7 Malitz Interpolation Theorem

7 Malitz Interpolation Theorem

THEOREM 8. (MALITZ[1969]). Suppose L has no function symbols. Let cp, $ be sentences of L,,, such that $ is universal and C cp + $. Then there is a universal sentence 8 of L,,, such that k cp -,8, k 8 + $, and every relation or constant symbol occurring in 8 occurs in both cp and $.

PROOF. We may assume without loss of generality that $ is quantifierfree, for the bound variables in $ can be replaced by new constants. Let S be the set of all finite sets s of sentences of M,,, such that only finitely many of the constants c E C occur in s, and s can be written as a union s = s, u s, where s, is quantifier-free, s, and s, have models, and: (*) There is no universal sentence 8 of M,,, with C As, + 8, k As, 7 8, and every relation or constant symbol (including the c E C) which occurs in 8 occurs in both s1 and s,. We claim that S is a inconsistent property. Let us check parts (C4) and (C7), and leave the rest to the student. (C4) Let Vxa(x) E s E S. We must show that for all c E C, { ~ ( c ) } u s E S. Let c E C and writes = s1 u s2 satisfying (*). Since s, is quantifier-free, Vxa(x) E s l . Let -+

s; = s, u {a(c)}, s; =

s2,

s'

=

s u'})C(.{

CASEI. c occurs in both s, and s,. Then s' E S because if I= A s; + 8, 8. k 17 S; -+ 8, then I= S, -+ 8, I= S, -+ CASEII. c does not occur in s, Then s' E S for the same reason as above, but time c does not occur in 8.

.

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MALITZ INTl3RPOLATION THEOREM

CASE111. c occurs in s2 but not in sl. Suppose s’# S. Then there is O(c) with C A si + e(c), C A s; + e(c). Hence C A s1 + (o(c). e(c)). Since c does not occur in sl, C A s1 4 Vx(a(x) + e(x)), when1 C A s1 -+ VxO(x). But C A s2 -+ Vxe(x), contradicting (*). Thi sI E

s.

(C7) Let {a(t),d = t } c s E S, where d E C and t is a constant. P show that s u {a(d)} E S. CASEI. a(t) E sl,d = t E sl. In this case C A s1 -+ a(d), and we si that s u {o(d)} E S by putting si = s1 u { d = t } , s; = s2. CASE11. a ( t )E s1,d = t E s2, d occurs in sl. Then s1 u {d = t} a1 s2 satisfy (*) because if C A (s1 u { d = t}) + 6 and C A s2 + 1 then C A s1 + (d = t + e), C A s2 + (d = t -+ 0). Therefore case applies, and s u {a(d)} E S. CASE111. o(t) E sl,d = t E s2, d does not occur in sl.If I= A (sl u { d e(d), then c A s1 -+ V X ( X = t + e(x)), t } ) .+ e(d), k A s2 -+ C A s2 -+ 1Vx(x = t + e(x)),’ contradicting (*) for s1,s2. Thus (*) satisfied by s1 u { d = t}, s2, and case I applies again. CASEIV. o(t) E s2, d = t E s2. In this case we see that s u { ~ ( d )E} by putting s; = sl,s; = s2 u { ~ ( d ) } . CASEV. o(t) E s2 ,d = t E sl,d occurs in s2. Similar to case 11. CASEVI. a(?) E s2 , d = t E sl,d does not occur in s2. If C A s1 4 e(a C A s2 u { d = t} + e(d), then C A s2 + Vx(x = t -+ 1e(x)), whence C A s2 + e(t). However, C A s1 4 e(t). This contradicts ( for sl,s2 . Then (*) holds for sl,s2 u {d = t } , and case IV applies. Let s E Sand let t be a constant. Let c E C not occur in s. We show th s u {c = t } E CASEI. t occurs in both s1 and s2. Then we see that s u {c = t} E by putting s; = s1 u { c = t } , s; = s2. CASEII. t does not occur in s2. Then again putting s; = s1 u { c = 1 s; = s, we see that s u {c = t } E S. CASE111. t does not occur in sl.Similar to case I1 with 1 and 2 reverse The above paragraph is the place where we would have difficulty L had function symbols, because some of the symbols in t might then o cur in s1 and others might occur in s2. By the Model Existence Theorem, every s E S has a model. Hen1 {p, I +} 4 S. This means that there is a sentence 8 of L ,,, of the requiri kind.4

s.

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MALITZ INTERF'OLATION THEOREM

DEPINITION. Let cr, cp be sentences of Lm1,. cp is said to bepreserved under submodels relative to cr iff whenever %, 23 are models of u, 58 is a submodel of %, and % is a model of cp, then 23 is a model of cp. THEOREM 9 (MALITZ[1969]). Let cp, u be sentences of L,,,. Thencp is preserved under submodels relative to D if and only if there is a universal sentence 8 of L,,, such that =! u -+ (cp ++ 8).

PROOF. We may assume that L has no function or constant symbols. For we may replace functions and constants by relations in the usual way and add to cr a sentence stating that those relations are functions or constants. Let C be a countable list of new constant symbols. By induction we define for each formula $ of L,,, a formula $ of M,,, as follows: If

"

+ is atomic, +" = $.

(A Y)" = A {$": $ E !PI. (1 $1" = 7 ($"). (VX$(X.. .))" = A $"(c.. .).

1

C6C

Thus $" is quantifier-free. Since cp is preserved under submodels relative to u, we have k

~7A

Cp -+ (0" + q").

By Theorem 8 there is a universal sentence 8 of L,,, such that b O A V + 8,

c 8 -,(8

-+

cp").

(u + cp) has no countable Since 8 A (0" 3 cp") has no model, 8 A model, since any countable model can be enumerated by C and thus (u" 3 cp"). From the Model Existence Theorem made into a model of it follows that 8 A (cr -+ cp) has no model at all, whence I= 8 + (u cp). Therefore I= u * (cp t,8).i -+

COROLLARY. Let u, cp be sentences of L,,,. Then cp is preserved under extensions relative to u if and only if there is an existential sentence 8 of L,,, such that I= cr -+ (cp t+ 8). PROOF. cp is preserved under extensions relative to cr if and only if (7 cp) is preserved under submodels relative to u.i

32

MALITZ INTERPOLATION THEOREM

"

Theorem 9 was first proved for first order logic L by LoS and Tarsk using methods quite different than the above proof. The corollary wai proved later by Henkin, and more generally by A. Robinson, for firs order logic. Several other known preservation theorems for L were gen eralized to L,,, by Makkai using the methods of the last two lectures.

PROBLEMS

1. (BARWISE, MALITZ).Suppose L has no function or constant symbols Let cp, be sentences of L,,, such that 4p is universal and k cp 3 $ Prove that there is a universal sentence 8 of L,,, such that k cp + 8 k 8 3 t+b, and every relation symbol which occurs in 8 occurs in both 4 and +. NOTE:To see why we assumed L has no function or constant symbols, consider the counterexample

+

k R(c) + (3x)R(x).

2. Let @ be a finite or countable set of sentences of L,,,, such that @ has no model. Then there are sentences 8,+, E L,,, for each rp E @ such that:

(i) {O,: cp E @} has no model. (ii) k cp -+ 6 , for each cp E @. (iii) For each cp E Qi, each relation, function, or constant symbol which occurs in 8, occurs in cp and in some member of Qi- (9).

3. (MAKKAI). Problem 2 above holds with (iii) replaced by: (iv) For each 4p E @, each relation symbol occurring positively (negatively) in 8, occurs positively (negatively) in cp and negatively (positively) in some member of @ - {cp}. 4. (BARWISE, MAKKAI,WEINSTEIN). Let 8 and 8 be countable models for L. Then can be isomorphically embedded in 8 if and only if every universal sentence of L,,, which holds in 23 holds in 8.

5. (CHANG,MAKKAI).Let 8, '23 be countable models for L. Then '23 is a homomorphic image of 8 if and only if every positive sentence of L,,, which holds in 8 holds in 23.

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MALITZ INTERPOLATION THEOREM

6. (MALITZ).For any two sentences cp, $ of L,,,, valent:

the following are equi-

(i) Any submodel of a model of cp is a model of $. (ii) There is a universal sentence 8 such that t cp

7. (BARWISE). For any two sentences cp, $ of L,,,, valent:

-+ 8

and I= 8 --* $.

the following are equi-

(i) Every countable model of cp can be embedded in some countable model of $. (ii) Every universal consequence of $ in L,,, is a consequence of cp. 8. (LOPEZ-ESCOBAR). Let cp, $ be sentences of L,,,.

Equivalent are:

(i) Any homomorphic image of a model of cp is a model of $. (ii) There is a positive sentence 8 in L,,,

such that cp F. 8, 8 I= $.