Completeness and persistence in the theory of models

Zeitaehr. f. math. Logik und Grundlagen d . Math. Bd. 2, S. 15-26 (1956)

COMPLETENESS APU'D PERSISTENCE IN THE THEORY OF MODELS

By ABRAHAM ROBINSON in Toronto (Canada)

1. Definitions and preliminary remarks In the present paper, we shall be concerned with the interrelations between

certain syntactical and semantic properties of b set of sentences K in the lower predicate calculus. We consider only closed sentences, i. e. sentences in which all variables are quantified. Such sentences may still contain (individual) constants which refer to specific individuals in the usual semantic interpretation of K in a model. Whenever we say that a structure M is a model of K we shall take i t for granted that all the relations (atomic predicates) of M occur in the sentences of K. Similarly, whenever we discuss sentences or predicates in connection with a given K we shall assume implicitly that the relations which occur in these predicates or sentences occur also in K . Finally, we shall take it for granted that the set K is non-empty and consistent, thus avoiding the separate discussion of trivial cases.

We say that the sentence X is defined in K if (in addition to all the relations of X , see above) all the constants of X occur also in sentences of K ; that X i u deducible from K i f K contains sentences Y,, . . . , Y,, such that (1.1) Y , A * * . A Y m 2 x

is a provable sentence of the lower predicate calculus; and that X is decidable in K i f either X or - X is deducible from K . Observe that this notion of decidability does not imply the existence of a concrete decision procedure.

The set of sentences K is said to be complete according to the standard defi. nition, if every sentence X which is defined in K , is also decidable in K . Two different modifications of this concept have been introduced by HENRIN [l] and by the present author [4] respectively. For a given set I' of (individual) constants, HENKIN calls the set of sentences K I'-complete i f for every predicate Q (5)

of one variable (i. e. well-formed formula with one free variable), such tha t Q (a) is deducible from K for all elements a of I', the sentence ( x ) Q(x) also is deducible from K .

For the second modified completeness concept we require the notion of a (complete) diagram. I n the semantic theory of thelower predicate calculus, a model M is determined by its set P of relations, its set C of constants, and the set of particular instances of the relations of P which hold between the elements of C (such as R ( a ) , &'(a, b , c ) , . . .). The diagram N of M is the set of atomic sentences S (a , 71, c) , . . . which hold in M together with the negations - S (a , b , c ) , . . . of such sentences if S ( a , b , c ) , . . . does not hold in M (compare [3], [5] ) . The set of sentences K is said to be model-complete if for every model M of K , the set K u N is complete, where N is the diagram of M .

16 ABRAHAM ROBINSON

2. Restricted completeness

There are other interesting notions related to that of completeness. I n order to introduce one of these, let us recall that for every sentence X , there exists a sentence X’ in prenex normal form and containing the same relations and constants as X , such that X = X‘ is a provable sentence of the lower predicate calculus. It follows that in order to ensure that the set K is complete i t is sufficient to require that every sentence X which is defined in K and which is in prenex normal form be decidable in K .

Now we may classify all sentences X which are in prenex normal form in a natural way according to the nuknber of blocks of quantifiers of the same type which appear in the prefix. Thus we shall say that the sentence X is of class 0 i f i t does not include any quantifiers a t all; and that X is of class 1 if i t is in prenex normal form and contains existential quantifiers only, or universal quantifiers only, or i f i t does not include any quantifiers a t all. We shall say that X , supposed in prenex normal form, is of class n , n 2 2 , if the number of changes of the type of consecutive quantifiers (from universal to existential or from existential to universal) does not exceed n - 1 . Thus, the sentence

(3 2) (Y) (4 (3w) &(x, ? / ? 2, w) 9

(4 (3Y) (2) (3w) &(2, Y, 2 , w )

(4 (Y) (34 (3w) &(x, Y, 2 , w)

where Q is free of quantifiers belongs to all classes n 2 3 while

belongs to the classes n 2 4 and

belongs to the classes n 2 2 . Predicates which are in prenex normal form can be classified in the same way.

We say that K possesses restricted completeness of order n (or briefly, is n-complete) if every sentence of class n which is defined in K , is also decidable i n K . For example, i t is an elementary but important fact that the diagram N of any model M is 0-complete. On the other hand, let N be the diagram of the field of rational numbers expressed in terms of the relations E ( z , y), S ( x , y, z ) , P ( z , y, x ) (“x equal to y”, ‘‘2 is the sum of x and y”, “ z is the product of x and y”). Then the sentence

( 3 4 P ( X , 292) (i. e. “2 has a square root”) is not decidable in N . Thus, N is zero-complete but not 1-complete.

Again, let H be a system of axioms for the concept of an ordered set. It is not difficult to formulate such a set H within the lower predicate calculus in terms of the relations E ( x , y) (“z is equal to y”) and &(x, y) (“x is smaller than y”), and without constants. Consider the sequence of sentences

x, = (3x1) (3 za) . * . (3 2,) [&(XI, z2) A &(%, 53) A * * - A &(%-I, 341 ( n = 2 , 3 , 4 , . . .),

COMPLETENESS AND PERSISTENCE IN THE THEORY Olr MODELS 17

and let K be the union of H and of { X 2 , X,, . . . , X,, . . .}. Then X, entails that there exist a t least n different elements and so K constitutes a system of axioms for the concept of an infinite ordered set.

Now let X be any sentence of the form

(2.1) where R is free of quantifiers. Such a sentence is said to be an existential sentence. We propose to show that if X is defined in K then i t is also decidable in K .

By assumption, X and R do not contain any constants, and contain no relations other than E ( x , y) and Q(x , y) . Represciit R in propositional normal form, as a disjunction of conjunction of atomic sentences or of the negation of such sentences, in the usual way. Next, replace the negation of any atomic sentence of the form E (2, y) by Q(x , y) v Q(y, x ) , and the negation of any atomic sentence of the form Q(x, y) by E ( x , y) v Q(y, x). Expand in accordance with one of the distri- butive laws so that there results a disjunction of conjunctions of atomic sentences (i. e. without negations). This may be written as

x = (3 X I ) . . . (3 xn) R ( x , , . . . , XJ (9% 1 O ) ,

m ‘- ( 3 x l ) ’ . ’ (3x17,) R ~ ( 2 , ( 1 , 1 ) 7 ” ‘ , X , ( n l , I ) ) , 2 = 1

where the predicates R,, i - 1, . . . , m are conjunctions of atomic formulae,

and where the symbol V Ri denotes the disjunction of these predicates. Next,

exchange the order of quantification and disjunction, omitting in each case the quantifiers whose variables do not appear in the predicate R, to which they belong. The result is

m

a = 1

m (3.2)

By examining the individual steps in the passage from (2.1) to (2.2), i t is not difficult to verify that the equivalence X = 2 is deducible from K . It follows that if we prove that Z is decidable in K , we shall thereby have shown also that X is decidable in K .

We consider the following alternatives : (i) a t least one of the Ri does not conta.in the relation Q (x, y) . In that ca.se,

the appropriate disjunct is of the form

( 3 ~ j ) (32 , ) . . . (32,) (32,) [ E ( x ~ , ~ k ) A * * . A E(x, , , xt)] and this holds in every model M of K as can be seen by taking an arbitrary but fixed element of M for x i , x g , . . . , x,, x t . It follows that both Z and X hold in all models of K and, accordingly, a.re deducible from K .

(ii) All the RL include atomic sentences which involve .Q. In that case, we eli- minate the relation E ( x , y) from the individual Ri in the following way. We consider a particular E (xi, xk) in Ri . If j = k we omit the term E [xi, xg) from the conjunction and - unless xj ocours elsewhere in Ri - the quantifier (3 xj) from the prefix to R,; if j < k we omit E (xi, x g ) and (3 xk) , and replace xk everywhere else in Izi by xi; and if k < j we omit E (xj , xn) and (3 xj) and replace xk everywhere

‘2 Ztschr. f . math. Logik

18 ABRAHAM ROBINSON

else in Ri by xj. Continuing in this way, we see that after a finite number of steps each Ri is reduced to a conjunction of atomic sentences of the form Q ( x , y). The resulting sentence

m

is then such that Z* 3 Z (and hence Z* = X ) is deducible from K .

a set of the for’m We call a predicate Rr in (2.3) cyclic if we can select from among its conjuncts

{Q(x, , , XIz)’ &(XI2, %,,I> * . . > &(x,,, x9J

(including the possibility that 1 = 1). It is not difficult to see that if Ri is cyclic then the disjunct in question cannot be satisfied in any model of K . On the other hand, if R: is not cyclic then some reflection shows that the i t h disjunct is satisfied in all models of K . We conclude that i f a t least one of the R: is not cyclic then Z* (and hence X ) is deducible from K , while if all the R: are cyclic then -Z* (and hence - X ) is deducible from K . Since the negation of a sentencc X is decidable whenever X is decidable, we have shown that every X of class 1 which is defined in K , is also decidable in K , K is 1-complete.

On the other hand, K is not 2-complete. Indeed, the sentence

(3 4 ( Y ) [E (299) ” Q (x Y)1 (i. e. “therc exists a first element”) is not decidable in K .

3. Pcrsisttincc Another concept of model-theory which has a bearing on completeness is that

of‘ persistence. A predicate Q(xl, . . . , 2,) is called persistent with respect t o a given set of sentences K , if for any set of constants a,, . . . , a,, which belong t o a model M of K , &(al, . . . , a,) holds in M only j f i t holds also in all other models of K which are extensions of M (see [2], Chapter 8). An equivalent definition is that for any model M of K , “&(a,, . . . , a,) holds in M” should entail “Q (a,, . . . , a,) is deducible from K u N ” where N is the diagram of M . We regard sentences as predicates of order 0. For these, the above definition of a persistent predicate reduces to tho definition of a persistent sentence as given in [5]. Thus, a sentence X is persistent with respect to R if for every model iM of K which satisfies X , X is deducible from K u N .

A predicate (in particular a sentencc - Bee [7]) ia called universal if i t is in prenex normal form and if it contains none but universal quantifiers (in particular, i f it does not contain any quantifiers a t all).

Consider the following conditions. (3.1) Every universal sentence X i s persistent with respect to K . (Note that X

(3.2) Every universal predicate Q is persistent with respect to K . (3.3) Every universal predicate Q which is defined in K (i. e. whose constants,

may contain constants which do not occur in K ) .

if any, occur in K ) , i s persistent with respect to K .

COMPLETENESS AND PERSISTENCE IN THE THEORY OF MODELS 19

We propose to show that any one of these three conditions entails the other two. It is clear that (3.2) entails (3.3) and also (since sentences are regarded as a class of predicates) that (3.2) entails (3.1). Now assume that (3.1) holds. Let,

&(xi, * * , xn) R(xi , * 3 xfi, b1, * * 9

be a universal predicate, where the constants of Q which do not occur in K , 6 , , . . . , b,, say, are distinguished on the right hand side. Suppose that &(a,, . . . , a,) holds for a set of constants a,, . . . , a, in a model M of K . Thus the sentence R(a,, . . . , ulL, b,, . . . , a,) holds in M and hence, by (3.1), holds in all extensions of M which are models of R . This shows that Q (2, , . . . , 2,) also is persistent with respect to K , and proves that (3.2) is satisfied.

Again, assume that (3.3) is satisfied, and let X be a universal sentence which, in additions to constants which occur in I<, contains also the constants a,, . . . , a,, X = Q (a,, . . . , a,&), say. Let M be a model of K in which X holds. Since Q(x,, . . . , 2%) is a universal predicate which is defined in K , (3.3) entails that Q (a,, . . . , a,$) = X holds also in all extensions of M which are models of K . It follows that K satisfies (3.1). Combining these results, we see that any one of the three conditions (3.1), (3.2), (3.3) singly, entails the other two. A set K which satisfies one, and hence all, of these conditions will be called pre-complete. (3.4) Theorem. Let Q(x, , . . . , xn) be a predicate which is persistent with respect

to a given set of sentences K . Then there exists an existential predicate

9 Y,), ' &*(xi, - 3 x J = @ ? / I ) . (3yJ R(x, , . . . , x,, ~ 1 , .

where R does not include any quantifiers - such that

is deducible from K . Thus, a predicate Q which is persistent with respect to ' K can be replaced by

another predicate Q* for which the fact that i t is persistent with respect to K (and with respect to any other set) is self-evident.

Proof of (3.4). For n = 0 (i. e. for sentences) the theorem ,is known (see [8] or [5] ) . For li 2 1 let a,, . . . , an be a set of constants which occur neither in Q nor in K : Then X = Q (a,, . . . , a,) is a sentence which is persistent with respect to K . Hence, by the result, just mentioned, there exists a sentence

(21) * * * ( x n ) [Q* (x i . * . 9 2%) Q (xi 9 * * * 9 xn)1

Y = @ Y , ) . . . ( ~ Y , ) Z ( Y ~ , . . . Y,,,), where Z is free of quantifiers, such that X = Y is deducible from K . It is concei- vable, a priori, that Z does not contain all the constants a,, . . . , a,, but if so we may add a provable sentence involving thesc constants, to Z in conjunction. Thus, we shall assume from the outset that .Z includes a,, . . . , a,&. We may then write

so that = R ( a , , . . . , a,,, Y,, . . . , Y,)

(3.5) Q(a,,...,an)__[(3~1)...(32/n,)R(al,...,an,~l,...,~,)l is deducible from K . But the constants a,, . . . , a, do not occur in either K ,

2'

20 ABKAHAM ROBINSON

Q(x, , . . . , z,J, or R ( x , , . . . , x ? ~ , y l , . . . , y J , by definition, and so by the rules of the predicate calculus, (3 .5) implies that the sentence

(21) ' * . (4 [ Q ( % 2 . . . > 2,) s3 [P Y1) . . ' (3 Y,) R(x1, . . . 3 x, 1 y1, . . 9 Ym)ll also is deduciblc from K . This proves (3.4).

(3.6) Corol lary. I t can be assumed that all the relations and constants which occur in the predicate Q* of Theorem (3.4) occur either in I< or in Q .

Although not stated explicitly in [B ] , a check on the proof given there shows that (3.6) holds for n -0 . It then follows from the above proof of (3.4) that the corollary holds also for general 11.

(3.7) Lemma. Suppose that the equivalence

(4 . . . (zn) [Q(x1, . . . x1J = Q* (xl, . . . , x7JI (n 2 1) (3.8) i s deducible from a given set of sentences K where Q and Q* m a y , or may not, include further quantifiers. Let

'11 92 . . . 991

be a n y sequence of quantifiers such that qt contains the variable x, (i = 1 , . . . , ?a). Then the equivalerrcr

i s deducible from K . q i . . . 9 , $ & ( ~ 1 , * . . 3 ~ n ) l 3 ki ~ 2 . + * 48, &*(XI, . * 9 3 xn)I

Proof. It is sufficient to show that, on the hypothesis of the lemma, the sentence

(3.9) [qi q z . * QR Q(x1 9 . . . 9 xnll 7 E P ~ q z . 3 . 411. Q* (51 9 . * 7 xJ1 is deducible from K . The conclusion of the lemma is then obtained by interchanging Q and Q* in (3.9) and combining the result with (3.9).

Let a l , . . . , alL be a set of constants which are not contained in either Q or Q* or the sentences of K . Then (3.10) &(a,, . . . 9 a,) = &*(a,, . . - , a,) is deducible from K , by (3.8). Suppose first that q, rules of the predicate calculus ensure that

and hence

(xJ. In that case. the

[(xn) &(a,, . . . 7 a, - I 1 %)I &* (a,. . . . f a,'- I f a,)

[(%I &(a,, . . * , % - 1 > %)I [(%) Q" (%> . . * , % - I , %)I are deducible from K . On the other hand, if qn = (32%) then we derive first that

&(a, , . . . , arb) 3 [ (3xn) &*(a,, . . - 9 a,-,, xJ1 is deducible from K , and afterwards that

[(3 2,) Q (a1 > * . . 3 at*-- 1 > zn)l 3 [(3 xn) Q* (a1 > . * > an- 1, x,)l also is deducible from K . In either case, we find that

[Qn Q (a1 9 . . f , %- 1 > 5 n ) l [a, Q* (a1 9 . . . , a,-, 9 z,)l is deducible from K . Repeating this procedure in succession for q n - , , we finally obtain the result that (3.9) is deducible from K .

. . . , q l ,


We now come to the principal result of this section. (3.11) Theorem. I f K is a pre-complete set of sentences then every predicute

Q(xl, . . . , xJ is persistent with respect to K . By the definition of a pre-complete set [see (3.2)] any uniaersal predicate is

persistent with respect to K . The point of the theorem is that this condition is already sufficient to ensure that the same conclusion applies to any other predicate.

Proof of (3.11). For every predicate Q ( x , , . . . , xn) there exists a predicate Q* (xl, . . . , xn) in prenex normal form and containing the same constants and relations as Q such that

is provable. It follows that in order to prove (3.11) we may suppose from the outset that Q (xl, . . . , xn) is in prenex normal form.

If Q(xl, . . . , xn) does not include any quantifiers a t all, or if i t includes only existential quantifiers, then the fact that Q is persistent is obvious. If Q (xl, . . . , x,?) is a universal predicate then the persistence of Q with respect to K is ensured by (3.2). This shows that all predicates of class 1 are persistent with respect to K . For the remaining predicates in prenex normal form, we shall prove the theoreni by induction on the class of the predicate. Suppose then that we have already proved (3.11) for all predicates of class rn 2 1 . Let Q(zl, . . . , 2,) be a predicate of class WL + 1 , but not of class wb, and suppose first that the last quantifier in the prefix of Q is universal. Then Q can he written as

where R is free of quantifiers and k 2 1 and the quantifier qL contains the variable y, (i = 1 , . . . , k - 1 ) , while the quantifiers after (3 y l ) are all universal. Then the predicate

( T I ) * * . (2,) [Q (xi 9 . . - 3 xn) Q" (21 9 . . * , x:n)l

(3.12) Q(z1, xn) == k i . . . q k - l ( 3 ~ n ) (Y~+I)... (YL) f i ( x i , . . . > X n Y i , YL)I,

S(%, . . * 7 %> y1, . * . 9 Y k ) = [ ( Y k + l ) . . . (Yz) R ( x , , . * * 7 x,, Y1, * * . , Y k , . . . > YJl is universal. It follows, by virtue of (3.2) that this predicate is persistent, and hence, by (3.4), that there exists a predicate (3.13) S"(Z,, . . ., x ~ , y l , . . . , ?/k) =:

= [(3 Y h + d . * * (3 Yp) l2* (IL.1, - * * 9 Z,, y1, * * . 9 Y k , * . 9 , yzJ, where R* is free of quantifiers such that

is deducible from K . Now substitute R* for S after (3yk) in (3.12). The result is (4 . . . (x,) (Yl) . - . (Yk) [ S ( Z l , * * * > 4 = S*(SlY . 9 Y k ) l

(3.14) &*(z,, . . . , xn) =:

= kl *%-I ( 3 Y k ) (3Y,+,). * * (3Yp) B*(Xl? * . Y x,, y19 * * , Yp)l. According to the lemma (3.7), the equivalence

is deducible from K , and it is therefore sufficient to prove that Q* (x, , . . . ,*xJ is persistent with respect to K . But the inspection of (3.14) shows that &* is of class m , and so Q* is indeed persistent with respect to K , by the hypothesis of the induction.

(xi) . - * (xn) [Q ( 2 1 , . - * , 2,) Q* (xi 3 . . 9 x,)l

22 ABRAHAM HOBIKSON

4. Tests for conipletcncss

In section 2, we gave an example of a set K which is 1-complete but not %corn- plete. Thus, the set in question is certainly not complete. There also exist sets of sentences which are pre-complete but not complete. For example, let K be a set of axioms for the concept of an algebraically closed field, formulated in terms of the relations of equality, addition and multiplication, ~E (x, y) , S (5, y, z) , P ( x , y , z ) , and without constants. X is not complete since the sentence

(4.1) (4 (Y) CS(Z, 2, Y) = S(Y, Y, Y)1 which is equivalent to the statement that the field is of characteristic 2, is not


decidable in K . At the same time, K is pre-complete. Indeed, let X be a sentence formulated in terms of the relations E , 8, and P, and in terms of a number of constants a1 , . . . , a , . Then if X holds in a model M of K , i t holds also in any other model of K which is an extension of 211 (see [3]). Thus X satisfies (3 .1) and is pre-complete.

The following theorem may be contrasted with the above remarks. (4.2) Theorem. Let K be a set of sentences which is both 1-complete and pre-

complete. Then K is mvnplete. Proof . Let X be any sentence which is defined in K . Then X is persistent with

respect to K , by (3.11). Hence, by (3.4) there exists a sentence Y which is existential and accordingly of class 1, such that X = Y is deducible from K . Moreover, by (3 .6) , we may assume that Y also is defined in K . Since K is 1-complete, i t follows that Y is decidable in K . But if so then X also must be decidable in K . This proves (4.2).

While the hypothesis of (4.2) is satisfied in many interesting instances it is not a necessary condition for completeness. Thus, let M be the ordered set of positive integers { 1 , 2 , 3 , . . . ). and let K be the set of all sentences formulated in terms of the relations E (2, y ) and Q ( x , y) (read “ 3 equal to y ” , “5 smaller than y ” ) and without constants. Then K is complete. At the same time, K is not pre-complete since the sentence

holds in M but i t does not hold in the ordered set (0 , 1 , 2 , 3 , . . .} which i s a model of K and an extension of M .

(4.3) Lemma. Given a set of sentewes K , suppose that for every model M of K , the set K u N is 1-complete, where N is the diagvam of M . Then, for every model M of K , the set K u N i s also pre-complete.

Proof . Let M be a model of K , N its diagram, and let X be a universal sentence which holds in a model &I’ of K u N with diagram N ’ . Since K w N‘ is supposed 1-complete and X holds in M’ i t follows that X must be deducible from K u N’ . Thus X holds in all extensions of M’ which are models of K u N ’ , K uN is pre- complete.

(4.4) Lemma. Civen a set of sentences K , suppose that for a particular model M of K , with diagram N , the set K u AT is pre-complete. Then K u N is also 1-complete.

Proof . Let M be a model of K , N the diagram of M , and suppose that K u N is pre-complete. Let X be a universal sentence which is defined in K u N . If X holds in M then i t must be deducible from K u N since K u AT is pre-complete. Suppose now that - X holds in M . If

(5) [E ( 1 I .) ” G? (1 7 4 1

x= (~l)...(~7~)~(Xl,...,~~),

X’ = (3 XI) . . . (3 X 7 J [ N Q (XI, . . . , Xn)]. where Q is free of quantifiers, define

Then - X zz X’ is provable, and X‘ is an existential sentence which holds in M . It follows that X’ holds also in all extensions of M , and hence is deducible from N .

24 ABRAHAM ROBINSON

We coriclucle that - X is deducible from Ii‘ w N ) and hence that in any case, X is decidable in K . Since all universal sentences which are defined in K u N are decidable in that set, the same must apply to all existential sentences. We conclude that K u N is 1-complete.

(4.5) Lemma. Let K be a pre-complete aet of smtences. Theta for every model M of K , with diagram X , K v N i s both pre-complete and l-complete.

Yroof.If RuiVisiiot prc-complete, then thereexistsasentence X - Q(al, . . . ,{~,J - \I here a,, . . . , a,’ are the constants of X which do not appear in K - such that, X holds in some modrl M’ of K v N but does not hold in an extension M” of 111’ which is a model of K v N . But both M ‘ and 111’’ are also models of K and so it follows that K is not pre-complete, contrary to assumption. This proves the first part of the lemma whilc the second part follows directly from (4.4).

(4.6) Theorem. I n order that a spt of sentences K be model-complete, i t i s necessary and sufficient that for ecery model M of K , the set K U N i s l-complete, where AT i s the diagram of M .

Proof . The necessity of the coridition is obvious. To prove sufficiency, consider any model M of K with diagram J. Then K W N is 1-complete and, by virtue of (4.3), K w N is also pre-complete. Theorem (4.2) then shows that K u N is completc. Since M is an arbitrary model of K we conclude that K is model- complete.

Theorem (4.6) leads readily to the model-completeness test of [4]. However, the method of proof employed in [4] is very different from that of the present paper. Nunlerous concrete applications of the test will be found in 161.

(4.7) Theorem. I n order that a set of sentences K be model-complete, it is necesswy and sufficient that f o r euwy model M of K , with diayram AT, the set K u AT i s p e - complete.

Proof . The condition is necessary. If I< is model-complete then K u N is 1-complete for all nioclels M of K , N being thv diagrmi of M . It then follows from (4.3) that every sct K v N of this description is iilso pre-complete.

The condition is sufficient for i t implies, by (4.4), that for every model M of I<, the set K v A r is also 1-complete. Hence K is model-complete, by virtue of (4.6).

A companion to Theorem (4.7) is (4.8) Theorem. In order that K he model-complete it is necessary and su f f i c imt

that K be pre-complete. Thus i t turns out that the notions of model-completeness and pre-completeness

coincide. Proof of (4.8). The condition is necessary. Let & ( G , , . . . , x,.) be a predicate

such that for some set of constants ccl, . . . ) a,, &(a,, . . . , n,) holds in a model M of K . Since K is model-complete, i t follows that &(a,, . . . , u,,) holds also in all extensions of &’ which are models of K .

The condition is sufficient. Indeed, i f i t is satisfied then K u N is 1-complete, for the diagram N of any niorlel M of K , by (4.5). Hence K u N is model-complete, by (4.6).


It is remarkable that the corresponding companion to (4.6) is not true. Thus, let K be a set of axioms for the concept of an infinite ordered set, as detailed in section 2 above. Then K is 1-complete, as proved in that section. But it is easy to see that K is not model-complete. Indeed, the ordered sets { 1 , 2 , 3 , . . .} and ( 0 , 1 , 2 , 3 , . . .} are models of K , the latter being an extension of the former, yet the sentence [x) [E (1, x) v Q (1 , z)] holds in the former mo-del but not in the latter.

Conversely, K may be model-complete without being 1-complete. Indeed, a t the beginning of this section we gave an example of a universal sentence (4.1) which is defined in a particular model-complete set K without being decidable in K .

5. Model-eompletencss and F-completeness

HENKIN’S definition of T-completeness was given in section 1 above. Now let K be a set of sentences and let M be a particular model of K , with diagram N . Then we shall say that K is M-complete i f the set K u N is I’-complete for the set T of all the constants of M .

(5.1) Theorem. In order that the set of sentences K be model-complete it is necessary and sufficient that K be M-complete for every model M of K .

Proof . Suppose that K is model-complete and let M be a particular model of K and r its set of constants. Suppose that for some predicate of one variable Q (x) , the sentence Q (a) is deducible from K v N for every a E r , where N is the diagram of M . Then Q (a ) holds in M for every a E r, and so (x) Q (x) holds in M. But K u N is complete and so every sentence which holds in M must be deducible from K v N . Hence [x) Q ( x ) is deducible from K v N , showing that K u N is M-complete.

Now suppose that K is M-complete for all models M of K , and consider a specific model M’ of K . Then we propose to show that the set K u N‘ is complete, where N is the diagram of M’. Let then X be any sentence in prenex normal form which is defined in K u N ’ . We have to show that X is decidable in K u N‘. The proof is by induction on the number of quantifiers in the prefix of X . If X is whithout quantifiers and holds in M’, then X is deducible from N’ and, a fortiori, from K u N’. If X does not holds in M’, then - X holds in M’ and is therefore deducible from K u N’.

Suppose that we have proved our assertion for sentences with n quantifiers in the prefix (n 2 0) , and let X = (2 ) Q [ z ) , where Q is defined in K u N , and contains n further quantifiers in the prefix. If X holds in M’ then &[a) holds in M’ for all constants a of M‘. Any such sentence &(a) is decidable in K v N’ by the hypothesis of the induction and so the sentence must actually be deducible from K uN‘. It then follows from the assumed M‘-completeness of K that (x) Q (x) is deducible from K u N’ . On the other hand, if - X holds in M , then (3 z ) [- Q ( z ) ] holds in M and so - Q (a) holds in M for some a E r. Then - Q (a) i s deducible from K u N’ , by the hypothesis of the induction, and so therefore are the sentences (3 z ) [- Q ( z ) ] and - X.

This completes the proof of (5.1). 3 Ztschr. 1. math. Logik

26 ABRAHAM ROBIXSON

Iteferonces [1] L. HENXIN, r-completeness. Proceedings of the International Congress of Mathematicans

[2] A. ROBINSON, On the Metamatheniatics of Algebra. Studies in Logic and the Foundations

[3] ------, On predicates in algebraically closed fields. Journal of Symbolic Logic vol. 19 (1954),

[4] , Ordered structures and related concepts. Symposium on the mathematical interpretation of formal systems, Amsterdam 1954. Studies in Logic and the Foundations of Mathematics, 1956, pp. 51-56.

in Amsterdam vol. IT, 1954, pp. 403-404.

of Mathematics, Amsterdam 1951.

pp. 103-114.

[j] -, On a problem of L. Henkin. To he published in the Journal of Synibolic Logic.. {F] --, Complete Theories. To be published in Studies in Logic and the Foundations of

[7] A. TARSKI, Contributions to the theory of models, I, 11. Indagationes Mathematicae vol. 16

[8] J. LO$. On thc extending of models (I). Fundamenta Mathematicae vol. 42 (1955),

Mathematics.

(1954)a pp. 572-581, 582-588.

pp. 38-54.

(Eingegangen rtin 23. Jxnnar 1956)

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Completeness and persistence in the theory of models