Models Constructed From Constants

CHAPTER 2

MODELS CONSTRUCTED FROM CONSTANTS

2.1. Completeness and compactness In this section, we prove the basic completeness theorem first proved by Godel (1930). The proof we give is due to Henkin (1949) and it applies to situations somewhat more general than Godel’s original proof. This extension was already noted by Malcev (1936). The result we prove is that every consistent set of sentences Tin a language 9has a model or, in other words, is satisfiable. The proof proceeds in two stages. We shall first show that T can be extended to another consistent set of sentences Tin an expanded language p,having certain desirable features. Then we show that every T having these desirable features has a model. It will make no difference which of the two steps we prove first.

DEFINITION. Let T be a set of sentences of 2’and let C be a set of constant symbols of 2’. (C might be a proper subset of the set of all constant symbols of 9.) We say that C is a set of witnesses for T i n 2 iff for every formula cp of 9with at most one free variable, say x , there is a constant c E C such that TI-

(Wcp --* 4 c ) .

We say that T has witnesses in 9 iff T has some set C of witnesses in 9. The meaning and usage of cp(c) should be quite clear here and in all succeeding places in this chapter: cp(c) is obtained from cp by replacing simultaneously all free occurrences of x in cp by the constant c. We shall be careful to use cp(c) only when it has been made clear from the context which variable x is to be replaced by c. Otherwise the notation ~ ( c would ) be ambiguous. For example, if cp is a formula with the free variables x , y , 61

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MODELS CONSTRUCTED FROM CONSTANTS

[2.1

we have to indicate whether q ( c ) is obtained from cp by replacing x by c or by replacing y by c. An alternative notation which is completely unambiguous is to write cp(c/x)for the formula obtained by replacing all free occurrences of x in cp by c. However, we prefer to use q ( c ) and rely on the context for clarity rather than use the more cluttered notation cp(c/x). LEMMA 2.1.1. Let T be a consistent set of sentences of 9. Let C be a set 1 9 1 1 , and let 9 = 9 u C be the of new constant symbols of power ICl = 1 simple expansion of 9 formed by adding C . Then T can be extended to a consistent set of sentences T in which has C as a set of witnesses in y .

Il9ipII. For each p < a, let c,, be a constant symbol which does not occur in 9 and such that c,, # c y if p < y < a. Let C = {ca : B < a}, 3’ = 2’ u C . Clearly 1 1 3 1 ’1= a, so we may arrange all formulas of with at most one free variable in a sequence q,, 5 < a. We now define an increasing sequence of sets of sentences of 9:

PROOF.Let

a =

T = T o c T , c ... c T , c ...,

(
and a sequence d , , t < a, of constants from C such that: (i). each T, is consistent in p; (ii). if c = [ + l , then T, = T, u {(3x,)q, -, q,(d,)}; xc is the free variable in qcif it has one, otherwise xc = u,; (iii). if 5 is a limit ordinal different from 0, then T, = U , < , T e . Suppose that Tc has been defined. Note that the number of sentences in T, which are not sentences of 2’ is smaller than a, i.e., the cardinal of the set of such sentences is less than a. Furthermore, each such sentence contains at most a finite number of constants from C. Therefore, let d, be the first element of C which has not yet occurred in T,. For instance, do = c o . We show that Tc+I = Tc u {(3x,)coc cp,(d,)} +

is consistent. If this were not the case, then

By propositional logic,

’

T, ( W P , * cp,(d,).

As dz does not occur in Tc, we have by predicate Icgic, T , I-

(VX,)((3Xc)cpc A 1cpc(xc))9

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COMPLETENESS A N D COMPACTNESS

63

which contradicts the consistency of T i . If 5 is a nonzero limit ordinal, and each member of the increasing chain T,, < 5 , is consistent, then obviously T , =-ui.sT, is consistent. This completes the induction. Now we let T = u c < z T s .It is evident that T i s consistent in and is an extension of T . Suppose that cp is a formula of 9 with at most the variable x free. Then we may suppose that cp = cps and x = xf for some g < a. Whence the sentence (34%

+

cpr(d,)

F. -I

belongs to T s + , and so to

The idea of the next lemma is just as simple, but its proof is more involved and tedious.

LEMMA 2.1.2. Let T be a consistent set of sentences and C be a set of witnesses Then T has a model % such that every elemelit of '?i is an interfor T i n 2. pretation of a constant c E C. PROOF.First, note that if a set of sentences T has a set C of witnesses in 9, then C is also a set of witnesses for every extension of T . Second, if an extension of T has a model a, then % is also a model of T . So we may as well assume that T is maximal consistent in 9. f o r two constants c, d E C,define C - d iff c = d ~ T . Because T is maximal consistent, we see that for c, d, e E C,

c

So

-

-

c;

if c if c

-

N

d and

-

d - e, then

d then d

c.

c

-

e;

is an equivalence relation on C . For each c E C , let

Z={d~C:d-c} be the equivalence class of c. We propose to construct a model 91 whose set of elements A is the set of all these equivalence classes Z, for c E C ; so we define ( 1 ) A = {Z : C E C } . We now define the relations, constants, and functions of a.

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(i). For each n-placed relation symbol P in 9, we define an n-placed relation R' on the set C by: for all c,, ..., c, E C , (2) R'(c, ... c,) iff P(c, ... c,) E T. By our axioms of identity, we have

-

t. P ( c l

... c,)

A

c1

= d , A ... A C,

3

d,

+ P(d,

... d,).

So is what is called a congruence relation for the relation R' on C. It follows that we may define a relation R on A by (3) R(Z, ... Z,) iff P ( c , ... c,) E T. By (2), the definition (3) is independent of the representatives of the equivalence classes P, , ..., P,. This relation R is the interpretation of the symbol P in PI. (ii). Now consider a constant symbol d of 9. From predicate logic, we have I- (3uo)(d= uo).

So (3uo)(d= u o ) E T , and, because T has witnesses, there is a constant c E C such that (d E c ) E T. The constant c may not be unique, but its equivalence class is unique because, using our axioms of identity,

(dE

CAd

C'+

C

C').

The constant d is interpreted in the model iY by the (uniquely determined) element i: of A . In particular, if d E C, then d is interpreted by its own equivalence class d in 8, because (d = d ) E T. (iii). We handle the function symbols in a similar way. Let F be any and let c , , ..., c, E C . As before, we have m-placed function symbol of 9, (3uo)(F(c, ... c,)

= u o ) E T,

and because T has witnesses, there is a constant c E C such that

( F ( c , ... c,)

= C)E

T.

Once more, we have a slight difficulty because c may not be unique, and we use our axioms of identity to obtain: k (F(c,

... c,) = c A c1 = d , A ... A c,

= d, A c = d )

+

F(dl ... d,)

= d.

This shows that a function G can be defined on the set A of equivalence classes by the rule

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COMPLETENESS AND COMPACTNESS

(4) G(E, ... Em)

=

F iff ( F ( c , ... c,)

= c) E T.

We leave the detailed steps of (4)to the reader. We interpret the function symbol F by the function G in the model ?1. We have now specified the universe set and the interpretation of each symbol of 9 in '$I, so we have completed the definition of thc model ?I. We have pointed out that the interpretation of each constant C E C i n ?I is its equivalence class C, and it follows that every element C E A is the interpretation of some constant c E C. We proceed to prove that '!I1 is a model of T. First of all, using (4)as the first step of an induction, we easily show that (5) for every term t of 2 with no free variables and for every constant cE

c,

?I C t = c if and only if ( t = c) E T.

Using the fact that C is a set of witnesses for T, we obtain from ( 5 ) : ( 6 ) for any two terms t , , f 2 of 9 with no free variables,

91 != t ,

5

t 2 if and only if

(tl

= t z ) E T,

(7) for any atomic formula P ( t , ... r,) of 9 containing no free variables,

?I != P ( t , ... ?,) if and only if P ( t l ... t n ) E T. Combining ( 6 ) and (7) will form a basis for proving: (8) for any sentence cp of 9,

31 C cp if and only if cp E T. (8) has an unusual proof in that it is proved by induction on the length of the sentences of 9. The reader will see that the reason why this could be done is because T is maximal consistent and has witnesses in 2. Without a great deal of trouble, we have for sentences cp, $ of 9 and

24 C

1 cp

if and only if

(1 cp) E

? l C c p ~ $ ifandonlyif

T,

(~~A$)ET.

Suppose cp = (3x)$. If 24 != cp, then for some 2. E A , 91 k $[;I. This means that % k $(c), where$(c) is obtained from $ by replacing all free occurrences of x by c. So $ ( c ) E T and because

I- Il/(c)

--*

(3x)$,

we have cp E T. On the other hand, if cp E T, then because T has witnesses, there exists a constant c E C such that

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MODELS CONSTRUCTED FROM CONSTANTS

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As T is maximal, $(c) E T , so (2( k $(c). This gives ?1 b $[?I and % ‘ i=cp. This shows that 91 is a model of T. -i

Note that a converse of Lemma 2.1.2 is very easily proved, and, in fact: LEMMA 2.1.3. Let C be a set of constant symbols of 9,and let T be a set of sentences of 2. If T has a model 3 such that every element of 31 is at1 ititerpretation of some constant c E C , then T can be extended to a consistent if; in 2f o r which C is a set of witnesses.

For the proof of Lemma 2.1.3, simply let T b e the set of all sentences of Y true in 91. The model BI constructed from the constants c E C of 2’ by taking suitable equivalence classes is said to be built up from the set C of constants of 2’. Since every a E A is the interpretation of some c E C , we see immediately that IAl < JCI.We now supply the proofs of three theorems from Chapter 1. THEOREM 1.3.21 (Extended Completeness Theorem). Let I; be a set of‘ setitences of 2.Then Z is consistent if and only if Z has a model.

PROOF.The consistency of Z if Z has a model is a straightforward argument. So assume I; is consistent. By Lemma 2.1.1, we consider extensions of I; and of 9 ( 1 1 9 1 1= 1 1 9 1 1 ) , so that has witnesses in p. By Lemma 2. I .2, let % be a model of 2. 3 is a model for the expanded language p, so let 23 be the model for 9 which is the reduct of 91 to P.Because sentences in Z do not involve the constants of 9 not in 9, we see that 2 ‘3 is a model of Z. -I COROLLARY 2.1.4 (Downward Lowenheim-Skolem-Tarski Theorem). Every consistent theory T in 2 has a model of power at most I l3’ll. PROOF.In the proof above we may choose ?I so that every element is a constant, and we have JBI = IA) < 1 1 9 1 1= Il2lpII. i Corollary 2.1.4 gives the original theorem of Lowenheim (1915): If a sentence has a model, then it has a countable (finite or infinite) model.

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COMPLETENESS AND COMPACTNESS

67

THEOREM 1.3.20 (Godel’s Completeness Theorem). A sentence of 2 is a theorem of 9 if and only if it is valid. PROOF.We need only concern ourselves with one direction of the theorem. then {o} iis consistent in 9. By If a sentence o is not a theorem of 2, Theorem 1.3.21, (1 o} will have a model in which o cannot hold. Hence o is not valid. i THEOREM 1.3.22 (Compactness Theorem). A set of sentences t; has a model if and only if everyjinite subset o f t ; has a model. PROOF.If every finite subset of t; has a model, then every finite subset of t; is consistent. So C is consistent and Theorem 1.3.21 shows that 1 has a model. i We conclude this section with a representative list of applications or consequences of the completeness and compactness theorems. Some additional exercises can be found at the end. COROLLARY 2.1.5. If a theory T has arbitrarily large jiriite models, then it has an injinite model. PROOF.Let T be a theory in 2’with arbitrarily large finite models. Consider the expansion 9’ = 2’ u {c, : n E w } , where c, is a list of distinct constant symbols not in 9. Consider the set T, of 9’ defined by

C

=T u

{ (c. i3

c,)

:n < m < w}.

Any finite subset t;’ of C will involve at most the constants c,, ..., c,, say, for some m. Let 91 be a model of T with at least m + 1 elements, and let a,, . .., a, be a list of m + 1 distinct elements of a. We can verify easily that the model (a, a,, ..., a,) for the finite expansion 9’‘= 9u { c o , ..., c,} of 2’is a model of Z’. So, by Theorem 1.3.22, C has a model. The reduction of this model to 9gives a model of T which is clearly infinite. i COROLLARY 2.1.6 (Upward Lowenheim-Skolem-Tarski Theorem). Zf T has infinite models, then it has infinite models of any given power

a a 1 1 ~ 1 .1 PROOF.The proof is similar to that of Corollary 2.1.5. Let cc, < < a, be a list of distinct constant symbols not in 9, and consider the set of sentences

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MODELS CONSTRUCTED FROM CONSTANTS

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= c,) : 5 < q < u } . Every finite subset C‘ of I; will involve at most a finite number of the constants c c . Hence any infinite model of T can be expanded to a model of C‘. By Theorem 1.3.22, Z has a model BI and by Corollary 2.1.4, this model is of power at most Z = T u {-I (cc

1 1 9u {Ce : 5 < u}II

= u.

On the other hand, the interpretations of the constants c, in 3 must give distinct elements of A . So a < IAl < u and IAJ = u. i A result first published by Skolem (1934) is the following:

2. I .I. There exist nonstandard models of complete nwnber COROLLARY theory.

PROOF.Recall from 1.4.11 that complete number theory is the set of all sentences holding in the standard model ( 0 , +, S, 0) of number theory. Since this theory has an infinite model, it has models of all infinite powers. A noncountable model of complete number theory clearly cannot be standard. i a,

A simple but powerful device in model theory is the method of d’ragrams. Let 91 be a model for 2’.We expand the language 2’to a new language

9,= 2’u

{c, : a €A }

by adding a new constant symbol c, for each element a E A . tt is understood that if a # b, then c, and c,, are different symbols. We may then expand 91 to the model

MA = (91, a),sA for 9,by interpreting each new constant c, by the element a . The diagram of M, denoted by A,,,, is the set of all atomic sentences and negations of atomic sentences of ~Y, which hold in the model PI,. If X is a subset of A , then we let 9,be the language 2’u { c , : a E and ?Ix = (a, a),,, be the obvious expansion of 94 to S x Iff . is a mapping from X into the set of elements B of a model 93 for 9, then (8, fa),,x is the expansion of 23 to a model for gXformed by interpreting each c, byfu. The method of adding new constant symbols for elements of a model is used again and again in model theory. The following proposition illustrates the usefulness of diagrams.

x}

PROPOSITION 2.1.8. Let and B be models for 9 a n d let f : A + B . Then the following are equivalent:

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COMPLETENESS AND COMPACTNESS

(a). f is an isomorphic embedding of 1 ' 1 into B. (b). There is an extension E 3 '21 and an isomorphism g : E that g 3 f . (c). (B,fa),,, is a model of the diagram of 3 .

'$3 such

PROOF.The implication from (b) to (a) is trivial. If (a) holds, one can extend the set A to a set C and extend the function f to a one to one function g from C onto B. Then define the relations of E by the rule

K L R [ c , . . . c,] iff BkR[gc, . . . gc,] , and similarly for functions. This will make (b) hold for and g. To prove the equivalence of (a) and (c), use the fact that by Proposition 1.3.18, for each formula q ( x , . . . x , ) and all a , , . . . ,a, in A ,

'211q [ a , . . . a,] if and only if 2'1, k q ( a , . . . a,) and '$3 k q [f a ,

. . . fa,] if and only if (B,fa),,,

L q(a,

. . . a,). i

Proposition 2.1.8 shows that the following three conditions are equivalent: (a') '21 is isomorphically embeddable in '23. (b') B is isomorphic to an extension of 3 . (c') B can be expanded to a model of the diagram of '11. In the special case that A C B and f is the identity mapping from A into B, Proposition 2.1.8 shows that '21 is a submodel of B if and only if 23, is a model of the diagram of 3 . COROLLARY 2.1.9. Suppose that Y has no function or constant symbols. Let T be a theory in 9 and 31 be a model f o r Y . Then (ZT is isomorphically embedded in some model of T if and only if every Jinite submodel of 91 is isomorphically embedded in some model of T. PROOF.We skip the easy direction and suppose that every finite submodel of I[ is isomorphically embedded in some model of T. We show that the set Z = T u A , is consistent. Every finite subset C' of Z contains at most a finite number of the new constants, say c,, , . .., cam.Because the language 9 has no function or constant symbols, the finite set A' = ( a , , ..., a,,,} generates a finite submodel 3' of 91. Let 8'be a model of T in which '11' is isomorphically embedded. We see without difficulty that C' c T u A,, . So, by Proposition 2.1.8, 8'can be expanded to a model of I',and hence L' has a model. By compactness, C has a model 8.By Proposition 2.1.8 again, the reduct of 8 to 9gives a model of Tin which '21 is isomorphically embedded. -I

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We next consider two applications from the theory of fields (see 1.4.9).

+,

COROLLARY 2.1.10. Let T be a theory in the language 2 = { ., 0, l } ? which has as models fields with arbitrary high finite characteristics. Then T has a model which is afield of characteristic 0. PROOF.Let T’ be the theory of fields and consider the set

C = T u T’ u { p l f 0 : all primes p ] . Recall from Chapter 1 that p l is our abbreviation for the term I + ... + I , p times, of the language 2. A finite subset C‘ of C will involve a highest prime, sayp. Let 2l be a model of T which is a field, so M is also a model of T’, and such that the characteristic of M is higher than p . Then M is a model of C’, whence by compactness, C has a model. This model is a model of T , is a field, and has characteristic 0. i COROLLARY 2.1.1 1. There exist non-Archimedean ordered fields elementarily equivalent to the orderedfield of real numbers. PROOF.An ordered field ( F , +, -,0, 1, < ) is Archimedean iff for any two positive elements a, b in F there is an n such that nu 2 6. This is not expressible in first-order logic. Let T be the set of all sentences of 9= { +, *, 0, I , < } holding in the ordered field of reals. Let c be a constant symbol different from 0 and 1. Let

C

=

T v (nl

< c:nEw).

For every finite subset C’ of C, there is an expansion of the reals to a model of C’. By compactness, C has a model in which c has an interpretation b. In this model, both 1 and b are positive; yet no finite multiple of 1 can exceed 6. -I Corollary 2.1.11 is the very beginning of a branch of model theory called nonstandard analysis. The model theory of nonstandard analysis will be developed in Section 4.4. Consider A , , the diagram of 9 introduced earlier. We see that Proposition 2.1.8 gives an intimate connection between models of A , and models in which 24 can be isomorphically embedded. By the positive diagram of \I1 we mean the subset of A , which consists only of atomic sentences (no negations of atomic sentences). We shall see that positive diagrams are associated with the following notion of homomorphic embedding. Given models 9 and 8’ for 9, 9 is homomorphic to 9’iff there is a function f mapping A onto A‘ satisfying the following:

2.11

COMPLETENESS AND COMPACTNESS

71

(i). For each n-placed relation R of (21 and the corresponding relation R‘ of a’, and all elements x , , ..., x , of A ,

if R ( x l ... x,,), then R ‘ ( f ( x , ) ...f(x,,)). (ii). For each m-placed function G of %’, and for all x l , ..., x , of A ,

(21

and the corresponding G’ of

f ( G ( x 1 ... xm)) = G ’ ( f ( x 1 )...f ( X m ) ) .

(iii). For each constant x of (21, f ( x ) is the corresponding constant of 31’. A function f satisfying the above is called a homomorphism of (21 onto 31‘. We write (21 z j (21’ to indicate that f is such a homomorphism; if it is not necessary to indicatef, we write (21 N (21‘ for (21 is homomorphic to a’. In this case we also say (21’ is a homomorphic image of (21. (21 is homomorphically embedded in 81’ iff (21 is homomorphic to some submodel of (21’. See Exercise 2.1.3 for some elementary properties of these notions. The next proposition corresponds to Proposition 2.1.8.

PROPOSITION 2.1.12. Let 3, 8 be models f o r 9. Then (21 is homomorphically embedded in 8 if and only if some expansion o j 8 is a model of the positive diagram of PI. COROLLARY 2. I . 13. Every partial order on a set X can be extended to a simple order on X .

PROOF.Suppose that d partially orders X . Let 31 = ( X , d ). Let ( c , : X E X } be distinct constants for x E X and let A be the positive diagram of PI. Let C =A

u ( c , f c, : x # y in X } u {a},

where CT is the sentence which expresses that < is a simple order (see 1.4.I). Let Z’ be a finite subset of C involving, say, the elements x l , ..., x,, and the corresponding constants. We need the following fact: (1) Every partial order < on { x l , ..., x,,} can be extended to a simple order d ’ on { x l , ..., x,,} so that < is preserved, i.e., if x i d x i , then xi

<’Xj.

The proof of ( I ) is not difficult and proceeds by induction on n. Assuming ( I ) , we see that ( { x , , ..., x , , ] , d ’) is a model of C’. By compactness, C has a simply ordered model ( Y , < ’), in which there is an element y , corresponding to each constant c,. Clearly the set { y, : x E X } is simply ordered by 6 ’ . If x < z, then y , < ‘ y z , and if x # z , then y,r # y,. Using the inverse of the 1-1 function y : x + y,, we can induce a simple order on X which extends <. -I

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12.1

EXERCISES 2.1.1. Show that there are also countable nonstandard models of complete number theory.

2.1.2. Prove the representation theorem for Boolean algebras (Proposition 1.4.4) by the method of diagrams. [Hint: (a). Every atomic Boolean algebra is isomorphic to a field of sets. (b). Every finite subset of a Boolean algebra generates a finite, therefore atomic, Boolean algebra, ( c ) . If B is isomorphically embedded in a field of sets, then % is isomorphic to a field of sets.]

2.1.3. Prove the following. The homomorphism relation N is reflexive and transitive. It is not symmetric nor antisymmetric. If 91 Y W , then IAl 3 ( B I . A sentence 0 is called posirioc iff it is built up from atomic formulas using only A , v , 3, V. t f 91 = 8,0 is a positive sentence, and 91 k c,then ‘5 k 0 . Compare this with Exercise 1.3.5. 2.1 .J. Prove the assertion ( I ) in Corollary 2.1.13. 2.1.5. Show that every ordered field is equivalent to a non-Archimedean ordered field. 2.1.6. Show that every group which has elements of arbitrarily large finite order is equivalent to a group which has an element of infinite order.

2.1.7. Show that every model of ZF is equivalent to a (countable) model ( A , E ) which has an infinite sequence

... E x ~ E xEX^. ~ Therefore every model of ZF is equivalent to a countable model which is not isomorphic to a transitive model. 2.1.8. Let ‘u = ( A , < , ...) be an infinite model such that < well orders A . Show that there is a model a’ = ( A ’ , <’, ...) equivalent to ?I such that < ’ is not a well ordering. 2.1.9. Show that every infinite model 91 for a language 9has a n equivalent model 23 of power II-YlI such that not every element of B is a constant of 23.

2.1.10. Let 9have no function or constant symbols. Let T be a theory in 9 and 91 be a model for 9. Then ‘u is homomorphically embedded in some model of T if and only if every finite submodel of is homomorphically

2.11

COMPLETENESS AND COMPACTNESS

73

embedded in some model of T. (This is a homomorphism version of Corollary 2.1.9.) 2.1.11. Let ?I be an arbitrary infinite model and let LY 2 )(-5?lI.Then there is a model 23 equivalent to 9 such that for every formula cp(x) with one free variable, if cp(x) is satisfied by infinitely many different elements of 23, then cp(x) is satisfied by LY different elements of 23. 2.1.12. A model 2 is said to bejnitely generated iff there is a finite set X c B which generates 2 (see Exercise 1.3.9). Let T be a theory in 3' and let 9 be a model for 9. Then 9-lis isomorphically embedded in some model of T if and only if every finitely generated submodel of is isomorphically embedded in some model of T. (Compare with Corollary 2.1.9.) 2.1.13 (i). If T , and T2 are two theories such that T , u T2 has no models, then there is a sentence cp such that T , t= cp and T , b 1 cp. (ii). If T , and T , are two theories such that for all a, ?l is a model of T , iff ?( is not a model of T,, then T , and T, are finitely axiomatizable. 2.1.14. Let T , c T , c T3 c ... be a strictly increasing sequence of closed theories i n 9. Show that their union T = u n < w T ,is , a consistent closed theory in 9 and it is not finitely axiomatizable. 2.1.15. Let T,, n E w , be a strictly increasing sequence of closed theories in a finite language 2.Prove that U,T , has an infinite model. 2.1.16. Let T be a finitely axiomatizable theory with only a countable number of complete extensions in a language 2.Prove that T has a finitely axiomatizable complete extension in 2. 2.1.17. Prove that every complete theory T in a countable language has a model 2l of power ~ 2 such " that for every 8 L T and every S B there is an R C A such that (8,S) is elementarily equivalent to (a,R ) . 2.1.18*. Let A be the theory of dense linear order without endpoints. Prove the following lemma (a), and then use (a) and the LowenheimSkolem-Tarski Theorem to give a simpler proof of Theorem 1.5.3 on the elimination of quantifiers for the theory A . (a). Let 2l and B ! be countable models of A , a l , . . . , a , E A , and b , , . . . , b, E B . If a , , . . . , a, and b , , . . . , b, satisfy the same arrangement, then (%, a , . . . a,) ( 8 ,b , . . . b,).

=

74

MODELS CONSTRUCTED FROM CONSTANTS

[2.1

2.1.19*. Let 2'= 0 be the language of pure identity theory. Prove the following lemmas (a) and (b), and then use (a), (b), and the Lowenheim-Skolem-Tarski Theorem to give a simpler proof of Theorem 1.5.7 on the elimination of quantifiers for pure identity theory. (a). Let 'u and 23 be models for 9 o f the same cardinality, a , , . . . , a , E A , and b,, . . . , b, E B . If a , , . . . , a , and b,, . . . , b, satisfy the same arrangement, then ('u, a , . . . a , ) (23, b , . . . b"). (b). Let q ( x , , . . x , ) be a formula of 9 a n d B(x, . . . x,) be an arrangement. Let S be the set of all cardinals a such that cp A 0 is satisfiable in a model of cardinality a. Then either S or its complement is a finite set of finite cardinals. 2.1.20. This and some of the following exercises are designed to show an alternative method of proving the extended completeness theorem for countable languages. A generalization of this method to noncountable languages is also given later. Let 9 be a countable language, and let T be a consistent set of sentences closed under t-. We aim to prove that T has a model. The starting point of our discussion is the countable Lindenbaum algebra By.We have already seen in Exercise 1.4.1 1 that Tcorresponds to a filter in Bo, the Lindenbaum algebra of sentences of 9. It is also easy to show that the set @ = {cp : T I cp

and cp is a formula of 9}

is a filter in B. For simplicity, we shall now operate in the quotient algebra B/@.In other words, the equivalence classes of this new algebra are given by sets of formulas

(+)

= {cp

: TI - cp-+),

with its unit element given by the set @, and its zero element given by {'p:

Tt- lcp}.

We denote this quotient algebra by BTand call it the Lindenbaum algebra of T . BTis obviously a countable Boolean algebra. 2.1.21. Let 'u be any Boolean algebra and let Y be a subset of A . The sum of Y , or the 1.u.b. of Y, is defined to be the unique y E A such that x < y for all x E Y (i.e. y is an upper bound for Y ) , and if z E A is any upper bound for Y, then y < z. We denote the sum of Y if it exists by V Y , or if the elements of Y are indexed by I , yi. In an entirely similar manner, we can define the

vis,

2.11

75

COMPLETENESS AND COMPACTNESS

product of Y . or the g.1.b. of Y, and denote it by A Y or AiEIy i . Sums and products of arbitrary Y c A do not necessarily exist. When they do exist, they satisfy the following identities (assume that y i exists):

viE,

( V Yi)+x iEI

(

V

iEI

yi). x

= =

V

i€I

V

iE1

(Yi+x), (yi.~),

These identities imply, of course, that the sums and products on the righthand side also exist. We leave the duals involving A to the reader. Let cp be any formula of 9. Let cp(k/p) be the formula obtained from cp by first replacing all bound occurrences of up in cp by u j , the first variable i n the sequence u o , u , , ..., not occurring in cp, and then replacing all free occurrences of r:, by u p . Show that in the Boolean algebra &,

v

(cp(k/p)) =

((3uk)(P)?

PEW

A ( d k l ~ ) =) ((Vukb).

PEW

Thus sums and products of certain sets of substitution instances of a single formula cp always exist and correspond to existential and universal quantification of cp. Note that the number of such sums (and products) in BT is countable. 2.1.22. An ultrafilter D on % is said to preserve the sum

v yi E D

V i e yi l iff

if and only if some y i E D.

I C I

Similarly, D preseroes the producr

A

A i s , yi iff

yi E D if and only if all y i E D.

i C I

Prove the following: Given a countable sequence of products A X o , A X , , ..., AX,, ... of B. Then there exists an ultrafilter D on B which preserves each product. [Hint: Pick a sequence x, E X , such that no finite product of elements of the form A X , + Z , is equal to zero. Now consider any ultrafilter D which has as elements all AX, X, .] There is also a corresponding result about countable sequences of sums.

+

2.1.23. Let D be any ultrafilter on '%IT which preserves all the products of Exercise 2.1.21. We shall now construct a model of T from the variables

16

MODELS CONSTRUCTED FROM CONSTANTS

[2.1

u o , u , , ... of 9. Since the procedure is quite similar to that of Lemma 2.1.2, we ask the reader to fill in all the details. First define equivalence by ui

*

u j iff ( u i E u j ) € D.

The equivalence classes are denoted by iji. Let c be a constant symbol of 9’Since . k (3u0)(c = uo), and D preserves sums, we see that for some i, (c = u i ) E D. Let the interpretation of c be the class i i i . Let t be any term of 9 (this includes the cases of function symbols and constant symb,ols) and up be a variable not occurring in t . Then k ( 3 u p ) ( t = up). Since D preserves sums, for some j , ( t = u j ) E D. Let the interpretation of the term t (defined on equivalence classes iji) be i i j . Finally, let P be a relation symbol of 9. We define the relation R by R(iji, ... ijiJ

iff ( P ( u , , ... uin)) E D.

In this way, we have defined in an unambiguous manner a model % for 2 with universe the set of equivalence classes i j i . Now prove by induction on the formulas q ( u , ... u,) of 2’that

PI t=

cp[ijo

... fin]

iff

(cp(u,

... u,)) E D.

To pass through the cases of V or 3, we again need the fact that Dpreserves sums. Since cp E T implies that ( c p ) E D, this shows that ?I is a model of T.

2.1.24. If the language 9is uncountable, then the number of sums and products corresponding to 3 and V in Bj7is also uncountable. Even though, in general, Exercise 2. I . 17 fails for uncountable sequences of products in an arbitrary Boolean algebra, there is, nevertheless, a version of it which holds for the algebra Bj7.This is because every formula cp contains only a finite number of symbols. The generalization of Exercise 2.1.17 is as follows (the proof is straightforward): Let 8 be a Boolean algebra, a be an infinite cardinal, and A X , , p < a, be a sequence of products of ?I. Suppose that for all fl < SI and all filters E on % generated by fewer than SI elements, whenever X, c E , then

A X f lE E.

Then there exists an ultrafilter D on % which preserves each product A X , . Using this result, a generalization of the proof in Exercise 2.1.18 can be given for noncountable languages 9. (A technical detail should be

2.21

REFINEMENTS OF THE METHOD

77

mentioned: Before proceeding with the proof we must first expand 9 to a language 9 with 1 1 9 1 1new constant symbols. This is apparently necessary, see the proof of Lemma 2.1. I .) 2.2. Refinements of the method. Omitting types and interpolation theorems In this section, we shall give two refinements of the method used in Section 2.1 to construct countable models with additional properties. The first refinement will lead us to the omitting types theorem. At the moment, the possible ramifications of this technique to noncountable languages and models are not yet fully understood. We shall mention only a couple of results for noncountable languages. The starting point of our discussion is the notion of a set C of formulas of 9 in the (free) variables x , , ..., x,. Here we are using x , , x 2 . ... as names for arbitrary free variables of 9. We could just as well use v m l ,v , , ~ ,..., but we abhor double subscripts. The following is a precise definition: C is a set of formulas of 9 in the (free) variables xl, ..., x, (symbolically, C = Z(x, ... x,)) iff x , , ..., x,, are distinct individual variables and every formula Q in Z contains at most the variables x i , ..., x, free. We now introduce the convention Q = o ( x I ... x,), as we did for cp = q ( u , ... v,). If u = u(xl ... x,), then the notation (21 k a [ a , ... a,]

means that the sequence a , , ..., a, of A satisfies Q in (21 (see the section on satisfaction). It is useful also to introduce the notation

i?I k C [ a , ... a,] to mean that for every u E I;, a , , ..., a, satisfies Q in 8;in this case we say that a , , ..., a, satisJies, or realizes, C in i?I. If c,, ..., c, is a sequence of constant symbols, then u(cl ... c,) denotes the sentence formed’by simultaneously replacing each free Occurrence of x i , 1 < i < n, in Q by the corresponding ci.Sometimes we shall replace just lome of the variables by constants. If m < n, the notation u(cl ... c,,,x,,,+, ... x,) is self-explanatory. For reasons explained in Section 2.1 {before Lemma 2.1.1), we must be careful to use the above notation only in a context where the list of variables x , , .. ., x, is given, A completely unambiguous notation can be introduced, but at great cost in readability. For example, we could use the notation

i?I C a[al/xl ... a,/x,] for i?I k o[al... a,],

78

[2.2

MODELS CONSTRUCTED FROM CONSTANTS

o(cl/xl

... C,/X,,,X,+~

... x,,) for

o(cl

... C,X,+~

... x,).

Let C be a set of formulas in the variables xl, ..., x,, and let be a model for 9. We say that % ' realizes C iff some n-tuple of elements of A satisfies C in a. We say that omits C iff '% does not realize C. The phrase C is satisfiable in has exactly the same meaning as 91 realizes C. C is consistent iff I: is satisfiable in some model. 2.2.1. Let T be Peano arithmetic and let C(x) be the set EXAMPLE

(0 f x, so f x, sso f x, ...}. An element is said to be nonstandard iff it realizes C(x). The standard model of T omits C(x), while all the nonstandard models realize C(x). EXAMPLE 2.2.2. Let T be the theory of ordered fields and let C(x) be the set

{Z

< x,z+z < x,z + z + z < x, ...}.

An element is said to be positioe infinite iff it realizes C(x). An ordered field omits C(x) if and only if it is Archimedean. The ordered fields of rationals and reals omit C(x). Non-Archimedean ordered fields were constructed in the last section using the compactness theorem. EXAMPLE 2.2.3. Let T be the theory of Abelian groups and let C(x) be the set {x f 0,2x f o,3x $ 0,...}.

Elements which realize C(x) are said to be of infinite order. An Abelian group which omits C(x) is said to be a torsion group. Thus in a torsion group, every element has a finite multiple which is zero. EXAMPLE 2.2.4. Here is an example of a set of formulas with infinitely many variables. Let T be the theory of partial order and let C be the set {XI

< xo, x2 < x i , XJ < x2, ...}.

A model % of T omits C iff 8 is a well founded partial ordering. A linear ordering omits C iff it is a well ordering. EXAMPLE 2.2.5. By a type T ( x l ... x,,) in the variables x , , ..., x,, we mean a maximal consistent set of formulas of 9 in these variables. Given any model !? and I n-tuple a , , ..., a,, E A , the set T(xl ... x,) of all formulas y(xl ... x,,) satisfied by a , , ..., a,, is a type, and, in fact, is the unique type realized by a , , ...) a,,. It is called the type ofal, ...) a, in '%.

2.21

REFINEMENTS OF THE METHOD

79

EXAMPLE 2.2.6. Let 21 be the ordered field of real numbers. Then any two distinct elements a, b E A have different types. For if a < b, there is a rational number r with a < r < b ; hence a satisfies x < r, while b does not. Thus % realizes 2" different types in one variable. The next proposition answers the question: When is a set of formulas realized by some model of a theory T? Its proof is a simple application of the compactness theorem. PROPOSITION 2.2.7. Let T be a theory and let C = Z(x, ... x,,). The following are equivalent: (i). T has a model which realizes C. (ii). Everyfinite subset of C is realized in some model of T. (iii). T u {(3x, ... x,)(al A ... A a,) : m < w , a,, ..., a, E C} is consistent.

We shall say that a formula a(xL ... x,) is consistent with a theory T iff there is a model 21 of T which realizes {a}, and we say that C(xl ... x,) is consistent with T iff T has a model which realizes C. Thus (i)-(iii) above are all equivalent to the statement that C is consistent with T. We now take up the question: When is a set 1 of formulas in x , , ..., x,, omitted in some model of a theory T? This is a more difficult question, and we need more than the compactness theorem to answer it. The key theorem of this section, Theorem 2.2.9, gives a necessary and sufficient condition for T to have a model which omits C. The w-completeness theorem 2.2.13 is one of a long list of consequences of it. We shall use Theorem 2.2.9 in the next section, and again later on. If Z , is a finite set of formulas, then there is no problem in determining whether C can be omitted, because the sentence Cp = (1x1

... X,)(Ol

A

... A a,),

where C = {a1, ..., a,}, and its negation icp express, respectively, that C is realized or omitted. Thus the interesting case is where C is infinite. Let us first take another look at Lemma 2.1.2. So far, we have only used the property that every element of 21 is the interpretation of a constant C E C in a simple way, to show that \ A ( < IC(. In this section, we shall make much more use of that property of 21. The central idea in dealing with our problem is the notion of a theory locally realizing a set of formulas. Let C = C(x, ... x,) be a set of formulas of 9. A theory T i n DEP is said to locally realize C iff there is a formula ~ ( x .,.. x,,) in Ysuch that:

80

[2.2

MODELS CONSTRUCTED FROM CONSTANTS

(i). cp is consistent with T. (ii). For all cr E C, T C cp -+ cr. That is, every n-tuple in a model of T which satisfies cp realizes C. We say that T locally omits C iff T does not locally realize Z. Thus T locally omits C if and only if for every formula cp(x, ... x,) which is consistent with T, there exists 0 E C such that cp A icr is consistent with T . For complete theories we have a simple proposition:

PROPOSITION 2.2.8. Let T be a complete theory in 2, and let C = C(x, . .. x,) be a set of formulas of 2. If T has a model which omits C, then T locally omits C. PROOF. The proposition may be restated as follows: If T locally realizes Z, then every model of T realizes C. Suppose T locally realizes C and let cp(x, ... x,) be a formula consistent with T such that T 1 cp + 6 ,cr E C. Let 9l be a model of T. Since T is complete, TI=( 3 x , ... x,)cp. So some n-tuple a , , ..., a, satisfies cp in 91. Then a , , ..., a, satisfies each cr E C, and hence realizes C in 91. -I The omitting types theorem is a converse of the above proposition. It holds, in fact, for arbitrary consistent theories in a countable language.

THEOREM 2.2.9 (Omitting Types Theorem). Let T be a consistent theory in a countable language 9, and let C(x, ... x,) be a set of formulas. If T locally omits C, then T has a countable model which omits C. PROOF. To simplify notation, let Z(x) be a set of formulas in one variable x. Suppose T locally omits Z(x). Let C = {c,,, c, , ...} be a countable set of new constant symbols not already in 2 and let 2''= 2 ' u C . Then 2'' is countable. Arrange all the sentences of 9' in a list cpo, q n l , cp2, .... We shall construct an increasing sequence of consistent theories T = To c T , c ... c T, c ... such that: (1). Each T,,,is a consistent theory of

2'which is a finite extension of T.

, ,,

(2). Either (P, E T,, or (1 cp), E T,, . (3). If (P, = ( 3 x ) J / ( x )and qnm E T,, then J/(c,) E T,, where cp is the first constant not occurring in T, or q m . (4). There is a formula ~ ( x E) Z(x) such that (1 ~ ( c , ) E) T,,

,,

2.21

81

REFINEMENTS OF THE METHOD

Assuming we already have the theory T,, we construct T,, as follows: . ..., r, Let T, = T u {O,, ..., O r } , r > 0, and let 0 = 0 , A ... ~ 0 Let~ cO, contain all the constants from C occurring in 0. Form the formula O(x,) of 2 by replacing each constant ci by x i (renaming bound variables if necessary), and prefixing by ] x i , i f m. Then O ( x m )is consistent with T. Therefore, for some ~ ( xE)Z(x), O(x,,) A ia(x,) i s consistent with T. Put the sentence ia(c,) into T,+,. This makes (4) hold. If cpm is consistent with T, u {a(c,)}, i put cp, into T,,, Otherwise put (1 cp), into T,, . This takes care of ( 2 ) . Lf cp, = (3x)$(x)is consistent with T, u (1 ~ ( c , ) } put , $(c,) into T,, . This takes care of (3). The theory T,,, is a consistent finite extension of: T,. Thus (1)-(4) hold for

,.

,

Tm.+

,

1*

Let T, = U,,,wT,,. From ( I ) and ( 2 ) we see that T, is a maximal conLet 23’ = (23, b o , b , , ...) be a countable model of T,,, sistent theory in 9‘. and let a’ = (a, bo,b , , . ..) be the submodel of 8’generated by the constants b,, b , , .... We then see from (3) that A = { b o ,b , , ...}.

Moreover, using (3) and the completeness of T,,, we can show by induction on the complexity of a sentence cp in 2” that

9l‘k cp,

% ’ k cp,

T, t= cp

are all equivalent. Thus &I’ is a model of T, and hence Finally, condition (4) ensures that 2l omits Z.-I

2l is a model of T.

When T is a complete theory, we see that locally omitting C(xl ... x,) is

a necessary and sufficient condition for T to have a model omitting Z. Here is a necessary and sufficient condition which works in general.

COROLLARY 2.2.10. Let 2’ be countable. A theory T has a (countable) model omitting C(x, ... x,) if and only if some complete extension of T locally omits Z(x, .,. x,,).

EXAMPLE 2.2.11. Consider the language X = { +, ., S, O}. We abbreviate I = SO, 2 = SSO, 3 = SSSO, .... By an w-model we mean a model 2l in which A = { 0 , 1 , 2 , 3,... } , that is, 2l omits the set {x f 0,x f I , x f 2, ...}. A theory T i n 2’ is said to be w-consistent iff there is no formula q ( x ) of 9such that

82

and

MODELS CONSTRUCTED FROM CONSTANTS

T + cp(O),

T b cp(I),

[2.2

T k ~ ( 2. . .) ~

T b (3x) icp(x). T is said to be w-complete iff for every formula cp(x) of 2 we have

T k cp(O), T k cp(l), T k cp(2), ... implies T 1 (Vx)cp(x). It follows from the omitting types theorem that:

PROPOSITION 2.2.12. Let T be a consistent theory in 9. (1). I f T is w-compfere, then T has an w-model. (ii). I f T has an w-model, then T is w-consistent. PROOF.(i). Weshowthat TlocallyomitsthesetC(x) = {x f 0,x f I , ...}. Suppose O(x) is consistent with T. Then TI=(Vx) 1 O(x) fails. By ocompleteness, there is an n such that not T 1 iO(n). Hence O(n) is consistent with T, so O(X)A i x f n is consistent with T. Thus T locally omits C(x). (ii). Trivial. i The w-rule is the following infinite rule of proof: From cp(0), cp(l),cp(2), ..., infer (Vx)cp(x), where cp(x) is any formula of 9. o-logic is formed by adding the w-rule to the axioms and rules of inference of the first-order logic 9and allowing infinitely long proofs. We have the following completeness theorem for w-logic.

PROPOSITION 2.2.13 (o-Completeness Theorem). A theory T in 9 is consistent in w-logic if and only i f T has an w-mode!.

PROOF.Let T' be the set of all sentences of 2 provable from Tin o-logic. Then Tis consistent in w-logic if and only if T' is consistent in 9. Moreover, T' is o-complete. Therefore T' has an w-model if and only if T' is consistent. i The formulation of w-logic above is aimed at studying the standard model of arithmetic. A useful generalization, which we shall call generalized o-logic, is aimed at studying ordinary models for first order logic enriched by a symbol for the set of natural numbers.

EXAMPLE 2.2.11'. Let 2' be a countable language which has among its symbols a special unary relation symbol N and special constant symbols

2.21

83

REFINEMENTS OF THE METHOD

0 , 1 , 2 , . . . . By an w-model for 2'we mean a model '2.l for 2'in which N is interpreted by the set w of natural numbers, and 0, 1 , 2 , . . . are interpreted by themselves. In an w-model, w is a subset of the universe A , but we allow A to contain elements outside of w or even to be uncountable. Let T , be the special set of sentences

TN = { N ( m ) : m < w } U { i m = n : m < n < w } which state that the natural numbers are distinct and belong to N . T N holds in every w-model for 2'. A theory T in 2'is said to be w-consistent iff there is no formula ~ ( x of) 2'such that

T NU T k cp(O), T , U T k cp(l), T , U T k p(2),

...

and T N U T k ( 3 x ) ( N ( x )A icp(x)).

T is said to be w-complete iff for every formula ~ ( x of ) 2" we have T , U T != cp(O), T , U T L cp(l), T , U T k p(2),

...

implies T N u T ( V x ) ( N x ) - , cp(x)).

The w-rule for 2' is the infinite rule: From cp(O), cp(l), cp(2), . . . , infer (Vx)(N(x)+ &)). By generalized w-logic we mean first order logic for the language 2'with T , added as an additional set of logical axioms and the w-rule added as an additional rule of proof. Propositions 2.2.12 and 2.2.13 take the following form for generalized w -logic. PROPOSITION 2.2.12'. Let T be a theory in 2'such that T , U T is consistent. (i). If T is w-complete, then T has an w-model. (ii). If T has an w-model, then T is w-consistent. PROPOSITION 2.2.13'. A theory T in 9' is consistent in generalized w-logic if and only if T has an w-model. The following example shows that the omitting types theorem fails for sets of formulas with infinitely many free variables.

84

MODELS CONSTRUCTED FROM CONSTANTS

[2.2

EXAMPLE 2.2.14. Let T be the theory of dense linear order without endpoints. Thus T is complete. Let Z(xox,xz ...) be the set {XI

< x0,xz < x1,x3 < x2, ...}.

As we observed before, a model 9l omits C if and only if is a well ordering. But T has no well ordered models, so no model of T omits C. However, T does locally omit Z, because if cp(xoxl ... x,,) is consistent with T, then cp A ix , , + < ~ x , , + ~is consistent with T. The omitting types theorem can be generalized to the case of countably many sets of formulas.

THEOREM 2.2.I5 (Extended Omitting Types Theorem). Let T be a consistent and for each r < o let .Y,(x, ... xn,) theory in a countable language 9, be a set of formulas in n, variables. If T locally omits each Z,, then T has a countable model which omits each C , . PROOF.Similar to the proof of the omitting types theorem. The only difference is that for each r the n,-tuples of new constants are arranged in a list: s:, s : t 1 , s : t z ,

....

The theories T, are built up so that for each r formula 0 E Z, such that (~(s:)) i E T,, -1

=

0, 1, ..., m, there is a

Here is a first application of the extended omitting types theorem. It uses the notion of an elementary extension which plays an important role in the rest of this book.

2.2.16. 8 is said to be an elementary exfension of 91, 91 < 9,iff (i). 23 is an extension of 'II,9I c 23. (ii). For any formula cp(x, ... x,,) of 9and any a , , ..., a,, E A , a l , ..., a,, satisfies cp in 'IIif and only if it satisfies cp in 8. When 23 is an elementary extension of PI we also say that BI is an elementary submodel of 8. A mapping f : A B is said to be an elementary embedding of '21 into 93, in symbols f: ?I < 93, iff for all formulas p(x, . . . x , ) of 2 and n-tuples a , , . . . , a , E A , we have % k p [ a , . . .a,] if and only if % k p [ f a , . . . f a , , ] .

2.21

REFINEMENTS OF THE METHOD

85

An elementary embedding of 'u into 23 is thus the same thing as an isomorphism of 8 onto an elementary submodel of 23. The following analogue of Proposition 2.1.8 is often useful.

PROPOSITION 2.2.17. Let 'u and 23 be models f o r 3 and let f : A+ B. Then the following are equivalent: (a). f is an elementary embedding of 8 into 23. (b). There is an elementary extension G > 8 and an isomorphism g : G G % such that g > f . (c). ('8, fa),,A is a model of the elementary diagram of 9. Proposition 2.2.17 shows that the following three conditions are equivalent: (a') 2l is elementarily embeddable in 23. (b') 23 is isomorphic to an elementary extension of 8. (c') '8 can be expanded to a model of the elementary diagram of 'u. In the special case that A C B and f is the indentity mapping from A into B , Proposition 2.2.17 shows that 'u is an elementary submodel of 23 if and only if B A is a model of the elementary diagram of 'u. Let us now consider the theory ZF, Zermelo-Fraenkel set theory. A model 23 = ( B , F ) of Z F is said to be an end extension of a model 9.l = ( A , E ) of ZF iff 8 ! is a proper extension of H and no member of A gets a new element, that is, if a e A

and b E B , then hFa implies b e A.

THEOREM 2.2.18. Every countable model ?1 = ( A , E ) of Z F has an end elementary extension.

PROOF. Let 9 be the language with the symbol e, a constant symbol ij for each a E A , and a new constant symbol c. Let T be the theory with the axioms WW, c

# ii, where

a EA.

T is consistent because every finite subset of T has a model of the form (91, a, c),,~. For each a E A , let Z o ( x )be the set of formulas

Z,(x) = { x E a } u { x f b : bEa}.

86

MODELS CONSTRUCTED FROM CONSTANTS

[2.2

It suffices to show that T locally omits each Fet C,(x). For then T has a model (23,a, c),, which omits each C,(x). We may also assume that A c B. 8 is an elementary extension of 2l because Th((U, a),,,) t T, whence (a, a),,, = (23, a),,,. 23 is a proper extension because c E B\A. Finally, 9 is an end extension because it omits each C,(x). To see that T locally omits each Z,(x), we note that a formula q(x, c) of 2' is consistent with T if and only if

Suppose q ( x , c ) is consistent with T, but q(x, C ) A ix z is not. Then q(x, c) A xzZ is consistent with T. Using the axiom of replacement in ZF, we see in turn that the following sentences hold in (%, a),,,:

Then for some bE A, q(6, c ) ~ 6 a Bis consistent with T, whence 5 6 is consistent with T. Thus T locally omits C,(x). -I

q ( x , c) A x

The omitting types theorem as it stands is false for uncountable languages. For example, let T be the theory with the axioms

in the language 2%' with constants {c. : u

c a,}u {d,, : n < w } .

Let T(x) be the set of formulas T(x) =

{X

f d, : n c o } .

Then T locally omits T(x). However no model of T omits T(x) because every model of T is uncountable but each model which omits T ( x ) is countable. A more complicated counterexample where the theory T is complete has been given by Fuhrken (1962).

2.21

REFINEMENTS OF THE METHOD

81

However, the omitting types theorem can be generalized to uncountable languages if we define the notion of ‘locally omits’ in the proper way. Let T be a theory and Z ( x , .. . x,) a set of formulas in a language 3 of power a. We say that T a-realizes Z iff there is a set @ ( x 1... x,) of fewer than a formulas of 9such that: (i). Q is consistent with T, (ii). T u @ ( x 1... x,) t C ( x , ... x,), that is, in any model 2i of T, any n-tuple which realizes 95 realizes C. T is said to a-omit C ( x , ... x,) iff T does not a-realize Z(xl ... x,,). Note that if Z has power less than a, then T a-realizes C trivially. Thus only sets of formulas of power a can ever be a-omitted.

THEOREM 2.2.19 (a-Omitting Types Theorem). Let T be a consistent theory in a language 9 of power a and let Z(xl .. . x , ) be a set of formulas of 2. If T a-omits Z, then T has a model of power < u which omits C. The proof is like the proof of the omitting types theorem. An important problem is to find a useful sufficientcondition for a theory in an uncountable language to have a model which omits a countable set of formulas. The a-omitting types theorem is of no help here since a countable set of formulas is never a-omitted when a > o. We now turn to the interpolation theorems of Craig and Lyndon.

THEOREM 2.2.20 (Craig Interpolation Theorem). Let cp, t,b be sentences such that cp t $. Then there exists a sentence 8 such that: (i). c p k B a n d B C $ . (ii). Every relation, function or constant symbol (excluding identity) which occurs in 8 also occurs in both cp and I). The sentence 8 will be called a Craig interpolant of cp, $. The identity symbol is allowed to occur in 8. The following example shows why this is necessary. EXAMPLE 2.2.21. In each of the following, cp and $ are sentences such that the identity symbol occurs in at most one of them, and cp i=$; however, cp, II/ have no Craig interpolant in which the identity symbol does not occur: (i). cp is ( 3 x ) ( P ( x )A iP ( x ) ) , II/ is ( 3 x ) Q ( x ) ; $ is ( 3 x ) ( P ( x )v 1 P ( x ) ) ; (ii). cp is (3x)Q(x), (iii). cp is ( ~ x y ) ( = x y), $ is (Vxy)(P(x)c-f P ( y ) ) .

88

MODELS CONSTRUCTED FROM CONSTANTS

[2.2

We shall see in an exercise, however, that in the Craig interpolation theorem, if the identity symbol occurs i n neither cp nor $, and if not k icp and not k I,+, then cp and I) have a Craig interpolant in which the identity symbol does not occur. 2.2.20. We assume that there is no Craig interpolant PROOFOF THEOREM 0 of cp and $, and prove that it is not the case that cp != $. To do this we construct a model of cp A i$. We may assume without loss of generality that 9is the language of all symbols which occur in either cp or Ic/ or both. Let 9, be the language of all symbols of cp, Y 2the language of all symbols of $, and 9, the language of all symbols occurring in both cp and $. Thus

Zl n P2= Y o ,

2,u 9,= 9.

Form an expansion 9' of 9by adding a countable set C of new constant symbols and let

9;= 9, u c,

9; = 2Yl v

c,

9; = ?Y2 u c.

The proof will resemble the proofs of the completeness and omitting types theorems, but the notion of a consistent theory will be replaced by the more general notion of an inseparable pair of theories. Consider a pair of theories T in Plio;and U in 9;.A sentence 8 of 2; is said to separate T and U iff

T ! = 8 and

U b 8 .

T and U are said to be inseparable iff no sentence 8 of 2;separates them. To begin with, we see that (1) {cp} and {I$} areinseparable. For, if O(c, ... c,,) separates {cp} and {$} i and u1 ..., u,, are variables not occurring in e(cl ... c"), then (Vu, ... u,,)O(u, ... u,) is a Craig interpolant of cp and $, contrary to our assumption. Now let 'Po9

cpI,cpZ, ..9$0,$19*2r...

be enumerations of all sentences of 9; and of 9;, respectively. We shall construct two increasing sequences of theories, { c p } = To c T , c T2 c ..., u, c u, c u, c ...

{l*} =

in -2'; and 9;, respectively, such that: (2). T,,, and U,,, are inseparable finite sets of sentences.

2.21

REFINEMENTS OF THE METHOD

89

(3). If T , u {cp,} and U , are inseparable, then q, E T,, 1 . If T,+ and U , u {JI,} are inseparable, then JI, E U,, 1 . (4). If cp, = (3x)a(x) and q+,, E T,, 1 , then o(c) E T,, for some c E C. If JI, = ( 3 x ) 6 ( x ) and $, E Urn+1 , then 6 ( d )E Urn+ for some d E C.

,

Given T , and U,, the theories T, + and then U,, are constructed in the obvious way. For (4), use constants c and d which do not occur in T,, U,,,, q, or $., Then inseparability will be preserved. Let

uw

Tcu = ( ) m < , T m ,

= Um
Then T, and U, are inseparable. It follows that T, and U, are each consistent. We must show that T, v U, is consistent. We show first that: ( 5 ) . T, is a maximal consistent theory in 9; , and U , is a maximal consistent theory in 9;. To show this, suppose q m $T, and ( - Icp,)$T,. Since T, u {cp,} is separable from U,, there exists 8 E 9; such that We see by the same argument that there exists 8' E 9; such that

T, t

7

p m --*

u, t 18'.

e',

But then

T, t e v e',

u,

(e v 09,

1

contradicting the inseparability of T, and U,. This shows that T, is maximal The maximality of U, is similar. consistent in 9;. Our next observation is: ( 6 ) . T, n U, is a maximal consistent theory in 9;. To prove (6), let a be a sentence of 9;. By (9,either a E T, or (1 u) E T,, and either a E U , or (1a) E U , . By inseparability, we cannot have CT E T, and (1a) E U,, or vice versa. Therefore either T, n U , k a or T , n U,, k ia. We are now ready to construct a m'odel. Let 23; = ( B l ,bo, b , , ...) be a model of T,. Using (4) and ( 5 ) , we see that the submodel 2l; = (a,, b,, b , , ...) with universe A , = {b,,, b , , ...) is also a model of T<,,. Similarly, U , has a model 9ii = (912, do, d , , ...) with universe A , = {do,d , , ...}. By (6), the 9; reducts of 81; and 91; are isomorphic, with b, corresponding to d,. We may therefore take b,, = d,, for each 11, whence '21, and '$I2 have the same z0 reduct. Let 91 be the model for 9 with 2,reduct 81, and P2reduct 9i2. Since cp E T,,, and (1$) E U,,, Yisamodel ofcpA71(1. -1

90

[2.2

MODELS CONSTRUCTED FROM CONSTANTS

We give two applications of the Craig interpolation theorem. The first application deals with ways of defining a relation. Let P and P’ be two new n-placed relation symbols, not in the language 9. Let Z ( P ) be a set of sentences of the language 9 u { P } , and let Z(P’) be the corresponding set of sentences of Y u { P ‘ } formed by replacing P everywhere by I“. We say that Z ( P ) defines P implicitly iff Z(P) u Z(P’)k

(VX,

... x , ) [ P ( x , ... x , ) tf P ’ ( x , ... x , ) ] .

Equivalently, if (91, R ) and (a, R ’ ) are models of Z ( P ) , then R = R‘. Z(P) is said to define P explicitly iff there exists a formula q ( x , ... x , ) of Y such that

Z(P)k

(VX,

... x , ) [ P ( x , ... x , ) ++ q(x1 ... X J l .

It is obvious that, if Z ( P ) defines P explicitly, then Z ( P ) defines P implicitly. Thus, to show that Z ( P ) does not define P explicitly, it suffices to find two models (a, R ) and (9lY R ’ ) of Z ( P ) , with the same reduct (If to 9, such that R # R‘. This is a useful classical method known as Padoa’s method. We now prove the converse of Padoa’s method.

THEOREM 2.2.22 (Beth‘s Theorem). Z(P) defines P implicitly Z(P)defines P explicitly.

if and

only

if

PROOF.We prove only the ‘hard’ direction. Suppose that Z ( P ) defines P implicitly. Add new constants c , , ..., c,, to 2. Then Z ( P ) u Z ( P ’ ) k P ( c , ... c,)

+

P’(C1 . * * c,).

By the compactness theorem, there exist finite subsets A c Z ( P ) , A ‘ such that A u A’ k P ( c l

... c,)

4

= Z(P’)

P ‘ ( c , ... c,).

Let $ ( P ) be the conjunction of all a(P) E Z ( P ) such that either a(P) E A or a(P’) E A ‘ . Then $ ( P ) A lf!/(P’)k P(C1 .. . C , )

+ P’(C1

... C,,).

Rearranging to get all symbols P on one side and all symbols P’on the other,

$(P)A P ( C 1 .. . C,) k $ ( P ’ ) + P’(C, . .. C,). Then, by the Craig interpolation theorem, there is a sentence U(c, ... c,) of 9u {cI ... c,} such that (1)

I/!’(P)AP(C, ... C,) k

o(C1

... C,,),

2.21

(2)

91

REFINEMENTS OF THE METHOD

qc,

... c,) c +(pi)-, P ’ ( c , ... c,).

But any model (H, R ’ ) for Y u {P’,c , , ..., c,,} is also a model for 9u {P, c, , ..., c,} when we interpret P by R’. Thus (2) implies

qC1 ... c,) c +(P)-,P ( C , ... c,).

(3)

Now (1) and (3) yield

+(P)c

(4)

... c,)

+,

qc,

... c,).

Since cl, ..., c, do not occur in +(P) (which is built from Z ( P ) ) , we have

$(P)

vxl ... X,[P(~,... x,)

+,

e(x,

... XJI,

where x , , ..., x, are variables not occurring in 8(cl ... c,). Therefore

THEOREM 2.2.23 (Robinson Consistency Theorem). Let 9, and 2,be two languages and let 9 = Yl n 9,. Suppose T is a complete theory in 9, und T , 3 T, T, 3 T are consistent theories in 9,, Y,, respectively. Then T I u T2 is consistent in the language 9, u 9,. PROOF.Suppose T , u T, is inconsistent. Then there exist finite subsets Z, c T , , C, c T2 such that C, u C, is inconsistent. Let 6,be the conjunction of C, and t~, the conjunction of C,. It follows that t ~ , =! it~,. By the Craig interpolation theorem, there is a sentence 8 such that t ~ , t 8,8 C i02, and every relation, function or constant symbol occurring in 8 occurs in both 0 , and 6,. Consequently, 8 is a sentence of Yl n Y 2= 2’. Now returning to T , and T,, we find that T , 18. Since T , is consistent, T , J i8, so T J i8. Moreover, T2 C i8, and, by the consistency of T,, T2 J 8 ; so T F , 8. But this contradicts the hypothesis that T is a complete theory in

2. -1 The Lyndon interpolation theorem is an improvement of the Craig interpolation theorem, but it holds only for languages which have no function or constant symbols. In order to state it, we need the notions of a positive and a negative occurrence of a symbol in a formula. In the following discussion we shall consider only formulas which are built up using the connectives A , v , 1,and the quantifiers V, 3. We do not allow the connectives 4,+,. [Strictly speaking, the language 9 was defined in Section 1.2 so that the only connectives are A and 1,and the only quantifier is V. The other con-

92

[2.2

MODELS CONSTRUCTED FROM CONSTANTS

nectives and 3 were introduced as abbreviations. Thus we now wish to avoid using the abbreviations +, -.I We now shall consider more closely the ways in which a symbol can occur i n a sentence. Let s be a symbol of 9,and let cp be a sentence of 9. Then s is said to occur positively in cp iff s has an occurrence in cp which is within the scope of an even number of negation symbols. The symbol s occurs negatively in cp iff s has an occurrence in cp which is within the scope of an odd number of negation symbols. Remembering that s may have several different occurrences in cp, we see that there are four possibilities: s does not occur in c p ; s occurs positively in cp; s occurs negatively in cp; s occurs both positively and negatively in cp.

-

The reason we do not want to use the abbreviations -+ and is that they contain ‘hidden’ negation symbols. For example, the sentence P(c) + Q ( c ) is an abbreviation of i( P ( c )A iQ(c)), so P occurs negatively but not positively in it, and the constant c occurs both positively and negatively in it. On the other hand, the abbreviations cp V $ = 1 (1cp A 1 $),

( 3 X ) c p = 1 ( V X ) 1 cp

will not cause any trouble in deciding whether a symbol s occurs positively or negatively, because they introduce exactly two ‘hidden’ negation symbols about cp, II/, and two is an even number.

+

THEOREM 2.2.24 (Lyndon Interpolation Theorem). Let cp, be sentences of 2 such that cp k Then there is a sentence 8 of 2 s u c h that: (i). cp k 8 and 0 k +. (ii). Every relation symbol (excluding equality) which occurs positively in 0 occurs positively in both cp and (iii). Same as (ii) for ‘negatively’.

+.

+.

The following simple example shows that we cannot find an interpolant 0 which satisfies (ii) and (iii) for constant symbols: (3X)(X

C A 1R ( X ) )

k

1 R(C).

Note that c is positive on the left, negative on the right, but must occur in any interpolant.

2.21

93

REFINEMENTS OF THE METHOD

PROOF OF THEOREM 2.2.24. The proof is obtained by making only a very few changes in the proof of the Craig interpolation theorem. We begin by assuming that there is no sentence 8 such that (i)-(iii) hold, and prove that cp A iI(/ has a model. Form the expansion 2” = 9u C as before. A formula is said to be in negation normalform (nnf) iff it is built up from atomic formulas and their negations using A , v , 3, V. Every formula is equivalent to an nnf formula. We assume that cp and are nnf formulas. Let a* denote the nnf of iQ. This time, the notion of an inseparable pair of theories is defined as follows. Let @ be the set of all nnf sentences a of 3‘such that every relation symbol which occurs positively (or negatively) in a also occurs positively (negatively) in cp. The set Y is defined similarly with respect to I(/. Let Y * = { u * : u E Y } . Two theories T c @ and U c Y * are said to be inseparable iff there is no sentence 8 E @ n Y such that T b 0 and U k i8. Using this notion we can apply the construction given in the proof of the Craig interpolation theorem to obtain a model of cp A $ * . This time we enumerate the sets of sentences 0 and T instead of the languages 2’;and Y;, and then construct T,, and U, as before. Some changes are needed in the rest of the proof because the sets @ and P ! are not necessarily closed under negation. Instead of proving that T , is maximal consistent, show that if u v 8 E T , then either u E T , or 8 E T,, and similarly for U , . Then show that T , and U, have the same equations and inequalities, and that the set A for all atomic and negated atomic sentences in T, U U, is consistent. Finally, let % be a model of A whose universe is the set of all constants, and prove by induction on complexity of formulas that is a model of both T , and U , and therefore a model

+

Of Cp A

$*. -1

A suggestion for further reading: The book “Building Models by Games” by Hodges [1985] gives an interesting treatment of a wide variety of applications of the Henkin construction in model theory. EXERCISES 2.2.1. Let T be a complete theory in a countable language, and let f ](XI), f 2 ( x 2 ) ,r3(x3),... be a countable set of sets of formulas such that each f , ( x , ) is consistent with T. Prove that T has a countable model which realizes each set f,(x,). 2.2.2. Let T be a complete theory. Show that T has a model

such that

94

MODELS CONSTRUCTED FROM CONSTANTS

[2.2

every set of formulas f ( x I ,x 2 , ...) which is consistent with T is realized in %. 2.2.3. Let ?A = ( A , < , +, ., 0, 1) be an ordered field. An element a E A is said to be finite iff there is an ti < w such that - n < a < n. Suppose that for any formula cp(x). if ?l C (3x)cp(x),then there is a finite a E A such that ?l k cp[a]. Show that ?[ is elementarily equivalent to an Archimedeanordered field. 2.2.4. Let T be a theory in a countable 2 ' and let Z ( x ) and d(y) be two sets of formulas of 9 which are consistent with T. Suppose that for every formula cp(x, y ) of 9 there exists O ( X ) E Z(x) such that for all S,(y), ..., S , ( ~ ) E d(y): if {cp, a,, ..., S,} is consistent with T, then {cp, S,, ..., S,, iO } is consistent with T. Prove that T has a model realizing A ( ) ? ) and omitting Z(x). 2.2.5. Let T be a complete theory in a countable language 9. Suppose that for each n < o,T has a model Yl,, omitting the set of formulas Z,(x). Prove that T has a model 91 which omits each Z,,(x). 2.2.6. Let 2 ' be a countable language and let 2''= 9u { P o , P , , ...} be a countable expansion of 9. Let T' be a maximal consistent theory in 2" and f ( x ) a set of formulas of 9. Suppose that for each n, the restriction of T' to 9u { P o , P I , ..., P,,} has a model which omits f ( x ) . Prove that T' has a model omitting T ( x ) . 2.2.7. Prove that there is an ordinal CL < o1such that every formula q of w-logic which has a proof has a proof of length less than a. 2.2.8. Show that the compactness theorem fails for o-logic. 2.2.9. Show that the Lowenheim-Skolem-Tarski theorem fails for models of T which omit Z. 2.2.10*. A model 8 of Peano arithmetic is said to be an end extension of 2l iff 8 is a proper extension of 91 and, for all b E B and a E A, if b < a, then b E A . Prove that every countable model of Peano arithmetic has an end elementary extension.

2.2.11*. Prove the following Restricted Omitting Types Theorem. Let 3' be a countable language and let T be a consistent V3 theory in 9, that is, a theory whose axioms are sentences of the form

2.21

REFINEMENTS OF THE METHOD

95

where cp has no quantifiers. For each n < w , let C, ( x , . . . x k ) be a set of universal formulas of 2. Suppose that for each n and each existential formula 8 ( x , . . . x k ) consistent with T, there is a formula ~ ( x . ,. . x k ) E C , ( x , . . . x k ) such that 8 A i f f is consistent with T. Prove that T has a countable model which omits each C , ( x , . . . x k ) . [Hint:The proof is similar to that of the Extended Omitting Types Theorem.] 2.2.12”. Deduce the Craig interpolation theorem from the Robinson consistency theorem. 2.2.13. Let 1,r be sets of sentences of Y such that Z u r is inconsistent. Then there exists a sentence 0 of Y s u c h that: (i). Z 1 8 and r k i8. (ii). Every relation, function or constant symbol which occurs in 8 occurs in some member of 1 and in some member of r.

2.2.14 (i). Show that the Robinson consistency theorem fails if T is not assumed to be complete. (ii). Show that the Robinson consistency theorem holds if the hypothesis that T is complete is replaced by the hypothesis that T is consistent, and for i = 1,2, T contains every consequence of Ti in 2’. 2.2.15. Prove the Craig interpolation theorem for formulas ~ ( x . ., . x,,), $(xl ... x,,). It can be deduced easily from the Craig interpolation theorem for sentences. 2.2.16. Assume Y has no function or constant symbols. Suppose that a set of sentences 1(P) of 2 u {P} defines P implicitly. Then there is a formula cp(x, ... x,) of 9 s u c h that: (i). C ( P ) t- P ( x , ... x,) cp(x, ... x,). (ii). Any symbol of Y which occurs in cp occurs both positively and negatively in Z(P).

-

2.2.17. Let 2” be an expansion of the language 2’and let P be an n-placed relation symbol in 2’\9. Let T be a theory in 9’. Suppose that for any model 2l for 9 and any two expansions a’, 2l” of % to models of T, the relations of 3’ and %” corresponding to P are the same. Prove that there exists a formula 8 ( x , ... x,) of 9 such that

96

MODELS CONSTRUnED FROM CONSTANTS

T F ~ ( x ... , x,,) ++ e(x,

[2.3

... XJ.

2.2.18. Let 9' be an expansion of 9and let T' be a theory in 9'. Suppose that each model for 9 has at most one expansion to a model T'. Prove that there is a theory Tin 9such that the models of T are exactly the reducts of the models of T' to 9.

2.2.19*. Show that the Lyndon interpolation theorem remains true when we add the conclusion: (iv). If cp is a universal sentence, then so is 8. Alternatively, it holds when we add: (iv'). If I(/ is an existential sentence, then so is 8. However, the theorem becomes false if we add both the extra conclusions (iv), (iv') at the same time. 2.2.20. Show that the Craig and Lyndon interpolation theorems hold with the following additional conclusion: (iv). Cf not I= icp, not I= cc/, and the identity symbol occurs in neither cp nor I(/, then the identity symbol does not occur in 8.

2.2.21*. Show that there is a model 'u of Peano arithmetic which has an infinite element x such that no y < x realizes the same complete type as x in 'u. 2.2.22*. Show that Peano arithmetic has two models 91 and 8 such that (B, +) but not 'u B. [Hint: Use Beth's Theorem.] (A, +)

2.2.23*. Let T be a complete theory in a countable language and let T(x) be a type over T which is consistent with T and locally omitted by T. Prove that T has a model in which infinitely many elements realize Q x ) . 2.2.24*. Let S be a set of fewer than 2" types T(x) which are maximal consistent with T and locally omitted by T. Prove that T has a countable model which simultaneously omits each T(x)E S. [Hint: Represent the Henkin construction by a binary tree.] 2.3. Countable models of complete theories In this section, we assume that 2 is a countable language. We shall embark on a thorough study of countable models of a complete theory. This study will give insight into what can be expected in general. Our study will center on two kinds of countable models, the atomic models, which are 'small', and the countably saturated models, which are 'large'. We begin with the atomic models.

2.31

COUNTABLE MODELS OF COMPLETE THEORIES

97

Consider a complete theory T i n 9. A formula cp(x, ... x,) is said to be coniplete (in T) iff for every formula @ ( x I ... x,) exactly one of Tkcp+@,

TCcp+i@

holds. A formula B(x, .. . x,) is said to be completable (in T) iff there is a complete formula ~ ( x ... , x,) with T b cp + 8. If O(x, ... x,) is not completable it is said to be incompletable. A theory T is said to be atomic iff every formula of 9which is consistent with T is completable in T. A model 9l is said to be an atomic model iff every n-tuple a , , ..., a, E A satisfies a complete formula in Th(9l). In this and the next chapter we shall frequently pause to illustrate our definitions with examples. We shall sometimes make assertions about the examples without proofs. These proofs usually involve a combination of standard algebraic results and the theorems in the first three chapters of this book. In Section 3.4, we return to the examples and supply proofs.

2.3.1. EXAMPLES (I). Let T be a complete theory and let co, cI, c2, ... be constant symbols of 9. Then any formula of 9 of the form Xo E C O A X !

CIA

... A X , ,

C,

is complete in T. If 31 is a model of T such that every element of A is a constant, then 8i is an atomic model. (2). The standard model of number theory is an atomic model. (3). Let T be the theory of real closed ordered fields. The ordered field of real algebraic numbers is the unique atomic model of T. For example, the ordered field of real numbers is not atomic. (4). Every finite model is atomic. ( 5 ) . Every model of pure identity theory is atomic. This gives an example of uncountable atomic models. (6). Every dense linear ordering without endpoints is atomic. (7). The following theory Tis a complete theory which has no conipletable formulas and no atomic models. The language Y has unary relation symbols Po(x), P , (x), .... The axioms of T are all sentences of the form (3X)(fi,(X)A

... A f i m ( X ) A l P j , ( X ) A ... A

1 fjn(X)),

where the i , , ..., i , , , , . j l ,...,J, are all distinct.

Our first theorem about atomic models is an application of the extended oniitting types theorem.

98

MODELS CONSTRUCTED FROM CONSTANTS

[2.3

THEOREM 2.3.2 (Existence Theorem for Atomic Models). Let T be a coiiipletc theory. Then T has a countable atomic model if and only if T is atoniic. PROOF.First assume that T has an atomic model '3. Let cp(x, ... xln) be consistent with T. Then, since T is complete, T 1 (3x,

... x,)cp(x,

... x,~).

Let a , , ..., a , € A satisfy cp, and let $(xI ... x,) be a complete formula satisfied by a , , ..., a,. Then we cannot have T t $ + icp. so we must have T t $ -+ cp. Hence cp is completable and T is atomic. Now assume T is atomic. For each n < w , let f , , ( x I ... xIt) be the set of all negations of complete formulas $(x, ... x,) in T. Then every formula cp(x, ... x,) which is consistent with T is completable, and hence cp A 7 ;I is consistent with T for some y E f , . Therefore T locally omits each set f,(xl ... x,). By the extended omitting types theorem, T has a countable model % which omits each f,. Then each a 1 2..., a, E A satisfies a complete formula, whence 9I is an atomic model. -I Returning to our examples, we see that complete number theory and the theory of real closed ordered fields are atomic, because they have atomic models. THEOREM 2.3.3 (Uniqueness Theorem for Atomic Models). IfN and 8 are countable atornic ntodels and ?I = 8,then N 2 8. PROOF.If BI or % is finite, then 91 g 23 is trivial. Let N and 8 be infinite and well-order the sets A and B with order type o.The proof will be our first example of a back and forth construction. We shall see many other proofs of this type later. Let a, be the first element of A and let cpo(xo)be a complete formula satisfied by a, in 91. Since $1 t (3Xo)Cpo(Xo), % k (3xo)'po(xo). Thus we may choose bo E B, which satisfies 'p,(xo). Now let 6 , be the first element of B\(b,), and let 'p,(x,x,) be a complete formula satisfied by b,,b, in 8. Then both % and 23 satisfy Vxo(cpo(xo) + (3x1)c~i(xo~i ))*

because 'po is complete. Therefore there exists lzl E A such that a,, a, satisfy 'p,(xox,). Next, let a, be the first element of A\(a,, a,}, and so on. Going back and forth o times, we obtain sequences aO,alra2,

.*.,bo,bi,b2,....

2.31

COUNTABLE MODELS OF COMPLETE THEORIES

99

By going back and forth we used up all of A and B, so

Moreover, for each n the n-tuples a,, ..., a,,-, and b,, ..., b , - , satisfy the same complete formula. It follows that the mapping u, -, b , is an isomorphism of 'i!l onto 8.-I

Our third result on atomic models shows that they should be thought of as 'small' models of T . First, we need to define the notion of a prime model. 'i!l is said to be a prime model iff ?I is elementarily embedded in every model of Th(M). ?I is said to be countably prime iff ?I is elementarily embedded in every countable model of Th(?i). THEOREM 2.3.4. Thefollowing are equicalent : (i). ?X is a countable atomic model. (ii). 91 is a prime model. (iii). 'u i s a countably prime model. PROOF.First assume that 'u is a countable atomic model and let T = Th(%). The proof that M is prime is one-half of the 'back and forth' construction. Let A = { a , , a , , a z , ...I and let % be any model of T . Let cpo(xo)be a complete formula satisfied by a,. Then T b (3x,)cp,, so we may choose b,E B which satisfies cpo(xo). Now let cp,(x,x,) be a complete formula satisfied by a,, a , . Then T 1 cpo(xo)-, (3x,)cp,(x,x,). Choose 6 , E B so that b,, 6 , satisfies cp, , and so forth. The function a,, -P 6 , is an elementary embedding of 9l into 23. Now assume 'II is prime. Then 'u is elementarily embedded in every countable model of T, so 91 is countably prime. '4ssume 2l is countably prime. Let a , , ..., a, E: A and let f (x, ... x,,) be the set of all formulas y ( x , ... x,) of 9 satisfied by a , , ..., a , . For a n y countable model 2' 3 of T, we have some elementary embedding f : ?I < W, whence f a , , . . . , f a,satisfies T ( x l ... x,) in B. Thus r is realized in every countable model of T. By the omitting types theorem, r is locally realized by T . Thus there is a formula q(x, ... x,) consistent with T such that T 1 cp -P y for all y E r. But, for each formula $(xl ... x,,), either $ E r or (1$) E r. Thus cp is complete in T. We cannot have T k cp -, 1 cp, so cp E f. Therefore ~ ( x ... , x,) is a complete formula satisfied by a , , ..., a,, in 91, and 91 is atomic. i

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12.3

We now turn to the study of 'large' countable models. Given a model '% and a subset Y c A , the expanded model (?I, a)oEywill be denoted by 'illy, and its language by Y Y . A model ?I is said to be w-saturated iff for every finite set Y c A , every set of forrnulas f ( x ) of PYconsistent with Th('Uy) is realized in ?ly. A model is said to be countably saturated iff it is countable and o-saturated. To gain some intuition, we shall list some examples of countably saturated models. Note that if ?l is o-saturated, then so is ?Iy for every finite subset Y c A. 2.3.5. EXAMPLES

( I ) . Every countable infinite model of pure identity theory is countably saturated. (2). The ordering of the rational numbers is countably saturated. (3). Let T be a theory in the language with only the constant symbols c,,, cl, ..., and axioms c i f c j , i < j < w . There are countably many countable models of T up to isomorphism; for each a < w , there is a model with exactly tl elements which are not constants. The model with zero nonconstants is the atomic model. The model with w nonconstants is the countably saturated model. (4). Let T be the theory of algebraically closed fields of characteristic zero. Again there are countably many countable models; for each a < o, there is a model of transcendence degree a over the rationals. The model of degree zero, i.e. the field of algebraic numbers, is the atomic model of T. The model of transcendence degree o is the countably saturated model. ( 5 ) . Every finite model is countably saturated. We need some additional notation for sets of formulas. Remember that a type in the variables x , , ..., x, is a maximal consistent set f ( x , ... x n ) of formulas. The set T'of sentences which belong to r is a maximal consistent theory; we call T' the theory o f f . If T E f,f is called a type of T. Given a model 3 of T and an n-tuple a,, ..., a, E A , the set of all formulas y ( x , ... x,) of 2 satisfied by a , , ..., a, is a type of T, called the type of a , , ..., a,. By a type of'ill we mean a type of Th(c2l). A formula cp(x, ... x,) is Consider a set of formulas C(x, ... x,) of 9. said to be a consequence of C, in symbols Z I. cp, iff for every model ?l and every n-tuple a , , ..., a,eA, if a , , ..., a, satisfies Z, then it satisfies 40. That is, 3 k C [ a , ... a,] implies 3 k q [ a , ... a,].

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COUNTABLE MODELS OF COMPLETE THEORIES

We let C(cl ... c,) denote the set of all consequences in 9 u {cl, of the set {fJ(cl ... c,) : b ( X , ... x,) E C } .

..., Cn}

The notation C(c, ... c,x,+ I ... x,) is defined in a similar way. Let 9‘ = 9u {cl, ..., c,} be a finite simple expansion of 9. There is a natural one-to-one correspondence between the types Z ( x , ... X , ) of 9 and the types F ( x , + , ... x,) of 2’. If Z(xl ... x,) is a type of 9, then

... x,)

Z’ = C ( c , ... c,x,+,

is a type of 9’. On the other hand, if f ( x , + Z(x, ... x,) = { ~ ( x... , x,) : C ( C ,

, ... x,) is a type of 9’, then

... c,x,+

... x,)

EF

}

is the unique type of 9such that Z‘ = F . (We leave the verification of this as an exercise.) One might wonder why we used only sets of formulas in one free variable in the definition of an o-saturated model. At first sight, it may appear that we would obtain a stronger notion by considering sets of formulas with finitely many free variables. The next proposition shows that we do not obtain a stronger notion in this way.

PROPOSITION 2.3.6. Let ?I be an o-saturated model. Then for each finite Y c A , each set of formulas F ( x l ... x,) of ZYconsistent with T h ( a Y ) is realized in V l y . PROOF.We argue by induction on n. The result holds for n = 1 by definition. Assume the result for n - 1 and let f ( x l ... x,) be consistent with Th(?ly). We may assume that F is closed under finite conjunctions. Let f’(xl

... x,-

= {(3x,)y(x,

... x,)

: E F}.

Then f ’ is consistent with Th(91r). By inductive hypothesis, there is an ( n - I)-tupIe a , , ..., a,- realizing f ’ in Yy.Let Y ’ = Y u { a l , ..., Q,- I } . Then Y’ is still finite. Moreover, the set T(cI ... C , - ~ X , ) is consistent with Th(91y,) because for each yl, ..., y m E F , (3x,)(y, A ... A )7, E F ’ . Since is w-saturated, there exists a, E A realizing f ( c I ... C , - ~ X , ) in 91yt. Then a , , ..., a, realizes F in ?Iy. -I Our three theorems below on countably saturated models will closely parallel our three theorems for atomic models. We shall prove an existence

102

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theorem, a uniqueness theorem, and a theorem showing that countably saturated models are 'large'. THEOREM 2.3.7 (Existence Theorem for Countably Saturated Models). Let T be a complete theory. Then T has a countably saturated model fi and only i f for each n < o,T has only countably many types in n oariables.

PROOF.Suppose first that T has a countably saturated model 81. By Proposition 2.3.6, every type of Tin n variables is realized in %. But no n-tuple can realize two different types in n variables. Therefore T has only countably many types. Now suppose that for each n, T has only countably many types in n variables. Add a countable set C = {cl, c,, ...} of new constant symbols to Y ,forming Ip'. For each finite subset Y

=

{ d , , ..., d,} c C,

the types T ( x ) of T i n 9, are in one-to-one correspondence with the types C ( x , . . . x , x ) of T in 9. Therefore T has only countably many types f ( x ) i n 9,. Also, there are only countably many finite subsets Y c C. Let

be an enumeration of all types of T in all expansions subset of C. Let v19 v2,

Y y ,

Y a finite

.'.

be an enumeration of all sentences of 9'. We form an increasing sequence T = TO c TI c T2 c

...

of theories of 2''such that for each m < o: (1). T,,,is a consistent theory which contains only finitely many constants from C. (2). Either (P,, E T,,,, or (1 9,) E T,,,, . (3). If q,,,= ( 3 x ) @ ( x ) is in T,,,+l, then @ ( c ) E T,,,,, for some C E C. ( 4 ) . If T , ( x ) is consistent with T,, , then T,,,(d)c T,,,, for some d~ C. The construction of T,,, is straightforward. The union T, = u , , , T , is a maximal consistent theory in 3'.Using (3) we see that T, has a model %' = (a, a , , a,, ...) such that A = { a l , a,, ...}. Thus % is a countable model of T. It remains to prove that % is o-saturated. Let Y c A be finite and let Z ( x ) be consistent with Th(%,). Extend C ( x ) to a type f ( x ) in Th(a,,).

,

,

,

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For some m, T ( x ) = T,(x). T , ( x ) is consistent with T, and hence with T,, . Then by (4), T,(ci) c T,, for some c, E C, and it follows that a, realizes T ( x ) in $?Iy. i

,

,

COROLLARY 2.3.8. If T is a complete theory with only countably many nonisomorphic countable models, then T has a countably saturated model. PROOF.Each type of T is realized in some countable model of T, and each countable model realizes only countably many types. Therefore T has countably many types. -1 THEOREM 2.3.9 (Uniqueness Theorem for Countably Saturated Models).

If % and 23 are countably saturated models and ?1 = 23, then % is isomorphic to 23. PROOF.The proof uses a back and forth construction which closely parallels the proof of the uniqueness theorem for atomic models. The only difference is that instead of working with complete formulas we work with types. Using countable saturation of (2I and 23, we obtain two sequences a,,a,,

such that

...,

A = { a , , a , , ...},

b,, b , , ..., B

=

{ b , , b , , ...},

and, for each n, a, realizes the same type in (?I, a,, ..., a,,- ,) as b, realizes in ('23, b,, ..., b , - , ) . Then

(9, a,, a , , ...) = ('23, b,, b , , ...), whence % 2 23 by the mapping a,,

-+

b,. -I

The 'dual' of a prime model is a countably universal model. A model '8 is said to be countably universal iff M is countable and every countable model 8 3 '21 is elementarily embedded in M. The next theorem shows that countably saturated models are 'large'. 2.3.10. Every countably saturated model is countably universal. THEOREM

PROOF.Let 23 be a countable model and (21 a countably saturated model, % ' = 8.Let B = {b,, b , , ...}. Using one half of the back and forth construction and the saturation of ?I, we obtain a sequence a,, a , , a 2 , ... in A such that (8, b , , b , , ...) = (%,a,, a , , ...). Then the mapping b,

-+

a, is an elementary embedding of

23 into %.

-I

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[2.3

For a related necessary and sufficient condition for countable saturation see Exercise 2.3.12. Example 2.3.12 shows that the converse of Theorem 2.3.10 fails. 2.3. I 1. Let T be the theory with infinitely many unary relations EXAMPLE P o ( x ) ,P l ( x ) , ..., and a double sequence of constants c i j , i, j < w. The axioms are ( V x ) i( P i ( x ) ~ P j ( x ) ) , i < j < w, Pdcij), i < w, c i j f cik* j
THEOREM 2.3.13 (Characterization of w-Categorical Theories). Let T he a complete theory. Then the following are equivalent: (a). T is w-categorical. (d). For each n < w , T has onlyjnitely many types in x , , ..., x,.

PROOF.The reader is advised to sit down before beginning this proof. We shall prove the equivalence of (a) and (d) by proving a chain of implications (a)

-+

(b)

-+

(c)

-+

(d)

+

(el

+

(f) + (a).

Each of the six equivalent conditions is interesting in its own right.

2.31

COUNTABLE MODELS OF COMPLETE THEORIES

105

Assuming (a), T is w-categorical, we prove: (b). T has a model 9t which is both countably saturated and atomic. Let 91 be the unique countable model of T. Then "21 is countably prime, so 91 is atomic. Since T has only one (hence countably many) countable models, it has a countably saturated model. Hence 'u is countably saturated. Now, assuming (b), we prove: (c). For each n < w , each type T(xl ... x,) of T contains a complete formula. Since 2l is w-saturated, the type r is realized in 9l by some n-tuple a , , ..., a,. Since 'ilis atomic, a , , ..., a, satisfies a complete formula y ( x , ... x,,). We cannot have ( yi ) E r, so y belongs to r. Assuming (c), we next prove: (d). For each n c w , T has onlyfinitely many types in x, , .. ., x,. To prove (d), let Z(x, ... x,) be the set of all negations of complete formulas ' p ( x , ... x,) in T . Then Z cannot be extended to a type in x, , ..., x,,, so Z is inconsistent with T. Therefore some finite subset (1 401 9

..., 7 v m }

c

Z

is inconsistent with T . Hence Tk

1 (1cpl A

. . . A 1 cp,),

whence T k cp,v...vcp,.

For each i < in, the set T i ( x , ... x,) of all consequences of T u f q i ) is a type of T. But in every model of T, every n-tuple satisfies one of the c p i , hence realizes one of the T i . Therefore r l ,r 2 ,..., rmare the only types o f T i n x , , ..., x,. Now we assume (d), and prove: , x,) (e). For each n < w , there are onlyfinitely inany formulas ~ ( x ... up to equioalence with respect to T. Given a formula cp(x, ... x,), let cp* be the set of all types T(xl ... x,) of T which contain cp. Then cp* = @ *implies Tk cp C I @. But there are only finitely many types of T i n x, , ..., x,, say m . Hence there are only 2" sets of types and therefore at most 2" formulas up to equivalence in T. From (e) we prove: (f). All models of T are atomic. To see this, let 2l be a model of Tand let a , , ..., a, E A . Let cpl(x, ... x,), ..., 'pr(x, ... x,) be a finite list of all the formdas satisfied by a , , ..., a,, up to equivalence in T. Then cp, A ... A cpr is a complete formula in T which is satisfied by a , , ..., a, in '$1. Hence % is atomic.

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MODELS CONSTRUCTED FROM CONSTANTS

[2.3

Finally, assuming (f), we see that any two countable models of T are atomic and elementarily equivalent, hence isomorphic. Therefore T is o-categorical. i The next theorem can often be used to show that a theory has an atomic model. THEOREM 2.3.14. Any complete theory T which has a countably saturated model has a countable atomic model.

PROOF.Assume that T has no countable atomic model. Then T is not atomic. Therefore T has a consistent incompletable formula ' p ( x , ... x").

For each consistent incompletable formula $(xl ... x,) of T, we may choose two formulas $ o ( x , ...x,) and J l l ( x ,...x,) each consistent with T such that

(1)

Tk$O+$?

Tk$,+$,

Tkl($oA$,).

and $ I are again incompletable. In this way we obtain a tree of incompletable formulas / 'Po0 ..*

$o

/ 'P\

/'Po\

\

/

'POI

..'

'PI0

...

'PI1

..*

\

Each infinite sequence so, sI,s2, ... of zeros and ones gives a branch Ts = (cp, cp,, (psosl, (psos1s2,...} of the tree. There are 2" branches. By ( I ) , each branch T s ( x , ... x,) is a set of formulas which is consistent with T, and any two branches are inconsistent with each other. Extending each branch Ts to a type of T, we obtain 2" different types. Therefore T does not have a countably saturated model. i The converse of the above theorem is false. For example, we have already seen that the theory of real closed ordered fields has a countable atomic model. But this theory has 2" types and therefore has no countably saturated model. Another example of a theory with a countable atomic but no countably saturated model is complete number theory. It is worth repeating here some of our examples of o-categorical theories: atomless Boolean algebras; the four complete theories of dense simple order; the theory of infinite pure identity models; the theory of infinite

2.31

COUNTABLE MODELS OF COMPLETE THEORIES

107

Abelian groups with all elements of order p ( p prime); the theory of an equivalence relation with infinitely many equivalence classes and each class infinite. We conclude this section with a surprising result of Vaught.

THEDREM 2.3.15. No complete theory T has exactly two nonisoniorpliic countable models.

PROOF.Assume T has exactly two nonisomorphic countable models. Our previous results show that T has a countably saturated model a and a countable atomic model 9I and that these two models cannot be isomorphic. Since '$3 is not atomic, it has an n-tuple b,, ..., b, which does not satisfy a complete formula. Our plan is to obtain a countable atomic model (6,cI ... c,) of the complete theory T' = Th((23, b, ... b,)) and show that the reduct Q is neither w-saturated nor atomic. Thus T will have at least three nonisomorphic countable models 91, '$3, 6. Since '$3 is countably saturated, ('$3, b, ... b,) is countably saturated. The theory T' thus has a countably saturated model, and therefore has a countable atomic model (a,cl ... c,). The reduct Q is a model of T. 6 is not atomic because the n-tuple c l , ..., c, does not satisfy a complete formula. It remains to be shown that B is not w-saturated. Because T is not wcategorical, it has infinitely many nonequivalent formulas. Therefore T' has infinitely many nonequivalent formulas. Hence no model of T' is both atomic and w-saturated. In particular, since (6,c l ... c,) is atomic, it cannot be w-saturated. It follows that 6 is not w-saturated. -I

EXERCISES 2.3.1. Show that if q ( x l ... x,) is a complete formula in T with respect to x , , ..., x,, then (3x,)q(xl ... x,-,x,) is a complete formula in T with respect to x l , ..., x,-,. 2.3.2. Show that for any model 2l the simple expansion (3, atomic model.

is an

2.3.3. Let c,, ..., c, be new constant symbols. Prove that for each n 2 m the map qx,

... X")

+qcl

... C,X,+l

... x,)

is a one-to-one mapping from the types in 9 in n variables onto the types in 2 ' u { c l , ..., c,} in n-m variables.

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MODELS CONSTRUCTED FROM CONSTANTS

[2.3

2.3.4. Suppose % -= 8. Show that an n-tuple a , , .,., a,,E A and an n-tuple b , , ..., b, E B realize the same type if and only if (%, a, ... a,,) = (% b , ... b,). 2.3.5*. Prove that a model BI is atomic if and only if for every finite subset Y of A, every element U E A satisfies a complete formula q ( x ) in Th(BI,). Use this to show that if % is atomic and Y c A is finite, then "& is atomic. 2.3.6. Prove that if % is elementarily embedded in 8, then every t y p e T ( x l .. . x,) which is realized in % is realized in '23. 2.3.7. Let Z(x, ... x,) be a type of a complete theory T. Prove that Z is realized in every model of T if and only if Z, contains a complete formula. 2.3.8. Prove that if a complete theory T has fewer than 2" types, then T has an atomic model. 2.3.9*. Prove that complete number theory has no countably saturated model. 2.3.10*. Prove that no ordered field is countably saturated. 2.3.1 1 . Prove that every complete theory which has a countably universal model has a countably saturated model. 2.3.12. Let be a countable model. Prove that PI is countably saturated if and only if for every finite subset Y of ?I, aIY is countably universal. 2.3.13. Show that every reduct of a countably saturated model to a sublanguage of 9is countably saturated. 2.3.14*. Let T be a complete theory and let BTbe the Lindenbaum algebra of T as defined in Exercises 1.4.10 and 2. I . 15. For n < o,let B,,, be the Boolean subalgebra of BT determined by the formulas q ( u , ... 0,- I ) . Prove that: (a). q ( o , ... o n - ] ) is consistent with T if and only if ( q ) # 0 in (b). q ( u , ... u , - ] ) is a complete formula in T if and only if ( q ) is an atom of 'B,,r. (c). T is an atomic theory if and only if each Bn,ris an atomic Boolean algebra. (d). T is o-categorical if and only if B,, is a finite Boolean algebra for each n < w . (e). T has a countably saturated model if and only if each 'Bn,Thas only countably many ultrafilters. [Hint:Show that types Z(uo ... u,- I ) of T correspond to ultrafilters in B,,T.]

2.41

RECURSIVELY SATURATED MODELS

109

2.3.15* (Ehrenfeucht). Let 9 = { < , c o , c,, ...} and let T be the theory of 9 which states that < is a dense simple order without endpoints and that c, < c,+, , n < o. T is easily seen to be complete. There are three kinds of countable models of T. If we identify the elements of the countable model with the set of all rationals, then one of the following three cases occurs: lim c,, = 03; n+m

lim c, <

03

and is a rational;

00

and is an irrational.

n+m

lim cn < n-rm

Determine which of these three models is countably saturated? Countable atomic? And neither? 2.3.16* (Ehrenfeucht). Modify the above example to obtain an example of a complete theory T with exactly n nonisomorphic countable models, n 3. [Hinr: Add n-2 I-placed relation symbols to 9 . 1 2.3.17**. Let T be a theory in a countable language 9. Prove that if T has more than w1 nonisomorphic countable models, then T has continuum many nonisomorphic countable models. This result disappears if the continuum hypothesis holds. It is an open problem whether the hypothesis of the result can be weakened to: T has uncountably many nonisomorphic countable models (assuming that the continuum hypothesis fails).

2.3.18*. Let ?I be a countably saturated model for an uncountable language 2'. Prove that there is a countable sublanguage Z fC Z such that for each formula cp of 2' there is a formula )I of T f such that

wP++*. 2.3.19*. Let 9I and 58 be w-saturated models for a countable language. Show that the direct product '2I x 58 is w-saturated. 2.3.20*. In a countable language, let T be a complete theory which is not w-categorical. Let r,,. . . , r, be consistent types over T. Show that T has a countable model 2l which realizes r,,; . . , r, but is not w saturated. 2.4. Recursively saturated models A recursively saturated model is, roughly speaking, a model which is saturated for recursive sets of formulas. The proofs of a number of early

110

MODELS CONSTRUCTED FROM CONSTANTS

[2.4

results in model theory were simplified by using the method of recursively saturated models. In this section we shall introduce the recursively saturated models, develop their basic properties, and give some illustrations of how they are used. There are a number of results in model theory which would be quite easy if every complete theory had a countably saturated model. But countably saturated models do not exist for complete theories with uncountably many types. However, countable recursively saturated models are often good enough, and they always exist for a complete theory in a countable language. To prepare for the definition we must first explain what is meant by a recursive set of formulas, or more generally a recursive set of expressions (where an expression is a finite sequence of symbols of 2’). Intuitively, a set of expressions is recursive if there is an algorithm which, given any expression cp, will produce the answer “yes” if cp belongs to the set and the answer “no” if cp does not belong to the set. A set of expressions is recursively enumerable if there is an algorithm which, given any expression cp, will produce the answer “yes” if cp belongs to the set and will never end if cp does not belong to the set. A t the beginning of a course in Recursion Theory several equivalent mathematical definitions of recursive set are presented. The principle that these mathematical definitions are equivalent to the above intuitive notion of a recursive set is called Church’s Thesis. In this course we will only need to know two things about recursive sets: (1) There are only countably many recursive sets of formulas. (2) In a recursive language, the set of all formulas described by a “finite scheme” is recursive. The intuitive description of recursive sets above will be enough to understand our treatment of recursively saturated models. However, for the sake of completeness we shall also give a precise definition of recursive set here. We shall choose a form of the definition which is particularly easy to apply to sets of formulas. We restrict our attention in this section to the case where the language 2 is countable. We may then take each symbol of 2’ to be an element of the set R ( w ) of sets of finite rank. Then each finite sequence of symbols of 2’ also belongs to R ( w ) . We begin with the notion of a recursive subset of R(w). DEFINITION. By a A,, formula, or bounded quantifier formula, we mean a formula in the language with only the E symbol and equality which is built from atomic formulas using only logical connectives and the

2.41

RECURSIVELY SATURATED MODELS

111

relativized quantifiers (Vx E y ) and ( 3 x E y ) . By a Z1formula we mean a formula which is built from A,, formulas using the positive connectives A , v , bounded quantifiers (Vx E y ) , ( 3 x E y ) , and existential quantifiers ( 3 x ) . A subset S of R ( w ) is recursively enumerable, or r.e., if it is definable in the model ( R ( w ) ,e ) by a XI formula &), that is, S = { a E R ( w ) : ( R ( w ) ,E ) k p [ a ] } . A subset S of R ( w ) is recursive if both S and R(w)\S are r.e.

The language 2 is said to be recursive if the sets of symbols of 2 and functions giving the number of places of symbols of 2 are recursive subsets of R ( w ) . That is, each of the sets { ( n ,u,,) : E w } , {c : c is a constant symbol of 9}, { P : P is a relation symbol of 2 }, { f : f is a function symbol of 2 } , { ( P , n ) : P is a relation symbol of 2 with n places}, { ( f , n ) : f is a function symbol of 9with n places} is a recursive subset of R ( w ) . For convenience we shall also include as part of the definition of a recursive language that each symbol of 2 is a natural number and that there are infinitely many natural numbers which are not symbols of 2. We shall restrict our attention in this section to recursive languages 9. Since R ( w ) is countable, every recursive language is countable and has countably many recursive sets of formulas. We can freely expand recursive languages by adding new symbols. Since any finite set is recursive, any expansion of a recursive language by finitely many new symbols is again a recursive language. Moreover, any expansion of a recursive language by a recursive set of new constants is again a recursive language, provided that there are still infinitely many natural numbers which are not used as symbols. If 9is recursive, then the set of all formulas of 2 is recursive, because there is an algorithm which decides whether an element of R ( w ) is a formula of 9.The set of all formulas cp(x,, . . . ,x,,) of 2 with at most x , , . . . ,x, free, and the set of all sentences of 2, are also recursive. We shall come across various other examples of recursive sets of formulas in this section. As a by-product of the Godel completeness theorem, the set of all proofs in 2 is recursive and the set of all valid sentences in 3 is r.e. We now give the key definition in this section. I

DEFINITION. Let 2 be a recursive language. A model ‘2l for 2’ is recursively saturated if for every finite set { c l , . . . , c,,} of new constant

112

MODELS CONSTRUCTED FROM CONSTANTS

(2.4

symbols, every recursive set T(x) of formulas of 2 ( c l , . . . , c,), and every n-tuple a , , . . . , a , of elements of A , if T(x) is finitely satisfiable in (a, a , , . . . , a , ) then T(x) is realized in (91, a , , . . . , a , ) . EXAMPLES 1 . Every o-saturated model is recursively saturated. 2. Every recursively saturated model of complete arithmetic has an infinite element, since the recursive set of formulas { n < x : n E o) is finitely satisfiable. 3. Every recursively saturated real closed ordered field has a positive infinitesimal element, since the recursive set of formulas {O
2.41

RECURSIVELY SATURATED MODELS

113

in which only finitely many constants from C occur, and let (91, ( 9 2 , .

.

*

be an enumeration of all sentences of 2‘.By carrying out the construction given in the proof of Theorem 2.3.7, we now obtain a countable recursively saturated model of T . -1

COROLLARY 2.4.2. Let 2 be a recursive language. Then every countable model i?l for 2’ has a countable recursively saturated elementary extension.

PROOF.Let 2‘ be a recursive expansion of 2’ with a countable set C of new constant symbols. Then l?l has an expansion ‘L” to 2’ in which every element of A is an interpretation of a constant. By Theorem 2.4.1, the elementary diagram T ’ of 9’has a countable recursively saturated model 23’. The reduct 23 of 8’to 2 is a countable recursively saturated elementary extension of 8.i We shall see in a later chapter that the above corollary has an analogue for uncountable models. In general, countable recursively saturated models are not unique in a complete theory. Some counterexamples are indicated in the exercises. The back and forth method is not able to prove uniqueness for recursively saturated models, but it is still quite powerful, as the next few results show.

DEFINITION. A model i?l is said to be w-homogeneous if for any pair of tuples a , , . . . , a, and b , , . . . , 6 , of elements of A such that

(a, a , , . . . , a , ) = ( 3 .b , , . . . , b , ) and any c E A there exists d E A such that

(a, a , , . . . , a , , c ) = (a, b , , . . . , b , , d ) . A countable w-homogeneous model is said to be countably homogeneous. The methods of the preceding section can be used to show that every w-saturated model and every atomic model is o-homogeneous. We leave the result for atomic models as an exercise. Our next proposition is that recursively saturated models are also w -homogeneous. This is somewhat surprising because the type of an n-tuple of elements of a model “2l is in general not a recursive set of formulas. The trick is that the property that

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[2.4

two n-tuples have the same type can be expressed by a recursive set of formulas, even though the type of each n-tuple is not recursive. PROPOSITION 2.4.3. Let 2’ be a recursive language. Then every recursively saturated model for 2’ is w-homogeneous. PROOF.Let 9 be a recursively saturated model for 2’ and let a , , . . . , a, and b,, . . . , b, be two n-tuples in A such that

(a, a , , . . . , a , ) = (8,b , , . . . , b,). Let c E A. Choose distinct new constant symbols corresponding to a , , . . . , a , , b,, . . . , b,, and c, and form the finite expansion 2” of 2’. Let T(x) be the set of all formulas of 2” of the form q ( a , ,.

*

>

c)*

q(b17.

. . b,, x ) ‘ 7

Then r ( x ) is a recursive set of formulas of 2” which is finitely satisfiable in the model 8’= (a, a , , . . . , a , , b,, . . . , b,, c).

By recursive saturation, T(x) is realized in a’, and this shows that % is w - homogeneous. -I If % is a countably homogeneous model, then a back and forth construction shows that whenever (91, a , , . . . , a , ) = (a,b , , . . . , b,), we have (a, a , , . . . , a , ) ZE (a, b , , . . . , b,). (This will follow from Exercise 2.4.5 and Proposition 2.4.4 below.) The back and forth construction can be captured in a more general context with the notion of a partial isomorphism. DEFINITION. Let and B be models for a language 2’. A partial isomorphism I : 71 B between 8 and B is a relation I on the set of pairs of finite sequences ( a , , . . . , a , ) , ( b , , . . . , b,) of elements of A and B of the same length such that: (i). 0 I O ; (ii). If ( a , , . . . , a , ) I ( b , , . . . , b,) then ( 2 I , a , , . .. , a , ) and (8, b , , . . . , b,) satisfy the same atomic sentences of .=Y(c,,. . . , c,); (iii). If ( a , , . . . , a , ) I ( b , , . . . , b,) then for all c E A there exists d E B such that ( a , , . . . , a , , c ) I ( b , , . . . , b , , d ) , and vice versa. Condition (iii) is called the back and forth condition. Thus \u is w-homogeneous if and only if the relation

(a, a , , . . . , a , ) = (a, b , , . . . , b,)

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115

is a partial isomorphism from to a. We shall now consider partial isomorphisms between two different structures.

PROPOSITION 2.4.4. (i) Any two finite or countable partially isomorphic models are isomorphic. (ii) Any two partially isomorphic models are elementarily equivalent. PROOF. (i) By a routine back and forth construction. (ii) Let I : % B. Show by induction on the complexity of formulas cp(x,, . . . , x , ) that if ( a , , . . . , a , ) I ( b , , . . . , b , ) then

91 k cp[a,,. . . , a,] iff B k cp[b,, . . . , b , ] . Then taking n = 0 we obtain 3 = 23. -I We shall next prove a partial isomorphism theorem for recursively saturated models which is analogous to the uniqueness theorem for countably saturated models. In order to obtain a partial isomorphism between two different recursively saturated models, the models must be recursively saturated “together”. To make this precise we introduce the notion of a model pair. To avoid complications we consider only languages which have no function symbols. In applications, the function symbols can be replaced by relations in the usual way.

DEFINITION. Let and ‘23 be two models for the recursive language 2’ which has no function symbols. The model pair @ , B ) is the model for the language 2 ” U 2’’ defined as follows. 2’’ is a recursive language obtained by replacing each relation symbol R of 2’ by a new symbol R’ with the same number of places, replacing each constant symbol c of 2’ by a new symbol c’, and adding one new unary relation symbol A . Identify each constant c’ and relation R’ with its interpretation in the model a. 2’ is formed in a similar way. Then (a,@)is the model

(a, B) = ( A U B , A , B , R’, R’, c’, c ’ ) ~ , ~ ~ ~ . to see that if (a, B) is a recursively saturated model pair,

It is easy then both 8 and 23 are recursively saturated models for 2’. However, it frequently happens that each of !? and I B is recursively saturated but the model pair (91, B) is not recursively saturated. For an example see Exercise 2.4.17. In order to extend the notion of a model pair to languages with function symbols in the natural way, a function symbol of 2’ should be interpreted by a pair of partial functions in (a, B), which are relations

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[2.4

but not functions in our treatment. For example, if F is a unary function symbol of 2, then F" should be interpreted by a partial function with domain A . For this reason we may as well replace function symbols by relation symbols in the original language 2.

THEOREM 2.4.5. Let 2 be a recursive language, and let % and B be elementarily equivalent models for 2 such that the model pair (?I, B) is recursively saturated. Then &I is partially isomorphic to 23. In fact, the relation

(a, a,, . . . , a,,)= (23, b , , . - . , b , ) is a partial isomorphism. PROOF.Let I be the elementary equivalence relation between n-tuples from 3 and B. We wish to show that I is a partial isomorphism. Since % = B ,the empty sequences are in the relation I. It is immediate that any pair related by I satisfies the same atomic formulas. We must verify the back and forth condition for the relation I . For each formula cp(x,, . . . , x,) of 2, define the formula '21 cp (x,, . . . ,x,) of 2"inductively as follows. For an atomic formula cp of 2, cp' is obtained by replacing each relation or constant symbol s of 2 by s91. The logical connectives are passed over with no change, and the quantifiers are relativized with the rules.

[(W#

= (Vx)[A(x)-+

$7,

[ ( 3 x ) q ] " = ( g x ) [ A ( x )A c p " ] .

The formula 'q is defined analogously. We see by induction that for any formula cp(x,, . . . ,x,) of 2 and any n-tuple a , , . . . , a, in A ,

91 L qo[a,,. . . , a,] iff (&I,B) L q " [ a , , . . . , a,], and similarly for E3. It follows that whenever a , , . . , , a, in A and b , , . . . , b, in B are such that

( 3 ,a,, and d E A , the set

. . . , a,) = (23,b , , . . . , b,)

r ( x ) consisting of

2"U 2 ' ( a , , .

B ( x ) and all formulas of

. . , a,,

d , b , , . . . , 6,)

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RECURSIVELY SATURATED MODELS

117

of the form is finitely satisfiable in the model pair ( a , B ) . Moreover, the set T(x) of formulas is recursive. It follows by recursive saturation of the model pair that there is an element e E B which realizes T(x) in (a, B). Then

(a, u,, . . . , a,,

d ) = (23, b , , . . . , b,, e ) .

Thus I satisfies the back and forth condition, and the proof is complete. i As a corollary we obtain a useful criterion for a theory to be complete.

COROLLARY 2.4.6. Let 2 be a recursive language. A theory T in 9 is complete if and only if for any recursively saturated model pair (a, B) of models of T , is partially isomorphic to 23. PROOF.If T is complete, then the models in the pair are partially isomorphic by Theorem 2.4.5. Suppose that T is not complete. Let a and 2 ‘3 be models of T which are not elementarily equivalent. Form the model pair (a, ‘$3). By Theorem 2.4.1 there is a countable recursively saturated model 6 elementarily equivalent to (%, 23). Using the relativized formulas (pa and q B from the preceding proof, we see that is a model pair (a’,B’)where %’=“I and B ’ = B . Then a’ and B’ are models of T but are not elementarily equivalent and hence not partially isomorphic. -1 There are several methods in model theory for showing that particular theories are complete. The method based on recursively saturated models and parital isomorphisms is quite powerful and easy to use. As an illustration of the method we obtain a complete set of axioms for the theory of the ordered group of integers under addition.

EXAMPLE 2.4.7. The following theory T is the complete theory of the ordered group of integers under addition, ( 2 ,+ , -, 0 , 1 , S ) . The constant 1 is not necessary but is included for convenience. The Abelian group axioms with + , -, 0. The axioms for linear order.

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MODELS CONSTRUCTED FROM CONSTANTS

[2.4

+ z s y + z.

xsy-,x

1 is the least element greater than 0. For each integer k > 1, the axiom that each x has a remainder modulo

k, ( V ~ ) [ k J x v k J x - .l .v. v k J x - ( k - l ) ] where k l x means ( 3 y ) [ k y = X I .

PROOF.Let ( 3 , B ) be a recursively saturated pair of models of T . Let Z be the relation such that ( a 1 , . . . ,a,) I

( 4 , . , b,) *

f

if and only if for each linear term P ( x , , . . . ,x , ) with integer coefficients, and each integer k > 1, (a)

3 k P ( a l , . . . , a , ) > 0 iff

B k P ( b l , . . . , b,) > 0

and (b)

3 k k I P(a,,

. . . , a,)

B k k I P ( b l , . . . , b,) . isomorphism from '21 to B. It iff

We shall show that Z is a partial is trivial that 0 I 0. Since each term in 2' is equal to a linear term with integer coefficients, ( a , , . . . , a,) I ( b , , . . . , b,) implies that ( a,, . . . , a,) and ( b , , . . . , b,) satisfy the same atomic formulas. It remains to prove that I has the back and forth property. Let ( a l , . . . ,a,) Z ( b l , . . . , b,) and let C E A . We must show that the recursive set T ( x ) of all formulas of the form

-

[P(Ul,. . . , a,, c ) > 01% [P(b,, . . . , b,, x ) > 0IB,

[kI P(al, . . . , a,, c)]'

++

[ k 1 P(b ,, . . . , b,, x)lm,

B(x) is finitely satisfiable in ( 3 , B ) . Using the axioms of T to simplify terms we see that it suffices to show that for all linear terms P ( x l , . . , ,x , ) and Q(xl, . . . ,x,) and natural numbers k > 1,1, rn such that

3 k P(a,, . . . , a,) < m c < Q(a,, . . . , a,) and c = 1 (mod k ) there exists x

E

B such that

23 k P ( b l , . . . , b,) < mx < Q(b,, . . . , b,) and x = 1 (mod k ) .

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RECURSIVELY SATURATED MODELS

119

If there are only finitely many elements between P( a , , . . . , a,) and Q ( a , , . . . , a,) in '21 then mc is equal to a linear term in ( a , , . . . , a , ) and mx may be taken to be the same linear term in ( b , , . . . , b,). Otherwise, since ( a , , . . . , a , ) Z ( b , , . . . , b , ) , there are infinitely many elements between P ( b , , . . . , b,) and Q ( b , , . . . , b,) in B. By the axioms of T , any infinite interval contains elements mx such that x = 1 (mod k ) . Therefore r ( x ) is finitely satisfiable in (a,'23). By recursive saturation, T(x) is realized in ('21, B), and thus the relation 1 is a partial isomorphism from '21 to '23. -I As another application of recursive saturation we give an easy direct proof of the Robinson Consistency Theorem (Theorem 2.2.23). We saw in an exercise in Section 2.2 that the Craig Interpolation Theorem follows quickly from the Robinson Consistency Theorem, and in fact the Robinson Consistency Theorem is only needed for finite languages. On the other hand, in Section 2.2 it was shown that the full Robinson Consistency Theorem is a corollary of the Craig Interpolation Theorem. Thus we only need a direct proof of the Robinson Consistency Theorem for finite languages. In this section it is more natural to prove it for recursive languages. 2.4.8 ROBINSON CONSISTENCY THEOREM (restated). Let 2,and 2, be two recursive languages and let 2' = 2, f l 2,. Suppose T is a complete theory in 2' and T , 3 T , T , 3 T are consistent theories in 2Zl, 2, respectively. U 9,. Then T , U T, is consistent in the language 9, PROOF.By replacing constant and function symbols by relation symbols in the usual way, we may assume that 2Zl and 2Z2 have only relation symbols. Let %?I be a model of T , and '23 be a model of T , . Form the model pair (in the natural extended sense for models of two languages)

('21, B) = ( A u B , A , B , R%,s" : R

E

9,, S E 3,).

The language of ('21, '23) is the recursive language 2': U 2':. Let (%', '23') be a countable recursively saturated model which is elementarily equivalent to ('21, @), whence '21' is a model of T , and '23' is a model on T,. Let (%?Io,B0) be the reduct of (a',@')to the smaller language 2Z"U2ZB. Then a,, and B,, are models of T and hence are elementarily equivalent models for 9, and ($?lo, @), is a recursively saturated model pair in the original sense. Therefore '21, and @, are partially isomorphic, and since

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MODELS CONSTRUCTED FROM CONSTANTS

[2.4

they are countable they are isomorphic by some isomorphism f . We may now expand 58' to a model '23'' for 2*U Z2by interpreting each relation symbol RE^^ by the f-image of its interpretation in '21'. Then 58" is a model of T , U T2 as required. -1 Recursively saturated 'models have a number of other applications, some of which are given in the exercises. We conclude this section with a general result about recursively saturated models, which shows that every countable recursively saturated model is "recursively saturated with respect to relations" as well as with respect to variables. We first need a useful lemma concerning the set of consequences of a recursive set of formulas.

2.4.9. Let 2 and 2' be recursive languages with 2 C 2' and let LEMMA T ( x ) be a recursive set of formulas of 2". Then there is a recursive set

Z(x) of formulas of 3'such that Z(x) and r ( x ) have exactly the same set of consequences with at most x free in the smaller language 2. PROOF.Let d , , d,, . . . be a recursive list of all deductions from T ( x ) of formulas of 2 i n x , and let cp,(x) be the conjunction of n copies of the formula proved by d,. Let S ( x ) be the set

Z(x)

= {'p,(x)

:n E w}.

Z(x) is clearly a set of formulas of 2 which has the same consequences in 2 as T(x).Moreover, Z(x) is recursive because one can decide whether a formula $ ( x ) belongs to 2 ( x ) by looking only at the deductions d , where m is at most the number of A symbols in $ ( x ) plus 2. -I THEOREM 2.4.10. Let 22 and 2" be recursive languages with 2 C 2' and let 9 be a countable recursively saturated model for 2. Then any recursive set r o f sentences of 3" which is consistent with the complete theory of '21 is satisfied in some expansion of '21. We remark that if 2' consists of 2 plus a recursive set of constant symbols, then the above theorem holds even for uncountable recursively saturated models a. To see this, let 9"be 2' plus the first n constant symbols and by Lemma 2.4.9 let be a recursive theory in 2"with the same consequences in 9, as r. Because 2l is recursively saturated there is an expansion of 3 by adding one constant which is a model of r,. Since

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any expansion of ‘21 by finitely many constants is again recursively saturated, the process may be repeated to obtain an expansion of ‘21 by countably many constants which is a model of r. The proof of the general case of Theorem 2.4.10 is more difficult and requires the hypothesis that ‘21 is countable.

PROOF OF THEOREM 2.4.10. Let 9;be an expansion of 9’ which has a new constant symbol for each element of A , and let $, O,, . . . be a list of all sentences of 9;. Let T be the elementary diagram of ‘21. Then r U T is consistent. Form a sequence of sentences . . of 9;such that

for each

11:

(1). k +,,+I+ +*, (2). is consistent with r U T. (3). If 0, is consistent with U T U {qn}then k 6,. (4). If 0, is of the form ( 3 x ) O ( x ) and is consistent with r U T U { +n} then there is an a E A such that k O(a).

+,,

r

+,,+,+

+,,

Recursive saturation is needed to show that the sequence can be . . . , +,, have been chosen to chosen to satisfy property (4).Suppose satisfy (1)-(4) and that 0, satisfies the hypothesis of (4). Let Y be the finite set of constants from A which occur in +n, and 0,. Then

r’(x>= r u {+,,I u { e ( x ) > is a recursive set of formulas of 9;which is consistent with T. By Lemma 2.4.9 there is a recursive set Z(x) of formulas of zywhich has exactly the same set of consequences as T ’ ( x ) in the language XY.It follows that Z(x) is consistent with T , and thus is finitely satisfiable in ?Iy. By recursive saturation there exists a E A which realizes Z(x) in STY. We claim that r’(a) is consistent with T. To prove this claim, suppose that k p(a) where p(x) is a formula of 9*. Existentially quantifying the elements of A - (Y U { a } ) , we may take p(a) to be a sentence of 2 y ( a ) and p(x) to be a formula of TY. Then r ’ ( x ) k p(x), and hence C(x)F p(x). Since a realizes 2 ( x ) in ?Iy, we have ?Iy k p [ a ] . Therefore p(a) belongs to T and the claim is proved. to be +, A O(a) in case (4), In view of the claim, we may take whence (1)-(4) will hold for n + 1 as required. The sequence of sentences +,,, II < w , may then be defined by recursion. It follows from (1)-(4) that % has an expansion ‘21’ to 2‘ such that 2‘1; is a model of r U T U { : II < w } , and in particular such that ‘21’ is a model of r. -I

r’(a)

+,,

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MODELS CONSTRUCTED FROM CONSTANTS

[2.4

As an example of the preceding theorem, let T be the theory of the ordered group of integers under addition from Example 2.4.7 and let r be the set of sentences with an additional function symbol for multiplication consisting of the commutative ring axioms and Peano’s axioms for the nonnegative elements. Then every countable recursively saturated model of T can be expanded to a model of T U r. The results in this section can be readily extended to the case of an arbitrary countable language 2 by modifying the notion of a recursively saturated model. A set S is said to be recursive relative to 2 i f there is an algorithm which decides whether or not an arbitrary input belongs to S but makes use of an “oracle” which will always correctly answer questions of the form “is cp a formula of 2?”. Everything goes through with only minor changes when the notion of recursive saturation is replaced by recursive saturation relative to 2.

EXERCISES 2.4.1. Prove that a complete theory in a recursive language which has continuum many complete types has continuum many nonisomorphic countable recursively saturated models. 2.4.2. Let T be the complete theory of models with countably many distinct constants and no functions or relations. Prove that all recursively saturated countable models of T are isomorphic. 2.4.3. Let T be the theory of divisible torsion free Abelian groups. Prove that all recursively saturated countable models of T are isomorphic. 2.4.4. Let ‘u be a countable model for a recursive language such that for each finite sequence ( a , , . . . , a , ) in A , every element of A realizes a recursive type in (a, a , , . . . , a n ) . Prove that 2I is elementarily embeddable in every recursively saturated model of Th(’u). 2.4.5. Let T be the complete theory of the model ( w , s ) .Prove that a model of T is countably homogeneous if and only if it is isomorphic to one of the following three models: the atomic model ( w , S ) , the countably saturated model formed by adding a countable dense set of copies of ( 2 ,S ) to the end of ( w , S ) , and the model formed by adding one copy of ( 2 ,S ) to the end of ( w , s ) .

2.4)

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RECURSIVELY SATURATED MODELS

2.4.6. Show that every atomic model is w-homogeneous. 2.4.7. If T is not w-categorical, then T has at least two nonisomorphic countably homogeneous models. 2.4.8. Show that if '21 is countably homogeneous and every type in finitely many variables in Th('21) is realized in '21, then '21 is countably saturated. 2.4.9. Let '21 and '23 be w-homogeneous models which realize exactly the same types in finitely many variables. Prove that '21 and '23 are partially isomorphic. 2.4.10*. Let '21 = ( A , s , . . .) be an o-homogeneous model for a countable language such that s well orders A . Prove that A has cardinality at most 2". 2.4.11. A model '21 is called almost w-homogeneous if there is a finite sequence a,, . . . , a , of elements of A such that ('21,ul,. . . , u n ) is w-homogeneous. Prove that if every countable model of theory T is almost o-homogeneous then every model of T is almost w-homogeneous . 2.4.12*. Let T be a complete theory in a countable language and let r ( x ) be a type such that: (i). T has a model which omits r, (ii). for every complete type 2(y l , . . . , y , ) over T , either

Z ( Y * , . . Y,> 3 r(Y;) for some i, or T has a model which realizes 2 and omits r. *

9

Prove that T has an w-homogeneous model which omits

r.

2.4.13. Let '21 be a countably infinite recursively saturated model for a recursive language. Prove that '21 has an automorphism which is not the identity function. 2.4.14. Let and let

be a recursively saturated model for a recursive language, (Pm(X1,...

,X,,Ylt...?Y,):m
124

[2.4

MODELS CONSTRUCTED FROM CONSTANTS

be an r.e. set of formulas. Suppose that for each m < o,

”CP,+,+

pm

and

a k v x , 3 y , v x 2 3 y 2 . . . vx,3ynCp,. Prove that

a ~ V x , 3 y , V x 2 3 y 2. .. v x , 3 y , A

cp,.

m
2.4.15. In an ordered field F, two elements x , y are said to realize the same cut over the rationals if for every rational number q we have q < x iff q < y . Let F and G be two ordered fields such that F C G and the model ( G , F, 0,1, +, s ) is recursively saturated. Prove that every element of G realizes the same cut as some element of F over the rationals. 0

,

2.4.16. Let (a, B) be a recursively saturated model pair. Prove that ‘u and B are recursively saturated. 2.4.17. Let T be a complete theory which has uncountably many complete types in one variable. Prove that T has countable recursively saturated models 91 and B such that the model pair (a,B) is not Use Theorem 2.4.5.1 recursively saturated. [Hint: 2.4.18. In a recursive language, let 3 and B be models such that B C l?l and the expanded model (a,B) is recursively saturated. Prove that the model pair (%,a)is recursively saturated.

2.4.19*. Let T be the theory in a language with countably many unary relations such that each nontrivial finite Boolean combination of relations is consistent. Show that every model of T is o-homogeneous, and that T has countably homogeneous models which are not recursively saturated. 2.4.20. Let I be a partial isomorphism from 9 to 8. Show that whenever ( a , , . . . , a, ) I ( b , , . . . , b , ) , the expanded models (a, a , , . . . , a , ) and (B, b , , . . . , b,*)are partially isomorphic.

2.41

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125

2.4.21. Let Y b e a recursive language with only relation symbols and let (8,23) be a countable recursively saturated model pair for 2. Prove that every existential sentence which holds in 8 holds in 23 if and only if '% is isomorphically embeddable in 23. 2.4.22. Let Y be a recursive language with only relation symbols and let (91, 23) be a countable recursively saturated model pair for 2. Prove that every positive sentence which holds in 8 holds in 23 if and only if there is a homomorphism of 8 onto 23. 2.4.23. Let Y be a recursive language with only relation symbols, let (a, 23) be a countable recursively saturated model pair for 2, and let U be a unary relation symbol of 2. The relatiuization cp" of a formula cp of Y to U is defined recursively as follows. If cp is atomic, then ( p u = cp, (VXcp)" = (VX)(u(X)+ cp"), ( 2 X c p ) " = (3X)(u(X)A 0"). Prove that 91 is isomorphic to the submodel of '23 with universe U' if and only if for every sentence cp of Y,'Ix F cp iff 23 L cpu.

2.4.24. Let 2' be a recursive language with only relation symbols, let ( 8 ,23) be a countable recursively saturated model pair for 2, and let E be a binary relation symbol of 2. A formula of 2 is said to be essentially existential if it is built up from atomic and negated atomic formulas using conjunctions, disjunctions, bounded quantifiers (VX)(XEY + cp), (34(XEY

A

cp),

and existential quantifiers. 23 is said to be an outer extension of K, and (5 is said to be a transitive submodel of 23, if '23 is an extension of CS and for every x E C and y E B , if 23 L y E x then y E C . Prove that 2 '3 is isomorphic to an outer extension of 8 if and only if every essentially existential sentence which holds in '% holds in '23.

2.4.25. By an arithmetical set we mean a subset of R ( w ) which is definable by a formula of first order logic in the model ( R ( w ) ,E ) . A model 8 for a recursive language 2 is said to be arithmetically saturated

126

MODELS CONSTRUCTED FROM CONSTANTS

[2.4

if for every finite sequence ( a l , . . . , a,) in A , every arithmetical set of formulas r(x) in Y ( a , , . . . , a,) which is finitely satisfiable in (a,a,, . . . , a , ) is realized in ( 3 ,a,, . . . ,a,). Prove that every countable model for Y has a countable arithmetically saturated elementary extension.

2.4.26. Use the method of recursively saturated models to obtain a complete set of axioms for the set of natural numbers with the order relation. 2.4.27. Use the method of recursively saturated models to obtain a complete set of axioms for the rational numbers with the order relation and addition (the theory of divisible ordered Abelian groups). 2.4.28. Let T be the theory with the binary relation s and unary function F and axioms stating that: (i). s is a dense simple ordering with no greatest or least element. (ii). F is an automorphism of the ordering S . (iii). For all x , x is strictly less than F(x). Prove using the method of recursively saturated models that T is complete. 2.4.29*. Use the method of recursively saturated models to show that the set of axioms for additive number theory described in Example 1.4.11 is a complete set of axioms for the set of natural numbers with zero, successor, and addition. 2.4.30*. Let 2 have one binary function symbol F and let T be the theory whose axioms state that F is a one to one function, every element belongs to the range of F, and 1Y

= 4x1,

.

* *

7

x,, y )

where y occurs in the term ~ ( x , ., . . , x , , y ) and the term is not a single variable. Use the method of recursively saturated models to show that T is complete.

2.4.31*. Let T be a complete theory in a recursive language L , and suppose that T has a complete extension T' in the language L U { c , : n < o} such that T ' has no atomic model. Then T has a model which is not recursively saturated.

2.51

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2.4.32*. Let 2 be the language with only the binary relation E and let 2” be the language with the binary relations E , S. Let ( R ( w ) ,E , Sat) be the model for 2” where Sat(x, y ) holds iff x is a formula of 2’and y is a tuple which satisfies x in ( R ( w ) ,E ) . Show that there is a recursive set r of sentences of 9’ such that ( R ( w ) ,E , Sat) k r, and for each countable nonstandard model % = ( R ( w ) ,E ) , % is recursively saturated if and only if % can be expanded to a model of r. 2.4.33*. Prove that a countable nonstandard model of complete arithmetic is recursively saturated if and only if it is the arithmetic part of some model of Zermelo set theory. 2.4.34*. Show that a countable nonstandard model of additive number theory can be expanded to a model of Peano arithmetic if and only if it is recursively saturated.

2.5. Lindstrom’s Characterization of First Order Logic

In this section we shall prove a result of Lindstrom which shows that first order logic is the only logic for which the Compactness Theorem and Downward Lowenheim Skolem Theorem hold. In order to state such a result we need a general notion of an abstract logic. This is the beginning of a subject called abstract model theory, which studies the relationship between various model theoretic results in arbitrary logics. There are many interesting logics which are richer than first order logic, such as logics with infinitely long formulas and logics with extra quantifiers. The theorem of Lindstrom shows that, even though there are richer logics, first order logic is of fundamental importance. No matter how we enrich first order logic, we must either give up one of the basic results which underlies our whole treatment of model theory, the Compactness Theorem or the Downward Lowenheim Skolem Theorem, or else go outside the notion of an abstract logic which is assumed in Lindstrom’s theorem. We shall not discuss logics other than first order logic here, and shall instead concentrate on results which characterize first order logic. To avoid long detours into side issues, we shall give a definition of abstract logic which is less general than is usually found in the literature, but is adequate for our present purposes.

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Recall that a language 2 is a set of relation, constant, and function symbols. In order so avoid some complications which would obscure the main ideas, we shall restrict our consideration in this section to languages 2 which have only relation and constant symbols, and no function symbols.

DEFINITION 2.5.1. An abstract logic is a pair of classes (1, k,) with the following properties, I is the class of sentences and k, is the satisfaction relation of the logic (1, k l ) . (i) Occurrence Property. For each cp E 1 there is associated a finite language 2p, called the set of symbols occurring in cp. The relation % k, cp is a relation between sentences cp of 1 and models % for languages 2 which contain Lfp. That is, if cp E 1 and % is a model for 2, then the statement '21 i=, cp is either true or false if 2 contains T q ,and is undefined if 2 does not contain 2+. (ii) Expansion Property. The relation '21Llcp depends only on the reduct of '21 to T9.That is, if '21 Ll cp and 23 is an expansion of '21 to a larger language, then B k l cp. (iii) Isomorphism Property. The relation % k l cp is preserved under isomorphism, That is, if '21 23 and '21k,cp then 23 k , q . (iv) Renaming Property. The relation '21k,cp is preserved under renaming. Formally, let p be a bijection (one to: one mapping) from a language 2 to a language p2which preserves the number of places of all symbols, and for each model 'illfor 2 let p'21 be the model for $3' induced in the obvious way by p. Then for each sentence cp E 1 with ZpC 2 there is a sentence pcp E 1 with ZPp = pZp such that for each model '21 for 2,'21 k l cp iff p% i=, pcp. (v) Closure Property. 1 contains all atomic sentences, I is closed under the usual first order connectives A , v , i, k, satisfies the usual rules for satisfaction for atomic formulas and first order connectives, and the set of symbols 2pbehaves as expected for atomic sentences and first order connectives. (vi) QuantiJier Property. 1 is closed under universal and existential quantifiers. That is, for each cp E I and each constant symbol c E T9there are sentences (Vx,)cp and (3x,)p in I with the set of symbols 2p- { c } such that: '21 k l (Vx,)cp iff for all a E A , ('21, a ) k, cp;

'21 L, (3x,)cp iff for some a E A ,

(a, a ) k, cp.

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(vii) Refutivization Property. For each sentence cp E f and relation R ( x , b , , . . . ,b,) with R , b , , . . . , b, not in 2+,, there is a new sentence cplR(x, b , , . . . ,b,), read cp refutivized to R ( x , b , , . . . , b,), which has the set of symbols 2+, U { R , b , , . . . , b,} and is such that whenever 23 is the submodel of a model % for 2+, with universe

B

=

{ a E A : R(u, b,, . . . , b , ) } ,

we have

(g,R , b , , . . . , b,) k l cp I R ( x , b , , . . . , b,) iff 23 k, cp. Two abstract logics are said to be equivalent if for each sentence of one of the logics there is a sentence of the other logic which has the same set of symbols and the same models. Two logics which are equivalent are alike except that the sentences are “renamed”, and may be considered as the same. We shall be interested in characterizing first order logic up to equivalence. Our notion of an abstract logic has no provision for free variables, and f should be thought of as a set of sentences rather than formulas. Notice that in the Quantifier Property (vi), a sentence in an expanded language with a new constant symbol takes the place of a formula with a free variable. To shorten our notation, we shall use the symbol 1 both for the logic and for the class of sentences of the logic. We use k for the satisfaction relation for ordinary first order logic, and k l for the satisfaction relation for an arbitrary abstract logic 1. The most familiar example of an abstract logic is the ordinary first order logic, usually denoted by f+,,+, = (f+,,+,, k). The Relativization Property (vii) for first order logic holds where cp relativized to R ( x , b , , . . . , b,) is the sentence formed by replacing each quantifier (Vx)+ in cp by ( V x ) [ R ( x ,b,, *

* *

, b,)+

$1

and ( 3 x ) $ in q by The class of sentences of first order logic, fm,+,, is a proper class because it contains sentences for all languages 2. However, for each language 2, the class of sentences cp E f w , , , with 2pC 2 is a set. By the closure and quantifier properties, every sentence of first order logic belongs to every abstract logic I, and for each model % and each first order sentence cp,

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‘21L)cp iff ‘21k cp. Many other examples of abstract logics have been studied extensively in the literature, but are outside the scope of this book. The book “Model-Theoretical Logics”, Barwise and Feferman (1985), has a large collection of articles surveying abstract model theory and the model theory for a variety of logics beyond first order logic. It should be emphasized that logics in the sense of Definition 2.5.1 deal with the same class of models as first order logic, and only the sentences and satisfaction relation may be different. This is a significant restriction which leaves a large loophole in Lindstrom’s theorem. There are many examples of logics in a generalized sense which study models with additional structure and thus do not fit within our framework. These include modal logics, programming logics, and logics for models with topologies and measures. Sentential logic and o-logic as described in this book are not examples of abstract logics in the sense of 2.5.1, because they also deal with different classes of models than first order logic. However, the notion of an abstract logic in Definition 2.5.1 is broad enough to explain what abstract model theory has to say about first order model theory. By a model of a set T of sentences of an abstract logic 1 we mean a model ‘21 such that 3 Ll cp for each cp E T. Two models 3 and B for the same language are said to be 1-elementarily equivalent if for each sentence cp E 1, 3 k I cp if and only if B k l cp. We shall now define two properties of abstract logics which correspond to the Compactness Theorem and the Downward Lowenheim Skolem Theorem.

DEFINITION. An abstract logic 1 is countably compact if for every countable set T C 1, if every finite subset of T has a model then T has a model. The Lowenheim number of 1 is the least cardinal a such that every sentence of 1 which has a model has a model of power at most a. By the Compactness Theorem, first order logic is countably compact. In fact, first order logic is fully compact, that is, every finitely satisfiable set of sentences has a model. The Downward Lowenheim Skolem Theorem shows that first order logic has Lowenheim number w . If the Lowenheim number of an abstract logic 1 exists, it must be at least w , because 1 contains every sentence of first order logic. For an arbitrary abstract logic, the Lowenheim number may not exist. However, the following simple result shows that it does exist if the logic does not

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have too many sentences. This simple result is of particular interest because it is one of the few natural results outside of set theory which uses the full power of the axiom scheme of replacement.

2.5.2. Let 1 be an abstract logic such that for each finite PROPOSITION language 2, the class {cp E 1 : Tq 2} is a set. Then the Lowenheim number of 1 exists. PROOF.Because of the Renaming Property, we may restrict our attention to the class I’ of sentences cp E I such that Zqbelongs to R ( w ) . By hypothesis, I’ is a set. For each cp E 1’, let a(cp) be w if cp has no models, and the least cardinal of a model of cp if cp has at least one model. By the axiom of replacement, there exists a cardinal a = sup{a(cp) : cp E l ’ } .

a is the Lowenheim number of 1. -I

We now prove a preliminary result which is an abstract version of Proposition 2.4.4.

PROPOSITION 2.5.3. Let I be an abstract logic which has Lowenheim number o. Then any two models which are partially isomorphic are 1-elementarily equivalent.

PROOF.Suppose 3 and 8 are models for a language 2 which are partially isomorphic by a relation I, but there is a sentence cp E 1 such that 3 !=[ cp but 8 k r i c p . By taking reducts we may assume that 2 = Zq. We may also assume that I is preserved under subsequences, that is, whenever ( a , , . . . , a,) Z ( b , , . . . , b , ) , any subsequence of ( a , , . . . , a,) is in the relation I to the corresponding subsequence of ( b , , . . . , b , , ) . Let A’ be the set of finite sequences of elements of A and let F : A’ X A + A’ and F’ : A’ X A’+ A’ be the functions

F((a,, F ’ ( ( a , , . . . ,a,,,),

. . . ,a,,), b ) = ( a , , . . . , a,,, b ) , ( b , , .. . , b , , ) ) = ( a , , . . . , a,, b , , . . . , b , , ) .

Form the expanded model 3”= ( A U A’, 3 , F, F’) where F and F‘ are ternary relations rather than binary functions. Define B‘, G, G’, and 8” analogously. By the Isomorphism Property, we may take 3 and 8 so

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that the sets A , A’, B , and B‘ are all disjoint from each other (with a different “empty sequence” in each of A’, B ’ ) . Since % is partially isomorphic to B, we may expand the model pair (%”,B”) to a model (3 = (a”,’@’,I). By the closure, quantifier, and relativization properties, there is a sentence I(, E 1 in the language of the model (5 which holds in & and implies that In particular, there is a first order sentence which holds in Q and implies that % z pB. (This sentence uses the symbols for F and F’. The atomic formula condition depends on the fact that 9 is finite and Z is closed under subsequences). Since 1 has Lowenheim number w , JI has a countable model &,,, from which we obtain models U,, and 8,for 2. %,, and B,,are countable models, and are partially isomorphic because Q, is a model of 4. By Proposition 2.4.4, a,, and B,,are isomorphic. But since & is a model of 4, U, k, cp and B,,Fl i c p , contradicting the Isomorphism Property. This completes the proof. -I

THEOREM 2.5.4 (Lindstrom’s Characterization of First Order Logic). First order logic is, up to equivalence, the only abstract logic which is countably compact and has Lowenheim number w .

PROOF.Let 1 be a countably compact abstract logic with Lowenheim number w . We must show that every sentence cp E 1 is 1-equivalent to some first order sentence 4, that is, for all a, % L, cp iff != JI. It is sufficient to consider models for a finite language 2. Let % and 58 be two models for 2. We shall define a “back and forth sequence” of relations I,, k E w , between U and ‘93. Let ( a l , . . . ,a,) and ( b l , . . . , b,) be n-tuples from A and B respectively. ( a l , . . . , a,) Z, ( b l , . . . , b,) means that ( a l , . . . , a,) and ( b l , . . . , b,) satisfy the same atomic formulas of first order logic. By induction on k we define ( a l , . . . , a,) Zk+l ( b , , . . . , b,) if and only if (1) For all c E A there exists d E B with ( a l , . . . , a,, c ) Zk ( b l , . . . , b,, d ) and (2) vice versa. We shall let % = k B mean that 0 Z, 0 where 0 is the empty sequence. Since 2 is finite and has no function symbols, for each k there is a finite set rk of sentences of first order logic in 9such that for all models % and B for 2, % =k 2 ‘ 3 if and only if % and B satisfy the same sentences of I‘,. Let cp E 1 be a sentence such that 9qC 9. It suffices to show that there is a k E w such that for all models % and B for 9, (3) ‘u = k B and % kl cp

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implies B k, cp. When (3) is proved, it will follow that cp is equivalent to a boolean combination of sentences in r,, and the proof will be complete. Suppose (3) fails for all k E w . Choose models a,, B, such that (4) ‘21, = k B,, ‘21, klcp, and Bk k / i c p . We next adjust the models 91, so that there is a model ‘21 for 2 which has each 3 , as a submodel. By taking a subsequence we may assume that all the ‘21, satisfy the same atomic sentences of 2. By the Isomorphism Property, we may also assume that all the have the same interpretations of the constants of 2. Then the union of the models ‘21, is a model ‘21 for 2 which has each a, as a submodel. We may also take the ‘21, so that each set A , is disjoint from w . As in the preceding proof, let ‘21” be an expansion of ‘21 with universe A U A‘ and extra relations A’, F, F’ where A’ is the set of finite sequences of elements of A , and the functions are given by F(a, b ) = ab, F ’ ( a , b ) = ab. We make similar assumptions for the B, and form the union B and expansion B”. We may now form an expanded model Q from the sequences of ’21, and 8, of the form

0 = (a”, B‘’,R , S, o,G , I), where R and S are relations such that for each k

A,

={aE

A : R(a, k ) } , B,

= { bE

E w,

B : S(b, k ) } ,

A; = { u E A’ : R(a, k ) } , B; = { b E B’ : S ( b , k ) } , and for each k E w , Z(k,a, b ) holds if and only if aZ,b. As in the preceding proof, by the Closure, Quantifier, and Relativization Properties there is a sentence 8 E 1 which holds in 6 and implies that ( w , s ) is a simple order with immediate successors and predecessors except for the first element, and (4) holds for all k E w . By countable compactness, 8 has a model 0-= (‘21-, Be,R - , Se,w ” , c e , I-), such that { w - , s ” )has a nonstandard element H . Then in terms of I ^ , (5) 91”- kl cp, BHeL/ i c p , and ‘21”- ~ ~ 2 It 3follows ~ ~that. the relation J between rn-tuples given by (a,, .

. . , u , ) J ( b , , . . . ,b,)

iff ( a l , . . . , ~ ~ ) l ~ ~ . -. .~, b,) ( b , ,

is a partial isomorphism between and BH-.But then by Proposition 2.5.3 and the hypothesis that 1 has Lowenheim number w , ‘?IH- and BH* are I-elementarily equivalent, contrary t o (5). Therefore (3) holds and the proof is complete. -I

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Theorem 2.5.4 shows that if we go beyond first order logic, we must give up one of three things: countable compactness, the Downward Lowenheim Skolem Theorem, or the properties of an abstract logic. There are good examples of each possibility. The logic 1( Q,) obtained from first order logic by adding the quantifier “there exist infinitely many” is an abstract logic which has Lowenheim number w but is not countably compact. Another such logic is the logic f,,,,,, which is like first order logic except that it allows countable conjunctions and disjunctions. The logic f( Q,) obtained from first order logic by adding the quantifier “there exist uncountably many” is an abstract logic which is countably compact but has Lowenheim number w , . There is a logic ltop(e.g. see Flum and Ziegler (1980)) which is not an abstract logic in the above sense because it deals with models with topologies, but is countably compact (in fact fully compact) and has Lowenheim number w . There are several other results which characterize first order logic in the style of Lindstrom’s Characterization Theorem, and use similar ideas in their proofs. Some of these results are stated as exercises. EXERCISES

2.5.1. The Hunfnurnber of an abstract logic 1 is the least cardinal a such that every sentence of f which has a model of power at least a has models of arbitrarily large power. (Thus the Upward Lowenheim Skolem Theorem shows that first order logic has Hanf number w ) . Show that every abstract logic 1, such that for every finite 9the class { cp E f : LfV C 9}is a set, has a Hanf number. This is the upward analogue of Proposition 2.5.2. 2.5.2. Prove that first order logic is the only abstract logic which has Lowenheim number w and Hanf number w . 2.5.3. An abstract logic 1 is said to pin down the ordinal a if there is a sentence cp E 1 and unary and binary relation symbols U and R in 9qsuch that cp has a model, and every model of cp is such that ( U , R ) is isomorphic to ( a , S ) . Show that if 1 is either countably compact or has Hanf number w , then f does not pin down w . 2.5.4. Prove that first order logic is the only abstract logic 1 such that any

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pair of partially isomorphic models is I-elementarily equivalent, and 1 does not pin down w .

2.5.5. A sentence cp in an abstract logic I is said to be 1-valid if 9l L I cp for every model 3 whose language contains zV.Prove that if an abstract logic 1 pins down o,then there is a finite 9such that the set of 1-valid sentences cp with YqC 2’is not r.e., and in fact is not even arithmetical. Hint:You may use the fact that the set of first order sentences which hold in the standard model of arithmetic is not arithmetical. 2.5.6 (Lindstrom’s Second Theorem). Prove that first order logic is the only abstract logic with Lowenheim number w such that for each finite language 2, the set of 1-valid sentences cp with zVC 2’is r.e. (or even arithmetical). 2.5.7*. Prove that first order logic is the only abstract logic 1 such that 1 has Lowenheim number w and the Robinson Consistency theory holds for 1.