. (ABf = ABC) it is to be expected that we should not be able to prove analogues of all the classical laws concerning monotonicity. However, if we consider predecessors of principal numbers for exponentiation we get results analogous to those in the preceding section for multiplication. Since we have defined C.O.T.s in terms of classes of sets of ordered pairs of natural numbers and since a classical method of defining exponentiation depends on consideration of finite (descending) sequences in a given ordering, we now define a (primitive) recursive function e which assigns a natural number to each finite sequence of elements in a representative of a C.O.T. which is indexed by a sequence in a representative of another C.O.T. DEFINITION

VII. 1. 1: A symbol of the form

b O •.• b n) ( a o '" an

where n > -1 and the a.; b, (i = 0, ... , n) are natural numbers is said to be a bracket symbol. If n = -1, the symbol is simply 0 which we call the empty bracket symbol and denote by O. We use upper case bold face letters (A, B, C, etc.) for bracket symbols. DEFINITION

VII. 1.2: e(O)

if n ?: 0 and A

= 0; =

(b o '" b

then e(A) =

n

ao '"

n

)

an

p{(ai,b i)

i= 0

where Pi denotes the i-th prime (Po

=

2).

+

1

JOHN N. CROSSLEY

238

THEOREM VII. 1.3: e is a one-one primitive recursive function from the set of all bracket symbols into J. Further, pe is recursive. PROOF. Left to the reader.

VII.2. We define exponentiation of C.O.T.s in this sub-section using the function e. DEFINITION VII. 2.1: If A is a linear ordering and a e C'A, then a is said to be the minimum element of A if b s C'A --+ eA. Clearly, if a linear ordering has a minimum element then it is unique. Notation. We write a = min(A) if a is the minimum element in the ordering A. If {bJ7 = 0 is a sequence of elements in C'B such that

then we write

o ::;;

i

< n

--+

B & b, + 1 i= bi'

--+ b2 --+ •.. --+

bn>s B.

DEFINITION VII. 2 .2: If A, B are linear orderings then E (A, B) is the set of all bracket symbols K such that K

=

bo ... bn) ( ao ... an where ("1m) (m < n --+ am e C'A & am i= min(A))

>

and
=

{
0 and B i= 0, then AB = 0; otherwise

= (b o .. , bm ) ao ... am

&K,K'eE(A, B) .&:K

& K'

ao

=o.V.

K i= 0 & [em ::;; n & ("Ir) (r ::;; m --+ a, = a; &

v (3r) ("Is) {(s < r

--+ as

=

a~

& «»; b;) e B & br i= b;.v.b r

b:)

= (b?

an

b,

=

b;n

& b, = b~)

= b; &


Since E(A, B) with the ordering induced by definition VII. 2.3 is equivalent to the classical definition of A raised to the power B (cf. [14J, p. 306 et seq.) and A B is a linear ordering of a subset of J, A B is a relation in our sense.

CONSTRUCTIVE ORDER TYPES, I THEOREM VII.2.4: If Ai ~ Bi (i

=

239

1,2), then A~l ~ A~2.

PROOF. The only non-trivial case is where Ai (or equivalently, A 2 ) is non-empty. Suppose Ai #- 0 and p: Ai ~ A 2 and q: B1 ~ B2 • Let r be the map defined only on pe by

reO) = 0, r(n) = e

q(bo) ... q(b m) ) ( p(ao) ... p(a m)

if n 13 pe and n

= e ( bo ... bm ) . ao ... am

r is partial recursive, since pe is recursive, and is one-one and onto since p, q are one-one and onto and e is one-one. The order-preserving property

follows from the classical case. DEFINITIONVII.2.5:AB = CRT(AB) where AsA and BsB'.J.We sometimes write "A exp B" for "A H" and "A exp B" for "A B" . By theorem VII. 2.4, A B is uniquely defined. THEOREM VII.2.6: (i) If A, B are co-ordinals, then A B is a co-ordinal. (ii) If A, Bare e.O.T.s, then A B is a e.O.T. (iii) (Parikh [12]) IfB is a well-ordering and A is a quasi-well-ordering, then A B is a quasi-well-ordering. (iv) (Parikh [12]) There is a quasi-well-ordering A such that T A is not a quasi-well-ordering where T 13 2. (v) If A B is a quord, and A #- 0, 1 and B #- then A and Bare quords.

°

PROOF OF (v). Suppose A 13 A and B I: B but A is not a quasi-wellordering. Then there is an infinite recursive descending chain, {aJ7'= 0, in A. Since B #- there is an element b in C' B and hence

°

is an infinite recursive descending chain in A B which is impossible. Suppose then that B is not a quasi-well-ordering, then there is an infinite recursive descending chain {bJ7'= 0 in B. Now since A #- 0, 1 there is an element a #- min(A) in C' A. Therefore

JOHN N. CROSSLEY

240

fe(b i ) }

t

00

i~O

a

is an infinite recursive descending chain in A B. This too is impossible and (v) is established. VII. 3. THEOREM VII.3.l: A exp (B+C) = AB.A C • PROOF. Let A GA, B GBand C GC where B)( C. Then B+ C is well-defined and by theorem II. 1. 4. (i) there exist r.e. disjoint sets /3, y such that C'B £ /3 and C'C £ y. C'(A exp (B + C)) = {e(K): KG E (A, B+ C)} and

KG E(A, B+ C)

+-+

K =

(eo

er ) & ('Vi) (a, i= min(A))

ao

a;

&

v

-+ .. , -+


... -+

v (3 s) (s < r & C'(A B and Kl


.

e


r)

B G C G

... -+

e

es )

G

C

& s + 1 -+ .,. -+ r ) G B). AC) = {j(e(K l ) , e(Kz)) : K l GE(A, B) & K z GE(A, C)}

G E(A,

B) +-+

es + 1 Kl = ( as + 1 and

er )

&
-+

es + z

•.. •..

-+ ... -+

er )

er) & ('Vi) (ai i= min(A)) ar G

B

K z G E(A, C) +-+

Kz =

eo ... es) & ('Vi) (a l i= min(A)) ( a o ... as

CONSTRUCTIVE ORDER TYPES,

241

I

Recalling that if r = s, then K 1 = 0 and e(K 1 ) = 0, and if s = -1, K 2 = 0 and e(K2 ) = 0, it follows that the map p defined by p(x) = l(x)* k(x) is order-preserving between A B • A C and A exp (B + C). Now let

.

.

15 = }(e(K 1 ) , e(K2 ) ) : K 1 =

t

(b o ... b

m)

ao

&K = 2

am

(co

a~

Cn)

a~

& (Vi) (ai' a; =f. min(A) &

b,

B

f3 & c, B Y)}

and let q be the map p with domain restricted to 15. Then q is partial recursive since 15 is r.e. Further q is one-one. For suppose q(x) = q(y) = z, say. Then z = pj(e(K 1 ) , e(K 2 )) = e(K) for some bracket symbols K 1 , K 2 , K. But K

=

(eo

ao

e

r)

a;

where a, =f. min (A) and e, B f3 U y, and there is precisely one number s such that -1 ~ i ~ s -+ e, B f3 & s < i ~ r -+ e, B y by the definition of 15. Therefore K 1 and K 2 are uniquely determined by K and our assertion is proved. This we have proved that q is a one-one, partial recursive, order-preserving map between A B • A C and A exp (B + C), i.e. is a recursive isotonism. The theorem follows at once from this. Notation. A O = 1; A n+ 1 = An.A. Corollary VII. 3 .2. If A is a co-ordinal, then AI" = An.

PROOF. If A = 0, the assertion is trivial. If A =f. 0, the reader will easily verify that N
242

JOHN N. CROSSLEY

in all other cases for A, B, C = 0 or 1, both sides are 1. We therefore assume A, B, C i= 0, 1. Let A e A, B e B, C e C, then

Cn) eE(A B, C) qn

(ABf = {
(C~

C~') s E( A B, C) &

qo

qn'

(Vr) (qr = e(Qr) & q; = e(Q;) & Q" Q; e E(A, B)

0 . v. D i= 0 & [en s n' & (Vr) (r s n --+ c, = c; & qr = q;))

:&: D

=

v (3r) ("Is) {(s < r

&

--+ Cs

« c., c;) s C & c, i=

= c; & qs = q;)

c; . v .

c, = c; & e AB +-+ (r,

s

t; & ("Is) (s S

v (3u) ("Iv) {(v < u

t, --+

--+

br• = b;s & a., = a;s))

b.;

= b;v &

a.;

= a;v)

& [«bru' b;u> e B & »; i= b;u) v (b ru = b;u & s A)]}.

Now

ABC = {
= (j(b~: c~) ao

: &: D

j(b~,: C~,)) e E(A, BC) an'

= 0 . v. D i= 0 & [en s n'

(Vr) (r

s

n

--+ j(b"

&

c.) = j(b;, c;) & a, = a;))

v (3r) ("Is) {(s < r --+ j(b., cs) = j(b~, c~) & as = a~) & «j(b" cr),j(b;, c;» s BC &j(b" c.) i= j(b;, c;) . v. j(b" c.) = j(b;, c;) & e

An]}.

CONSTRUCTIVE ORDER TYPES, I

But

(j(b r , c.), j(b;, c;» s BC +-+ (c r , c;) & C & c, #- c; . V.

and

c, = c; & (b" b;) & B

Now let p be the partial recursive function defined only on

{e(X) :X= (idoo

s. =

in) & (Vi) (d s pe)}, i

d;

(where we recall that pe is recursive) by (p(O) = 0 and) Co

.•.

Cn

))

p ( e ( e(Qo) ... e(Qn) = e (j(b oo, co)

aoo

where

j(b omo, cO)j(b 1 •0 , c1 ) a omo

•••

a1 , o , "

j(b 1,m" c1 )

•••

a1,m,

... j(bno, cn) ... j(bnmn, Cn») ... ano . .. a nmn

Using the definitions of (ABf and ABC given above the reader will readily verify that p is order-preserving, one-one and onto, from which it follows that p: (AB)c ~ ABC. Taking C.R.T.s completes the proof. As in the classical case, we do not have in general, ACBc = (ABf.

VII. 4. We now introduce principal numbers for exponentiation and show that predecessors of principal numbers for exponentiation satisfy [the analogues of] the classical laws for exponentiation. DEFINITION VII .4.1: A co-ordinal A > 1, is said to be a principal number for exponentiation if

1~ B < A We write £ nentiation.

--+

BA

=

A.

(exp) for the collection of all principal numbers for expo-

244

JOHN N. CROSSLEY

THEOREM VII. 4.2: All principal numbers for exponentiation are infinite co-ordinals whose classical ordinals are limit numbers. PROOF. Left to the reader (cf. theorem IV. 4.9). The condition in definition VII. 4. 1 is stronger than the condition: 1 ~ B, C --+ Be < A. This will be shown later in a manner analogous to that referred to in § VI. 2 by proving that if 2A = A, then W divides A. THEOREM VII. 4 . 3: W is a principal number for exponentiation. PROOF. It suffices to prove that, if N I; In> then W ~ N W . Let N = {(x, y): 0 ~ x ~ y < n}, then clearly N I; In. If S I; of, then s is expressible in the form

where for all i, 0 defined by

~

a, < n. Let f be the (partial) recursive function

°)

f(s) = e r r - 1 ... ( a r a r - 1 '" a o where columns with bottom entry

°

have been omitted.

E·g·f(n 2 .3+n.0+2) Then, if u, v I; of and u a; and b, may be zero,

(2 0)

=e 3 2 .

= nrar+ ... +a o and v = n'br+ ... +b o' where

and

(1)

(We remark that the fact that a" a; _ l' . . . and b" b, _ l' . . . may be zero does not affect the ordering.) But the ordering ~ given by (1) is precisely the ordering in N W of the bracket symbols

(a,r

0 ) and ao

(r .. , 0 ) b; ... b o

where columns with bottom row zero have been omitted. Clearly, one-one. Hence f: W ~ N W and the theorem is proved.

f

is

245

CONSTRUCTIVE ORDER TYPES, I

Corollary VIl.4.4. 2w = W. THEOREM

VIIA.5: If A > 1, then A B = A C

-+

B =

c.

Let A G A, B G Band C G C, and suppose p: A B ~ A': Then A ..... AC and since A B and AC are well-orderings, it follows that p is an extension of the unique minimal isotonism, Pc, between AB and A C• Now, classically, e > 1 & e r = e.1 -+ r = ..1. Therefore there is an isotonism qe (not necessarily partial recursive) such that qe: B ..... C. Now the map PROOF. I) B

defined only on E( A, B) is an isotonism between A B and Ac. Hence by theorem IV. 5.3, p is an extension of r: Since A > 1, there is a non-minimum element, say a O, in C' A. Let p' be the map p with domain and range restricted to

then p' is partial recursive. Further, if p'

(e (:0))

is defined then its value is e

(~o)

for some y. Now let q' be the map

then clearly q' is partial recursive") and agrees with qe on theorem IV. 5.3). q' is one-one, since

c-s (again by

I) We are here using a similar extension procedure to that used in the proof of theorem VI. 2.4. 2) (x)o = exponent of (po =) 2 in the prime factorization of x.

246

JOHN N. CROSS

(e (:0)) = p (e (~O)) 2

2j ( u, q'(x» + d. 3X 1 . . . . . P:" &

--. p'

=

j

( u' , q'(y»

+ -: 3Y1 •

. ..

.

p~m

for some u, u', d, d', n, m, XI' .•. , X n' YI, ... , Ym where d, d' = 0 or 1. O But by the definition of p', any image of p' is of the form 2 j ( a , b) + I and hence d = d' = 1, n = m = 0, u = u' = a O and p'

and p'

(e (:0)) = 2

(e (~o)) =

j

( aO, q'(x»

2j (a

O

, q' ( Y»

+

1

+ 1.

Therefore

from which it follows, since p' and e are one-one, that X = y. Thus we have shown that q' is a recursive isotonism between Band C. This completes the proof. THEOREM

but A :F B.

VII.4. 6: There exist co-ordinals A, B, C such that AC = BC

PROOF (as in the classical case). Let A = 2, B = 3, C = W. Then by theorem VII.4.3, 2w = s". THEOREM PROOF. A

VII.4. 7: C > 1 & A < B --. C" < CB.

< B -. (ElD) (D :F 0 & A+D

= B). Hence by theorem

VII.3.1, C = C" + D = CA.. CD. Now CD:F 0 since C:F 0, hence (3E) (CD = 1+E). Hence CB = C\1+E) = CA+CA.E by theorem VI. 1.6 and C A ~ CB. But B

CA. = C B

-.

CA. E = 0 -. E = 0 -. CD = 1 -+ D = 0

which is a contradiction. This completes the proof.

CONSTRUCTIVE ORDER TYPES, I

247

VIIA.8: (i) If A, C> 1, then A < A C • (ii) If C > 0, then A s A C • LEMMA

PROOF. (i) Since C > 1, there is a D # 0 such that 1 +D = C. Therefore A C = A 1+ D = A.A D by theorem VII.3.!. Now IADI > 1, ,by classical arguments, hence there is an E # 0 such that AD = 1 + E. Hence A C = A(l+E) = A+AE where AE # 0, i.e. A < A C • (ii) follows at once. THEOREM

VII 04.9: There exist co-ordinals A, B, C such that A < B

but A C $ B C •

Let A = 2, B = V and C = W, then by theorem VIIA.3, Wand by lemma VIIA.8, V < V W = B C • Now if A C ::s; B C , then by theorem 11.5.4 and the transitivity of ::S;, Vand Ware comparable, which contradicts the construction of these co-ordinals. PROOF.

AC

=

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 VI.3.2). We have, however, theorem VIlA. 11 which is analogous to theorem VI. 3. 3. VII.4. 10: If E is a principal number for exponentiation, then A, B < E -+ A B < E and conversely if A, B > 1. LEMMA

The assertion is trivial if A, B ::s; 1. Otherwise, if E is a principal number for exponentiation, then A < E -+ A E = E and similarly for B. Hence A IBE) = E. Now B < E and therefore there is a C # 0 such that B+C = E. Therefore E = A IBE) = A(B+C) = AB.A c . But A C > 1, since C # 0; hence A C = 1 +D for some D # O. It follows that E = A B(l+D) = AB+ABD where ABD # 0, i.e. A B < E. Conversely, suppose A, B > 1 and A B < E. Then by lemma VII.4. 8 .(i), A < E. Since E is a principal number for exponentiation, E = A E = (ABl = ABE. By theorem VII 04.5 it follows that BE = E and hence by theorem VI. 2.3. (ii) B < E. PROOF.

THEOREM VII.4. ll : If there is a principal number for exponentiation, E, such that B, C < E (or equivalently B C < E or B, C::s; l) then A < B-+ AC::S;~.

PROOF.

By the transitivity of ::s; and lemma VII. 4. 10, A C < E and

248

JOHN N. CROSSLEY

BC < E. Hence by theorem 11.504, A C and BC are comparable. Now, classically, F < Ll -+ t" ~ Lltl>, hence AC < BC -+ A < B. THEOREM

VII 04.12: If A, B, C are co-ordinals, A C < B C

-+

A < B.

If C = 0 then there is nothing to prove. Otherwise, by lemma VII.4.8, A s A C and B s B C and therefore, by theorem 11.5.4 and the transitivity of ~, A and B are comparable. Hence by the ciassical theorem cpr < tpr -+ cp < tp, we have I A I < I B I and hence A < B. PROOF.

THEOREM

VIlA. 13: There exist co-ordinals A, B, C such that I < A C ~ B C but A $ B.

(as in the classical case). Let A = 3, B = 2 and C = W, then by theorem VII 04.3 (proof), A C = BC = W. PROOF

THEOREM

VII A. 14:

If B, C are

comparable and A > I, then

A B < AC PROOF.

-+

B < C.

By theorem VII 04.7.

THEOREM VIlA. 15: If there is a principal number for exponentiation, E, such that A C < E, then

I < A B < AC

-+

B < C.

PROOF. 1 < A < A implies A, B, C are all ~ 1. By lemma VII 04.10, if A C < E, then A, C < E and BC < E -+ B, C < E. Hence by theorem 11.504 and the transitivity of ~, Band C are comparable. Hence by theorem VII .4.14, B < C. B

C

VIII. Natural well-orderings up to

w(J)w

VIII.t. We showed in § IV that the finite co-ordinals are unique but that for each infinite classical ordinal F there exist c mutually incomparable co-ordinals of classical ordinal F. We now go on to give criteria for collections of co-ordinals which contain precisely one representative for each member of a given collection of classical ordinals. Using these we can give simple criteria for recursive well-orderings to be natural well-orderings, in the sense that if two recursive well-orderings are of the same classical ordinal, then they are recursively isomorphic provided

CONSTRUCTIVE ORDER TYPES, I

249

they are of not too large an ordinal and they are both natural wellorderings. By theorem 1.4.4 it is sufficient to describe co-ordinals which contain such natural well-orderings. In this section and the next we work in a slightly more general context: we do not assume that all our wellorderings are recursive, though it will turn out that they are. In [4] we shall extend our results much further as announced in [21]. DEFINITION VIII. 1. 1: I) If d' is a collection of co-ordinals, then d' is said to be T -unique if

IA I = IB I
A, Bed' &

-+

A = B.

d' is said to be strictly Fvunique if d' is T-unique but not A-unique for any A > T. By theorem IV. 3.6 it follows that d' is strictly T-unique if d' is Tunique but not (T + I)-unique. Corollary VIII, 1.2.

f(? is

strictly co-unique.

PROOF. Immediate from corollaries IV. 2.2 and IV.4. 2. We now give two proofs of the following theorem. The first proof does not use multiplication except in the form A. w. 2) The first three lemmata are common to both proofs. THEOREM VIII. 1.3: The collection £"( +) of all principal numbers for addition is strictly wW-unique. LEMMA VIII, 1 .4:

If A is a quord, then B+A

=

A

+-+

B.w

s

A. 3 )

PRooF.4 ) Suppose B. w ::; A, then there is a co-ordinal C such that B,w+C = A. ThereforeB+A = B+(B.w+C) = (B + B.w)+C = B.w+ C (by theorem II.4.1.(vii» = A. Now suppose B + A = A. If B = 0, then the assertion is trivial. If A = 0, then B = 0, hence we may assume A t= 0 t= B. By hypothesis I)

This definition is adapted from [10],

2) Since we may define W by recursion, thus W = Lr», W"+ I = W" .co, 3) Bu» :S A may also be written (3C) (B, W C = A) which brings out the

+

similarity with theorem VIII. 2.2. 4) This Iheorem can also be proved for co-ordinals using a technique similar to thai in the proof of theorem VIII.2,2.

250

JOHN N. CROSSLEY

there exist quasi-well-orderings A, B and a recursive isotonism f such that f: B+ A ~ A where B)( A. Let (X = CA, proof only.

P = CB.

We introduce the following notation for this 00

Poo

=

Boo

=

(xo

= {x: (Vn)f-" (x)

Ao

=

u

"=0

j" + 1 (P),

A [Poo, s

(X)},

A [(Xo'

We shall prove: 1)

(xo

("\

2)

(xo

u

Poo = 0, Poo = (x,

3) Boo e s.»,

4) x e (xo -+ f(x) = x, 5) x e Poo -+ f(x) # x, 6) Boo)( Ao, 7) Boo+Ao = A.

1) If x e Poo, then x = j"(y) for some n > 0, some yep. Hence f-n(x) is defined and t (X; so x t (xo. 2) Since f maps P u (X onto (x, x s (X implies either ('
or (3n) [F"(x) s P].

I.e. x e (X -+ x s (xo v x e Poo. Conversely, x s (xo -+ X = fO(x) s (X and x s Poo -+ x = j"(y) for some yep, some n > 0, i.e. x e (x. 3) Since B)( A there is a partial recursive function p such that if x e p u (X then xs

(X +-+

p(x) = 0 & x s

p +-+

p(x) = 1

(by theorem II .1. 5). We now use p to calculate a function g such that x s Poo

-+

g(x) = j( r-"(x), n -1)

CONSTRUCTIVE ORDER TYPES, I

251

where n = 11,{r'(x) s P & (\'s) (s < r

Step A. Calculate j-I(X). If a value (say) P(XI)'

-+

j-S(x) e oe}.

XI

is obtained, calculate

Three cases arise: 1. No value is obtained for XI or XI is defined but no value is obtained for p(x I ) ; 2. XI is defined and p(x l ) = 0; 3. XI is defined and p(x I ) = 1. We proceed according to cases. Case 1. g(x) is undefined. Case 2. Repeat step A with

Xl

replacing x,

Case 3. g(x) = j(xl, n) where n is the number of times case 2 has arisen in the computation and X I is the value most recently obtained in performing step A.

g is clearly partial recursive. Suppose g(x) = g(y), then g(x) = j(xl,n) = g(y) for some Xl = j-"-I(X) = j-"-l(y). But is one-one, therefore X = Y and g is one-one. We now show g maps Pw onto p. 00. By the definition ofg, g(pw) 5;; p. 00. If j(x,n)ep.oo then f"+l(x)epw and g(f"+I(X» =j(x,n); hence p.oo 5;; Pw. Next we show that g is order-preserving between Bw and B. ca. It suffices to show that if (xo, Yo) s Bw and Xo = rex) and y = f"(y) where x, yep and 0 < r < m -+ rex) e oe and 0 < S < n -+j'(y) e oe, then I ~ m < nor 1 ~ m = n & (x, y) s B. If m > n, then since j is one-one and order-preserving, (jm - "(x), y) s B+ A. But yep and r - "(x) e oe which contradicts B) (A. Hence In ~ n. If m = n, then (x, y) s B+ A where x, yep. We conclude (x, y) e B. This completes the proof of 3).

r:

4) Since A is a quasi-well-ordering and A o 5;; A, A o is a quasi-wellordering. Now j maps oeo = C' Ao onto oe o since X e oeo -+ j-I(X) e oe o & j(x) e oeo which implies oe o 5;; j(oeo) 5;; oe o' But j is order-preserving, hence by theorem III. 1.6, j = 1 on oe o ' 5) x e Pw -+ x = f"(y) for some n > 0, some yep. Since j is one-one, x = j(x) impliesr"(x) = j-" + lex). Butj-"(x) e p andj"?' + I(X) e oe and p n oe = 0 since B )( A. Therefore j(x) ¥ x. 6) Since j is partial recursive, bj is r.e. If x e Pw, then by 6) j(x) ¥ x.

252

JOHN N. CROSSLEY

If XC Ci o, then by 5) f(x) = x. Hence Cio, {J(O are contained in the disjoint r.e. sets {x: x C fJf&f(x) ¥- x} and {x: x s fJf&f(x) = x}. Hence by theorem 1I.1.4.(i) B(O)( A o . 7) By 6), B(O + Ao is well-defined. By 2), C(B(O + Ao) = Ci. By definition B(O ~ A and A o ~ A. It therefore suffices to prove that {JwXCio ~ A and A ~ Bw + A o . If x e {Jw and y s Cio then(3n) (f-n(x) c {J) but ("In) (f-n(y) c «). Hence
and only

if; B <

PROOF.

A

~

B. co ::;; A.

Immediate from definition IV. 4.8 and lemma VIII. 1. 4.

LEMMA VIII. 1.6: If A cYt'( +), then A = wn < A.

wn for some 11, or

i

for all n,

PROOF. If A = 1, the assertion is trivial. If A > 1, then by lemma VIII. 1.5, 1. t» = W::;; A. If A ¥- W, then W < A. Now suppose W n < A (where n > 0). Since A is a principal number for addition, by lemma VIII .1. 5 W n • w = W n + 1 ::;; A. Hence either A = W n for some n or for all 11, W n < A.

LEMMA VIII. 1.7: If P is a principal number for addition, then P. w is a

principal number for addition and there is no principal number Q such that P < Q < P.w.

The first part is a restatement of theorem IV. 4 .10. Suppose Q e.Yf'(+) and P < Q, then by lemma VIII. 1. 5, P. w ::;; Q; hence PROOF.

Q 1:: P.w.

LEMMA VIII. 1.8: W n is a principal number for addition for every n,

CONSTRUCTIVE ORDER TYPES, I

253

PROOF. If n = 0 or 1, then the assertion is trivial. Suppose n > 0 and W n is a principal number for addition, then by lemma VIIL1. 7, W n + 1 = W n • w is a principal number for addition. Hence the lemma is proved by induction. PROOF OF THEOREM VIn. 1 .3 (FIRST VERSION). By lemmata VIII. 1.6 and VIII .1. 8 a co-ordinal A of classical ordinal < W W is a principal number for addition if, and only if, it is of the form Wn • Hence £( +) is wW-unique. Now let V, V' be two incomparable upper bounds for {W n : n e Y} constructed as in corollary V. 2.3. Then 1V I = I V' I = co", Now A < V --. A < W n < V for some n, and similarly for V'. But A < W n--. A+W n = W n and therefore A+V = V and A+V' = V', i.e. V and V' are principal numbers for addition. Thus £( +) is strictly wW-unique. LEMMA VIII.l.9: (i) W m < W n if m < n, (ii) Ifn ~ 1,1+ W n = W n ,

(iii)

If m <

n, W m+ W n = W n.

PROOF. (i) If m < n, then n = m+(n-m). Hence by theorem VII.3.1, W n = Wm+(n-m) = WmW n- m = W m(I+E) [for some coordinal E] = W m+ WmE. Now I W m I < I W n I, hence W m < W n. (i i) By (i), if n ~ 1 then W:s; W n and hence by lemma VIII. 1 .4, 1+ W n = W n • (iii) W m+ W n = Wm(l + W n - m) = wmW n - m = W n if m < n. DEFINITION VIII .1.10: A co-ordinal C (an ordinal T) is said to be a polynomial in W (polynomial in co) if C (T) can be expressed in the form C = W n .a n+ ... +ao = p(W) (r = w n .a n+ ... +a o = p(w)) where the a, are natural numbers and an ¥= O. The degree of p(8p) is n and the rank of p (rk(p)) is the number of non-zero ai' We observe that I p(W)

1

= p(w).

LEMMA VIII. 1 . 11: If p( W) is a polynomial in W of degree < n, then p(W)+ wn = W n • PROOF by induction on the rank of p. If rk(p)

=

1, then p(W) = Wma m

JOHN N. CROSSLEY

254

for some m

~

0, some am #

p(W)+ W n = W n if op

< n,

o.

Applying lemma VIII .1. 9. (iii) am times,

Now assume the lemma holds for rk(p) = m -1 > o. Then peW) = = i!r{a r # O}. Then rk(q) = rk(p)-l. By am applications of lemma VIII. 1. 9. (iii), peW) + wn = q(W) + W n and by the induction hypothesis, q(W)+ W n = W n. q(W)+ Wm.a m where m

LEMMA VIII. 1. 12: If n > 0, then A < nomial in W of degree < n.

wn

if, and only if, A is a poly-

PROOF. By lemma VIII .1.11, peW) < wn if op < n. Now if A < W n, i A I = p(w) for some polynomial in ca. Hence by corollary IV. 2 .7, A

=

peW).

LEMMA VIII .1.13: WWand W V are principal numbers for addition. PROOF. Since n < V, there is a U such that V = n + U. Then W n+ W V = W n+ W n + U = W n(1+ W u) = WnW U since 1 < U and U W< (using lemma VIII. 1.4). Hence Wn+W V = WnW U = n W + U = W V , and W n < W V for every n. Similarly W n < W W for

w

every n. Now every ordinal < W is represented by a polynomial in wand hence by corollary IV. 2.7 and lemma VIII .1.12 we also have, conversely, A < W V --+ A < W n for some n, and similarly for W w . Therefore if A < WV W

A+ W

V

=

A+(W

n+

W

v)

=

(A+ W

n)+

W

V

=

W

n+

W

V

=

W

V

for large enough n (and similarly for W w). Thus W V and W W are principal numbers for addition. PROOF OF THEOREM VIII .1.3 (SECOND VERSION). By lemmata VIII .1. 6, VIII .1.11 and VIII .1.12, every co-ordinal of the form W n is a principal number for addition and there are no other co-ordinals which are principal numbers and have ordinal < co". Hence £( +) is wW-unique. By lemma VIII. 1.13, W W and W V are principal numbers for addition. But W W = W V --+ W = V by theorem VII.4. 5, which contradicts the definitions of W, V. Hence £( +) is strictly wW-unique. It follows at once from theorem VIII. 1.3 that the collection of predecessors of principal numbers of ordinal < W W contains precisely one

CONSTRUCTIVE ORDER TYPES,

I

255

co-ordinal for each ordinal < W W and is closed under addition by theorem IV.4.11. We close this section with an example of a principal number for addition whose classical ordinal is not a (classical) principal number for addition (v. § IV.4). Example VIII .1. 14. Let p, V be as given in § IV. 4. Let IJ( = C'W v and let U = {(x, y) : x, yea & x S y}. Then IJ( is r.e., clearly, but is

not recursive. For

IJ(

recursive implies

{x: (3y) (y = e(~)) &yelJ(}

= p

is recursive, which contradicts the choice of p. U is ofclassical order type t» and W V and U are strictly disjoint (§ 11.1) but clearly not (even r.e.) separable. Hence W V +- U is well-defined, but does not belong to CR T(W v ) + CR T( U), and is of ordinal W W + co. Let P = CR T

(Wv+-U).

Now if p = {v;}:"= 0 where i < j --+ Vi < vj and Vn = V [ {Vi: i < n}, then Vn s nand C'WVn is recursive. Now CRT(WVn) = W n and by theorem II .1. 6 it follows that W n < P for every n. However, W V {: P since W V + B = P implies that W V + U s P which is a contradiction. By theorem IV. 3.6 we similarly have W V + n {: P for all n. Since A < P --+ I A I < W W + w it follows that A < P --+ A < W n for some n. Therefore A+P = A+(Wn+Q) [for some Q since W n < P] = (A+ Wn)+Q = Wn+Q [by lemma VII1.l.8] = P. Hence Pis a principal number for addition. I P I is not a classical principal number for addition since W W < wW+w but wW+(ww+w) > co", VIII. 2. In this section we prove a multiplicative analogue of theorem VIII.l.3. LEMMA VIII.2.1: LI =f. 0 & T > LIT'

--+

r > I",

PROOF. Immediate from theorem 2, p. 292 in [14]. THEOREM VII1.2.2: If A is a co-ordinal, then BA = A ~ B W divides A, i.e. ~ (3C) (A = BWe). PROOF. B WC=A-+BA=B 1 + WC=B wC=A by theorem IVA.4.(i).

256

JOHN N. CROSSLEY

Conversely, suppose BA = A. We may assume that A > 1, since otherwise there is nothing to prove. By hypothesis there exist well-orderings A, B and a recursive isotonism f such that A e A, B e Band f: A

Let

IX

= C' A,

/3

~

BA.

= C' B. We also write

"a
"I a I" for "I CRT(A [{x: x
8 IX,

f(a) = j(b, at) where b 8

/3 and

fear) = j(b, a, + 1) for some b 8

at 8

IX

/3.

Since f is order-preserving,

la 1 = I B I . I a 1 1+ A for some A < I B

I·

Hence lal~IBI.lall and by lemma VIII.2.1, lal~lall. Similarly, 1a, I ~ I a, + 1 I· It follows that, since A is linear,
n(x) = /lr(xr = x; + 1) [= Pr{(lfY(x) = (If)' + l(X)}] is always defined if x 8 IX. If a 8 IX and n(a) = n, then

I an I = I B I· I an I + A where A < I B I·

But I B

I . I an I z I an I since B "# 0 and therefore A

= 0 and j'(c.) = j(min(B), an)'

257

CONSTRUCTIVE ORDER TYPES, I

We observe that, for any x, if n > n(x) then (lft(x) = (If)n(x)(x).

Let C = A[{x: Lf(x) = x} and let D = O(Bw ) where g is the partial recursive function, defined only on pe, which maps only bracket symbol images ofthe form e (no n l bno bn ,

...

•••

ns) where n i , bn , I> of and no > n l > n2 > ... > ns bn,

~0

onto

n l+ln l nl-I min(B) bn , min(B)

no no-lno-2 ( e bno min(B) min(B)

n, ns-I bn, min(B)

0 ) min(B) .

I.e. g(x) inserts the missing positive integers in the top row of e -I(X) and in the columns where an integer was missing inserts min(B) in the bottom row and takes the image under e of the resulting bracket symbol. It is clear that g is one-one, so D is well-defined. We shall now show that A ~ D. C from which it follows at once that A = B W C where C = CRT(C). Let

.((n(X)-1

hex) = J e kf(lf)"(X) -

...

1 ••.

i

0)

., .

kf(lfi(x) ... kf(x) , (if)

n(x») (x)

.

Clearly, h is partial recursive. Suppose hex) = hey), then kh(x) = kh(y) and 111(x) = Lh(y). Hence (If)"(X)(x)

Now, since e is one-one we have and hence, for 0 :c::; r < n(x),

n(x)

= (Lf)"(Y).

=

(I)

n(y)

kf(lf)r(x) = kf(lf)'(y).

(2)

Putting r = n(x)-I in (2) and using (I) we have f(lf)"(X) - I(X)

But f is one-one, hence (If)n(x) - I(X)

= f(lf)n(x) = (If)"(X) -

I(y).

I(y).

258

JOHN N. CROSSLEY

Now assume where s < n(x) = n(y). Then by (2) with r = s-l

(If)S(x)

(kf) (If)' - I(X)

and using (3)

= (If)S(y)

(3)

= (kf) (If)' -

I(y)

f(lf)S - I(X) = f(lf)' - I(y).

By the one-one property off, (If)s - I(X) = (If)s - I(y)

and by induction it follows that x = y. I.e. h is one-one. It is clear that h maps C' A onto C' D. C and it only remains to prove that h is order-preserving. Suppose a
a,
+-+

f(a j )
+-+a j + 1
or

a, + 1 = a; + 1 &

where b, + Hence

1

b, + 1 < B b; + 1

= kf(aJ

a
+-+

an


an = a~ an

=

& b; < B b~ or

a~ & an -

I

=

..... or +-+

an

=

an


an

a~ & ... &

al

a~ -

=

I

& b; -

a~ &

I


b~ -

I

or

b, < B b',

a~ or

= an &

b; < B b~ or (4)

..... or since f(aJ

= j(a

an = a~ & bn = b'; & ... & b 2 = b; & b l
+ I'

b, + I)'

259

CONSTRUCTIVE ORDER TYPES, I

Now h(a)


hea') +-+ an an


=

or

a~ & (3r) (1 :::;

s < r ...... b.

=

(5) b~ & b,

< B b~).

(4) and (5) are equivalent since, as we observed above, if n > n(a), then an = an(a) and b; = min(B). Finally, h maps C' A onto C' D. C for suppose

x

=j

(

e

II (

bn

•.. •••

0) ) bo

,c e C'D. C,

where c e C'C (and b, e C'B).

=

Let ao

c, a, + 1

= r:

(j(b r, ar»'

Then ao e C' A and if a, e C' A, then a, + 1 e C' A. In particular, an e C' A and an = h - '(x), We have therefore proved h: A ~ D. C and the theorem is established. LEMMA VIII.2.3: A co-ordinal A is a principal number for multiplication if, and only if,

o<

B < A +-+ B W divides A.

PROOF. Immediate from theorem VIII. 2.2 and definition VI. 2.1. 1 We observe that Wwo = W 1 = W, (Wwn)w = W wn. w = W wn+ by theorems VII. 3.3 and VII. 3.1. LEMMA

either A

VIII. 2.4: If A is a principal number for multiplication, then n n n ww for some n, or for all n, Ww divides A and Ww < A.

=

PROOF. By theorem VI. 2.2, if A is a principal number for multiplication then 2 < A. Hence 2A = A and by theorem VIII.2.2, 2w divides A. But by corollary VII .4.4, 2 w = W. Therefore, if I A I = w, A = W. Otherwise Wdivides A and W < A by theorem VI.2.3.(ii). wm Now suppose W < A for m < n (where II > 1). Then, by lemma m 1 VIII.2.3, if A is principal (Wwm)w = Ww + divides A. By theorem n wm 1 wn VI.2.3.(i), W + :::; A and hence, if I A 1= ww , A = W or, for all r, w w W • divides A and W • < A.

Corollary VIII.2 .5. If A, B are principal numbers for multiplication

260

JOHN N. CROSSLEY

and B < A then A = B < A.

r:

LEMMA

VIII. 2 . 6:

PROOF. p'(W) a

wn

for some n, or for all n, B

=

WOP'.a+q(W)

where

q

p(W)+p'(W)

is a polynomial in Wand =

p'(W).

VIII.2.7: If Dp < op', then WP(W). WP'(W)

=

W p'(W).

By theorem VII.3.3, WP(W). WP'(W) = lemma then follows from the previous one. PROOF.

LEMMA

VIII.2.8:

WP(W)+P'(W).

The

If ap < op', then WP(W)+ WP'(W)

PROOF.

divides A and

Ifp( W), p' ( W) are polynomials in Wand ap < ap', then p(W)+p'(W) = p'(W).

#- O. By lemma VIII. 1. 11, LEMMA

wn

=

Wp'(W).

By lemma VIIL2. 7,

WP(W)+WP'(W)

=

WP(W)+WP(W).WP'(W) =

WP(W) {l+W P'(W)}.

By lemma VIIA.8.(ii), W::::;: WP'(W), hence by lemma VIII. 1.4, = WP'(W). Therefore WP(W)+ WP'(W) = WP'(W).

1+ WP'(W)

LEMMA

VIII. 2.9: If a co-ordinal A is of the form A

=

WPI(W).at

+ ... + WPe(W).a e +

q(W)

(6)

where Pt, ... , Pe and q are polynomials in W such that Pt(W) > P2(W) > ... > peCW) at #- 0 and Pt(W) > W, then

A < Wwn and A+ Wwn = wwn if apt < n. Conversely, if A < wwn for some n, then A is expressible in the form (6) where apt < n. PROOF. We prove the two parts simultaneously. Suppose 0 < A < W then I A I has Cantor normal form (cf. e.g. [14], p. 320)

all. at + ... + roT•. a e + q(ro).

where T i > T 2 > ... > T c-

w:

(7)

CONSTRUCTIVE ORDER TYPES,

261 wO' For each i, T, is a polynomial in ro, since otherwise roT; ~ ro which contradicts A < Wwn. Now to every ordinal of the form (7) there corresponds naturally and in a bi-unique way a co-ordinal of the form (6) (i.e. under the mapping p(ro) -+ p(W)). In order to prove the lemma it therefore suffices by virtue of corollary IV. 2 .7 to prove that A + W W" = W w" where n > 0Pl = degree of the polynomial I', (in co), Suppose oq = m - 1, then A+ Wwn = WPl(W).al +

=

WPl(W).al +

I

+ WPe(W).ae+q(W)+ Wwn +q(W)+(Wm+ Wwn)

by lemma VIII. 2 . 8 = WPl(W).al + ...

= Now by

e

i

L ~

1

WPl(W).al + ...

+ WPe(W).a e+ W m+ Wwn + WPe(W).a e+ Wwn =

by lemma VIII. 2 .8 C, say.

a, applications oflemma VIII.2.8 we have C = Wwn.

LEMMA VIII. 2. 10: (i) (3n) (A < WW'') +4 A < Www.

+4

A < WWv,

(ii) (3n) (A < WW)

PROOF. (i) Let V= n+U, then I U 1= co, By lemma VII.4.8, W ~ W U , hence by lemma VIII. 1. 4, 1 + W U = W U • Now Wwn. WWV = W exp (W n+ W v ) = W exp (W n+ W n +u) = W exp (W n • {1 + W u } ) = W exp (W n . W u) = W exp (W n + U ) = WWV. Hence by lemma VIIA.S, Wwn < Wwv. wn Conversely, suppose A < Wwv, then I A 1< ro for some n. But by lemma VIII. 2.9 there is a co-ordinal A' of the form (6) such that I A' I = I A I and A' < Wwn. Hence by corollary IV.2.7, A = A' and A < WW". (ii) follows at once by substituting 'w' for 'V'. (In this case, U = W.) THEOREM VIII. 2. 11; The collection £(.) of all principal numbers for wO' multiplication is strictly ro -unique. PROOF. By lemmata VIII. 2.4 and VIII. 2.9 every principal number for wn wO' multiplication of classical ordinal < ro is of the form W and con-

262

JOHN N. CROSSLEY n

versely, alI the co-ordinals Ww are principal numbers for multiplication. Hence£{.) is co",W-unique. w V Ww and WW are principal numbers for multiplication, since by the w w v V n. n. proof of lemma VIII. 2.10, Ww Ww = Ww and Ww Ww = WW • ww wv wn Further, A < W or W implies A < W for some n; hence, since w n ww all the Ww are principal numbers for multiplication, A. Ww = W v WV and A. Ww = W • wv But WW,W = W implies, by theorem VII.4. 5 (twice), W = V which is a contradiction. Therefore£"(.) is strictly co"'w -unique, THEOREM VIII .2.12: £' (exp) c £' (.) c £' (+). PROOF. By theorem VII.4. 2 every principal number for exponentiation is infinite. Suppose Pe£(exp), then by lemma VII.4.10, A < P-+ AA < P and hence (AAy = P = A P• Hence if A > 1, then by theorem VII. 4.5, AP = P and hence P is a principal number for multiplication. Now suppose P E £(.), then W :==::; P by lemma VII. 2.4, hence by lemma VIII. 1.4, I+P = P. Therefore if 0 < A < P, P = AP = A(1+P) = A+AP = A+P. I.e. Pe£(+). W W E £(.) - £(exp) since for every F < co'" there is a co-ordinal C < W W but to" is not a (classical) principal number for exponentiation. W 2 E £( +) - £(.) by similar argument. Hence alI the inclusions are strict. THEOREM VIII. 2. 12 indicates how we might extend our classes of co-ordinals to get uniqueness up to higher ordinals. We shall present results obtained by this approach in [4] and [21]. Appendix At. In many theorems concerning (classical) ordinals use is made of the theorem

If a well-ordered set ex is similar to a subset of a well-ordered set then ex is similar to an initial segment of p.

p,

The proof of this theorem requires the axiom of choice. Accordingly, it is not surprising that its analogue fails for C.O.T.s and co-ordinals.

CONSTRUCTIVE ORDER TYPES, I

263

In fact, we have made use of this fact in giving counterexamples to analogues of classical laws like A < B --+ AC :s Be. DEFINITION AI.I: A::s 8 if there is a recursive isotonism from A onto a (linearly ordered) sub-relation of B, i.e. if A ~ A' S; 8. Clearly, if Al ~ A 2 , 8 1 ~ 8 2 and Al ::S 8 1 , then A 2 ::S 8 2 , DEFINITION A I .2: A ::S B if there exist A e A and B e B such that A ::S 8. We write A <. B if A ::S B and A ;f. B. THEOREM AI. 3: (i) A ::S A,

(ii) A ::S B & B S C --+ A ::S C, (iii) A < B --+ A -< B, (iv) there exist co-ordinals A, B such that (a) A -< B but A -cI: B, (b) A ::S B & B S A but A ;f. B.

PROOF (i) - (iii) Left to the reader.

(iva) Let A = V B = (ivb) Let A = V, B = contains an infinite Clearly, V [u ~

W, then clearly V ::S Wand V;f. W. W. Now by Post's lemma ([13], p. 291) p = C'V (naturally ordered) recursive proper subset a, W. Hence W -< V.

DEFINITION AI.4: A e.O.T. A is said to be quasi-finite if A and A* are quords. We write :F for the collection of all quasi-finite C.O.T.s. THEOREM AI.5::F is partially ordered by:::;. PROOF. Suppose A ::S Band B::s A where A, B e:F. If A or B = 0 then A = B = O. We may therefore assume A;f. 0 ;f. B. Suppose g : A ~ 8 1 s; Band h: 8 ~ Al s; A, then f: A ~ Al s; A where f = hg, and Ai ;f. 0. Since A, Al are linear orderings, for every x e C'A either
on:F.

264

JOHN N. CROSSLEY

References [I] H. Bachmann, Transfinite Zahlen (Berlin 1955). [2] P. Bernays and A. A. Fraenkel, Axiomatic Set Theory (Amsterdam 1958). [3] A. Church and S. C. Kleene, Formal Definition in the Theory of Ordinal Numbers. Fund. Math. 28 (1936) 11-21. [4] J. N. Crossley, Constructive Order Types, II. (To appear). [5] M. Davis, Computability and Unsolvability (New York 1958). [6] J. C. E. Dekker, The Constructivity of Maximal Dual Ideal in Certain Boolean Algebras. Pacific J. Math. 3 (1953) 73-101. [7] , An Expository Account of Isols, Summaries of talks (Summer Institute of Symbolic Logic, Cornell 1957) pp. 189-199. and J. Myhill, Recursive Equivalence Types, University of California [8] Publications in Mathematics, n.s, 3, no. 3, 67-214. [9] S. C. Kleene, Introduction to Metamathematics (Amsterdam 1952). [10] G. Kreisel, Non-uniqueness Results for Transfinite Progressions. Bull. Acad. Polon. Sci 8 (1960) 287-290. [II] J. McCarthy, The Inversion of Functions defined by Turing Machines, Automata Studies. Annals of Maths. Studies, no. 34 (Princeton 1956) 177-181. [12] R. J. Parikh, Some Generalizations of the Notion of Well-ordering (Abstract). Notices Amer, Math. Soc. 9 (1962) 412. [13] E. L. Post, Recursively Enumerable Sets of Positive Integers and their Decision Problems. Bull. Amer. Math. Soc. 50 (1944) 284-316. [14] W. Sierpinski, Cardinal and Ordinal Numbers (Warsaw 1958). [15] R. Smullyan, Theory of Formal Systems. Annals of Maths. Studies, no. 47 (Princeton 1961). [16] A. Tarski, Cardinal Algebras (New York 1949). [17] , Ordinal Algebras (Amsterdam 1956). [18] J. S. UIlian, Splinters of Recursive Functions, JSL 25 (1960) 33-38. [19] A. N. Whitehead and B. Russell, Principia Mathematica, Vol. II 2nd ed. (Cambridge 1927). [20] K. Schutte, Predicative Well-orderings, these Proceedings. p. 280. [21] J. N. Crossley and R. J. Parikh, On Isomorphisms of Recursive Well-orderings (Abstract). JSL 28 (1963) 308.