Recursively Presentable Prime ModelsAuthor(s): Leo HarringtonSource: The Journal of Symbolic Logic, Vol. 39, No. 2 (Jun., 1974), pp. 305-309Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272643 .

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THE JOURNAL OF SYMOLIC LOGIC Volume 39, Number 2, June 1974

RECURSIVELY PRESENTABLE PRIME MODELS

LEO HARRINGTON

It is well known that a decidable theory possesses a recursively presentable model. If a decidable theory also possesses a prime model, it is natural to ask if the prime model has a recursive presentation. This has been answered affirmatively for algebraically closed fields [5], and for real closed fields, Hensel fields and other fields [3]. This paper gives a positive answer for the theory of differentially closed fields, and for any decidable H,-categorical theory.

?0. To review some definitions. The language of a theory T is denoted by L(T). All languages will be presumed countable. An x-type of T is a set of formulas with free variables x, which is consistent with T and which is maximal in this property. A formula with free variables x is complete if there is exactly one x-type containing it. A type is principal if it contains a complete formula. A countable model of T is prime if it realizes only principal types. Vaught has shown that a complete count- able theory can have at most one prime model up to isomorphism.

If T is a decidable theory, then the decision procedure for T equips L(T) with an effective counting. Thus the formulas of L(T) correspond to integers. The integer a formula p(x) corresponds to is generally called the G6del number of p(x) and is denoted by rq,(x)y. The usual recursion theoretic notions defined on the set of integers can be transferred to L(T). In particular a type r is recursive with index e if {rid; p e r} is a recursive set of integers with index e.

A structure W is recursively presentable if .1 is countable and if there is an enumeration ao, al, . . . of 1- such that the complete diagram of <Kd, ao, a1, . .. > is decidable, that is, if {p(ao, . . ., an); a? F p(ao, . . ., an)} is a decidable theory.

If this review has been inadequate, see [6] and [7].

?1. Recall Vaught's theorem on the existence of a prime model (see [7]). If T is a countable complete theory, then T has a prime model iff every formula con- sistent with T is a member of a principal type. An effective version of this is the following:

THEOREM. Let T be a complete decidable theory. The following are equivalent: (a) T has a recursively presentable prime model. (b) There is a recursive function f such that given a formula qp(x) consistent with T,

f(rq,(x)1) is an index for a recursive type r,, rF contains qp(x), and r, is principal. PROOF. (a) => (b) Obvious. (b) => (a) (Notice that we may infer from (b) that for any formula p(x) consistent

Received February 28, 1973. ? 1974, Association for Symbolic Logic

305

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306 LEO HARRINGTON

with T there is a complete formula which implies p(x). But (b) does not assert that such a complete formula can be found effectively from rp(x)'.)

The prime model will be constructed by an effective Henkin argument. Let xo. xl, . . . be the free variables of L(T), and let Ho, H1, . . . be the Henkin

formulas of L(T). We may arrange matters so that H,, is a formula with free variables xo, . . ., x,,. (Thus we may let H,,(xo . . ., x,,) be the formula 4(x0, . . ., x,) v Vx(- p(x0,. . ., xn - , x)) where p(x0,.. . ., Xn) is the nth formula of L(T).)

Let c0, cl,... be a sequence of new constants and let I * = T U {Hn(co, * * . cn); n e <D}-

Our aim is to construct a complete decidable extension , of X' (which therefore gives rise to a recursively presentable model of T) with the property that for each n there is a formula ,n(xo, . . ., xn) such that c0,.. ., cn realize rF0 (which clearly implies that the model 9 gives rise to is prime).

91 will be constructed in stages. 91, will be the part of 91 constructed by stage s. Of course, As, will be a finite collection of sentences, and *' u As, will be consistent.

At each stage s there will also be a sequence of formulas T3(xo, . x,Xn), n = 0, 1, . W will be the sth approximation to the desired formula 4pn.

For notational purposes let O(c0, . . ., CN,) be the conjunction of 94!, and let

G%(X0) I ., x,) = 3X%+, * Xm[Ho(Xo) A ... A Hm(Xo.. ., Xm) A 6s(Xo,..., XN)],

where m = max (n, N,). Stage 0. to& = 0, qn(Xo -* . ., Xn) = O0n(X3 . *, Xn). Stage s + I. Let s code in some canonical way a triple of integers <n, m, t>.

This tells us to look at the mth formula with free variables xo,..., x ,. Let (XOY - - -., X%) be this formula. If +(xo,.. ., xn) Fo rap or if Ae' U 91s F b(co, .., cn), or if

T, Gsn-1(Co - ., Cn,-,1) M 32x[8n(co . . .,c Cn1, x) A b(co, * * *, c -.,, x)], then leave everything the way it is and go on to stage s + 2.

Otherwise let 91S,+ = 9, U {(c0, . . ., cn)}, and let

V, + 1(xo . . xi) = V,(xo, . . ., xi), i< n, = s+(xo, . . ., xi), i n.

Notice that (i) T, Gs(co.. ., c,) CFpV(co,..., cC),

(ii) &~n(xos , xn) e- r,,i..

Let 91 = aY u(U, 91,). CLAIM. For each integer k there is an s such that,for alls' _ s, (p;(Xo,..., Xk) =

(Pk4(Xo, . . ., Xk), and such that, for all b(xo, . . ., Xk) e rF,, 91 F /(co0,**, ck). (This claim clearly completes the proof of the theorem.) PROOF OF CLAIM. Assume the claim is true when k = j, for all j < n. We will

prove the claim for k = n. For each j < n there is a stage a such that Ti = hi' for all s' > a. Since rF is principal it contains a complete formula. So there is a stage a' such that, for all s' > a', Ojs'(xo, . . ., xj) is a complete formula.

Thus there is a stage s such that, for all s' > s and for all j < n, my = m>' and 8js' is a complete formula.

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RECURSIVELY PRESENTABLE PRIME MODELS 307

For any s' > s, if s' codes K], m, t> for some j < n, then since G1' is complete, nothing is done at stage s' + 1. Thus, for all s' _ s, a4 = p'.

For any formula b E rp0 and any s' > s, T u {6n(Co,.. , cn), f(co, ..., cC)} is consistent (since by (ii) both sn' and b are in rFS), and therefore since 0 _ is com- plete, T, On'- (co.. .Cn -c1) F 3x[sn(co, .*Cn -c1, X) A b(Co0,.**, Cn - 1, X)]. So if b(xo,..., xJ) is the mth formula in the variables xo,..., xn and if s' > s codes <n, m, t> for some t, then, by the construction, ' u s + 1 F (cO, * * , cCn)-

?2. We are now in a position to demonstrate that a few assorted decidable theories have recursively presentable prime models.

(a) For a theory T, a formula 8(x) of L(T) is algebraic if, for some n, T F 3 5 nx8(x). A type is algebraic if it contains an algebraic formula.

The following fact is fairly familiar. COROLLARY 1. If T is a complete decidable theory with a prime model, and if the

prime model realizes only algebraic types, then the prime model has a recursive presentation.

PROOF. Given p(x) a formula consistent with T, by assumption there is an algebraic formula @(x) such that T F 8(x) -- p(x). Find such a 8(x), and let r, be a recursive type containing 6(x). Since 6(x) is algebraic, r, is principal. This procedure can be carried out effectively, and so, by the theorem, T has a recursively present- able prime model. E

(b) A theory T has the finite basis property if, for any collection S of atomic formulas in n variables (atomic in the sense of syntactical complexity), there is a finite subset F of S such that T, F F S.

Using [8] as a guide, call a quantifier free formula p(x) in L(T) constrained if, for every atomic formula 8(x), either T, p(x) F 8(x) or T, p(x) F - 6(x). Consider the following property of theories T: (*) every.quantifier free constrained formula of L(T) is complete.

Notice that any substructure complete theory (see [7]) has property (*).

LEMMA 2. Let T be a complete decidable theory, and assume that T is model complete (see [7]), has the finite basis property, and satisfies property (*). Then T has a recursively presentable prime model.

PROOF. Let 4(x) be consistent with T. Since T is model complete we can find a quantifier free formula +(x, y) such that T F p(x) +-+ 3yb(x, y).

Construct a collection S of atomic formulas in the variables x, y as follows: SO = 0. Given Sn, let 8(x, y) be the nth atomic formula in the variables x, y. If T U Sn u {+(x, y), 8(x, y)} is consistent, let Sn + 1 = Sn u {8(x, y)}. Otherwise let Sn + l = Sn. Let S = Un Sn.

By the finite basis property there is a finite subset F of S such that F generates S. For any atomic formula 8(x, y), either 8(x, y) E S or T u S u {0(x, y)} F - O(x, y). Thus the conjunction of F u {+(x, y)} is a quantifier free constrained formula and so by property (*) it is complete. Thus S u {0(x, y)} generates a principal type r, and so {3yO(x, y); 8(x, y) E r} also generates a principal type. Let IF, be this type. The procedure of finding r,, from T was effective, and thus by the theorem T has a recursively presentable prime model. Z

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308 LEO HARRINGTON

(c) A. Robinson has shown that the theory of differential fields of characteristic O (DFO) has a model completion, called the theory of differentially closed fields of characteristic 0 (DCFO). DCFO is complete and substructure complete. L. Blum has provided DCFo with a recursive set of axioms, and has shown that, for any

i DFO, V has a prime differential closure, that is, DCFO U (diagram of d) has a prime model. (For an exposition of DCFo see [7].) For any .0 k DFO, DCFo u (diagram of d) has the finite basis property (see [4]). If a? is a computable differ- ential field of characteristic 0, then DCFo u (diagram of a?) is decidable. Thus by Lemma 2 we have

COROLLARY 3(i). The differential closure of a computable differential field of characteristic 0 is itself computable. E

C. Wood has shown for each prime p that the theory of differentially perfect fields of characteristic p (DFp) has a model completion called the theory of differ- entially closed fields of characteristics (DCF,). Let .W k DF,. In [8] it was shown that DC',, U (diagram of d) has a prime model, called the prime differential closure of -d. It was also pointed out in [8] that DCF, u (diagram of a) has the finite basis property and satisfies property (*). So again by Lemma 2 we have

COROLLARY 3(ii). The differential closure of a computable differentially perfect field of characteristic p is itself computable. [1

(d) A theory is Ml-categorical if it has exactly one model of power Ol. Baldwin and Lachlan [2] have shown that any X1-categorical theory bears a remarkable resemblance to the theory of algebraically closed fields.

COROLLARY 4. If T is an Ml 7J categorical decidable theory, then every countable model of T is recursively presentable.

PROOF. By the results of [2], it is clear that any countable model of T may be viewed as the prime model of some other Ml-categorical decidable theory T'.

The rank of a formula is defined in [1], in which it is shown (but not explicitly stated) that the function rg,(x)y' -* rank (cp(x)), qp(x) a formula of L(T'), is a recur- sive function with range the integers.

Given p(x) consistent with T', construct P,0 as follows: ro = {qp(x)}. Given ',, if +(x) is the sth formula in the variables x, and if both b(x) and _+(x) are con- sistent with rs, u T', then let

rs + = rs u {+(x)} if rank (+(x) A Ars) < rank (Arp), = rP u { -n(x)} otherwise.

Let rP be the type generated by Us rPs,. Clearly,, r, is just an effective version of the usual construction of a principal

type containing q.(x) (as in [7]). So, as in the usual construction, rP, is principal. E [For the sake of completeness I should mention here that A. Lachlan has

devised a more straight-forward proof of Corollary 4. It goes as follows: By [2] any countable model of T may be viewed as the prime model of some

Ml-categorical decidable theory T', and in fact we may assume that T' has a strongly minimal formula D(x) and that the prime model of T' has dimension 0. Given 4(x) a formula consistent with T, construct a recursive type rP containing p(x), with the property: For any formula b(y, x), if By(D(y) A y(y, x)) e rP, then, for some algebraic formula 0(y), By(D(y) A 0(y) A y(y, x)) E rP. This can be done since

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RECURSIVELY PRESENTABLE PROIE MODELS 309

the prime model of T' has dimension 0. The prime model over a realization of rF has only algebraic solutions of D(y), and thus this model has dimension 0. So Pr, is realized in the prime model, and therefore rP, is principal.]

?3. To conclude, a few comments might be in order about possible improve- ments of our theorem on the existence of a recursively presentable prime model. There are two ways one might attempt to strengthen this theorem, namely:

(1) If T is a complete decidable theory with a prime model, then the prime model has a recursive presentation.

(2) If T is a complete decidable theory with a recursively presentable prime model, then there is a recursive functions such that, given a formula p(x) consistent with T, f(rp(x)') is the Godel number of a complete formula @(x) ,and T F @(x) p(x).

As one might suspect, neither strengthening can hold; both have counter- examples. (There is even a counterexample to (2) which satisfies the hypothesis of Corollary 1.) But the counterexamples to (1) and (2) which I know of, though both of finite similarity type, are definitely not finitely axiomatizable. So, although it seems unlikely, (1) or (2) may just hold for finitely axiomatizable theories. It would also be interesting to find natural counterexamples to (1) and (2). In particular, it is not known if DCFo satisfies (2).

REFERENCES

[1] J. T. BALDWIN, aCr is finite for Xl-categorical T, Transactions of the American Mathe- matical Society, vol. 181 (1973), pp. 37-51.

[2] J. T. BALDWIN and A. H. LACHLAN, On strongly minimal sets, this JOURNAL, vol. 36 (1971), pp. 79-96.

[3] Y. L. ERSHOV, Numbered fields, Proceedings of the 3rd International Congress for Logic, Methodology and Philosophy of Science, North-Holland, Amsterdam, 1967.

[4] I. KAPLANSKY, An introduction to differential algebra, Hermann, Paris, 1957. [5] M. 0. RABIN, Computable algebra, Transactions of the American Mathematical Society,

vol. 95 (1960), pp. 341-360. [6] H. ROGERS, JR., Recursive functions and effective computability, McGraw-Hill, New York,

1967. [7] G. E. SACKS, Saturated model theory, Benjamin, New York, 1972. [8] C. WOOD, Prime model extensions for differential fields of characteristic p # 0, this

JOURNAL (to appear).

SUNY AT BUFFALO

AMHERST, NEW YORK 14226

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