Countable Models With Standard Part

COUNTABLE MODELS WITH STANDARD PART M. MORLEyI Cornell University, Ithaca, N. Y., USA

One may consider co-logic as the study of those models where one specified part is isomorphic to the natural numbers, (w, <). In MORLEY (1970) we generalized this to '21-logic where (co, <) is replaced by an arbitrary structure '21. In this paper we restrict ourselves entirely to countable models. The most interesting result is Theorem 3 which characterizes countable models of ~o-categorical PC classes. 1. Preliminaries

The reader is assumed familiar with the notion of a predicate calculus with equality and the rudiments of model theory about such. In this paper all languages and all models will be countable. If L is a language and '21 = (A, Ri ) (i E l) is a set with relations R, corresponding to the relation symbols of L then '21 is an L-structure. Th('21) is the set of sentences satisfied by '21. If s is a sequence of length n in '21 then the type of s is the set of formulas with free variables among v o, ... , V n _ I satisfied by s. If '21 is an L-structure and U a unary relation on '21 then '21 relativized to U('211 U) is the subsystem with universe U. If '21 = (A, R;) (i E l) and J c I then (A, R i ) (i E J) is 121 reduced to J('2111). If T is a sentence in L, U a unary relation in Land J a subset of the relations in L then the class { there is an an such that (ao, ... , an> has type p. If Th('21) = Th(Q3) and both are saturated then '21 is isomorphic to Q3 (see VAUGHT, 1961).

The

1

This work was partially supported by NSF Grants GP-22719 and GP-21869.

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If one enlarges L by adding a new constant for each element of '2l, then the set of sentences in the enlarged language valid in 'l1 is the diagram of '2l, D('2l). 2. The syntax model For each L-structure '2l we describe a new structure, Syn('2l) which is intended to represent formally the formulas satisfied in '21. Syn('2l) will be composed of three disjoint parts M, N, P such that: (i) Syn('2l)I M and properly restricted is isomorphic to A; (ii) P is the set of finite sequences of elements of M; (iii) N is the natural numbers with the usual arithmetic; (iv) there is a binary relation lh(x, n) which holds for x E P and n EN if the length of x is n; (v) there is a ternary relation y(x, i, y) which h~lds if x E P, i EN and y E M is the ith element of x; (vi) there is a binary relation Satsf(y, n) which holds if yEP satisfies formula number n under some fixed Godel numbering of the formulas of L. For a fixed language L, the class {Syn('l1); '2l is L-structure} is not an elementary class since there is no set of axioms which would guarantee that N would be isomorphic to the natural numbers. However, it is well known that there is a finite set of axioms whose models all have an initial segment isomorphic to the natural numbers with their usual arithmetic and such that the recursive sets are numeralwise definable therein (see, e.g., KLEENE, 1952). We call this initial segment of N the standard numbers. By adding the usual inductive definition of the satisfaction predicate to this theory we may obtain a theory we shall call Syn(L) which has the following properties: (i) Syn('2l) is a model of Syn(L) for every L-structure '2l; (ii) if <;S is a model of Syn(L) then <;S I M and restricted to the relations of L is an L-structure, say <;S'; and further (iii) the subsystem of <;S with universe M together with the standard part of N and those elements of P with standard length is isomorphic to

Syn(<;s'). Notice that when L has only a finite number of relation symbols then Syn(L) is finitely axiomatizible. Since the recursive sets are numeralwise

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definable we obtain the result (MAKKAI, 1964) that a PCI! set with a recursively enumerable set of axioms is a PC set. 3. '21-models Suppose T is a set of sentences in L, U a unary relation in Land J a subset of the relations of L. Then a model
if <:.8 is saturated.

if and only

PROOF: Suppose <:.8 is the unique countable model in some PCI! class. That is, there is a consistent theory T such that for every countable model
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Let p be such a type. We enlarge the language by adding an n-ary function symbol! corresponding to p. Let T p be all the universal closure of all formulas of the form:

3vnC ..... C(vo, ... ,vn_1,!(VO ,

•••

,vn_1))

where C is an element of p relativized to V. Then Th(CS) relativized to V together with T p for each p consistent with Th(CS) are the axioms for the desired PC" class. We now turn to the question of which I.2l belong to the greatest equivalence class. As mentioned above these are exactly those 21 such that I.2lDef(w, -c). THEOREM 2. I.2l Def (w, <) if and only if I.2l is not saturated. PROOF: If I.2l is saturated then by Theorem 1 it is in the last equivalence class and hence does not define (w, <). Suppose I.2l is not saturated. Then there is a sequence s of length n in I.2l and an n+ l-type p which is consistent with the type of s but not realized by any sequence extending s. Let eo, 1?1, ... be an enumeration of p. We enlarge Syn(l.2l) by adding a new binary relation H == {(x, m); x E P, lh(x) = n + 1 and em is the least element of p not satisfied by x}. Call this enlargement of Syn(I.2l), CS. I assert Th(CS) defines (w, <) from 1.2l. That is, if G: is a model of Th(CS), G: IM is isomorphic to 'Zl, then G: IN is the standard natural numbers. To show this, notice that it follows from Th(t:.8) that if a subset of N definable using parameters from G: has an upper bound then it has a least upper bound. Using the sequence s of length n in I.2l we define: X

== {n EN; 3x(x extends s & H(x, n))}.

All the elements of X are standard but, since p is consistent with the type of s, X is cofinal in the standard natural numbers. Thus if G: IN has a nonstandard number it would have a least upper bound for the standard numbers. But this certainly contradicts Th(CS). Thus the hierarchy determined by the relation I.2l Deft:.8 has only two equivalence classes corresponding to saturated and unsaturated models. 4. Finite definability

Suppose we were to restrict our definition of I.2lDefcs by requiring that the theory T be finitely axiomatizible. The corresponding hierarchy 2No equivalence classes and no greatest one. However, it will have a least

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equivalence class corresponding to those structures which are the only countable model in some PC class. We wish to characterize the \21 which appear in the least class. By Theorem 1 we know \21 must be saturated. From MAKKAI (1964) we know that the theory of every PC class is recursively enumerable so Th(\21) must be recursive. The sufficient conditions are stronger. THEOREM 3. (i) '21 is the unique countable model in a recursively axiomatized if \21 is saturated and every finite sequence occurring in \21 has a recursive type. (ii) If \21 has only a finite number of relations then we may replace PC d by PC in the above.

FCd class if and only

PROOF: Part (ii) follows from (i) by the remark at the end of Section 1. Suppose T is a recursively axiomatized theory such that for every countable model 6: of T, 6:I U and suitably restricted is isomorphic to \21. Then \21 is saturated by Theorem 1. That every n-type occurring in \21 is recursive is a special case of the following proposition (by letting Q be formulas relativized to U). PROPOSITION. Suppose T is a consistent theory, Q a set of formulas with free variables among Va, ••• , V n _ 1 and :E a maximal consistent subset of Q. If Q is recursive in T but :E is not then there is a model of T in which no sequence satisfies all the formulas of :E. The proof of this proposition is a straightforward generalization of the proof of the usual omitting type theorem (see VAUGHT, 1961) and will not be given. Conversely, suppose \21 is saturated and every n-type occurring in \21 is recursive. We wish to find a recursively axiomatized PC d class such that \21 is its unique countable model. Our axioms will be in three parts. The first part is Syn(L) where L is the language of \21. The second part is Th(\21) relativized to M. The third part will say \21 is 'recursively saturated'. Since all recursive predicates are numeralwise definable in Syn(L) we may find a formula defining a binary function T(e, n) where for e fixed the values of T enumerate the eth recursively enumerable set. If s E P has length m and a E M let sea be the element of P of length m + 1 formed by adding a to s. Then the third part of our axioms is the following one: 'Vs'Ve3x'Vk'Vy3i (Satsf(s*x, T(e, k») v(i

~k

& iSatsf(s*y, T(e, i»))).

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Then every model of T relativized to M is saturated and hence isomorphic to 21. It is interesting that Theorem 3 requires only that the types be recursive not that there be a uniform recursive way of giving all the types. It is true but not trivially so that there are countable saturated 21 with Th(21) recursive but some of the types occurring in 21 nonrecursive. I want to express my appreciation to Robert Soare for several helpful conversations concerning this fact. The preceding theorem has applications to countable models of ~I­ categorical theories. It has been shown (MARSH, 1966) that a countable model of such a theory will be saturated if it contains an infinite set of indiscernibles. From this it is easy to find a PC" class with axioms recursive in the original theory whose only countable model is the saturated model of the original theory. COROLLARY. If T is recursive and ~ I-categorical then every type consistent with T is recursive. After finding the above corollary I learned of an earlier and more general result of HARRINGTON (1971). His result is: THEOREM. If T is recursive and ~ I-categorical then every countable model of T has a recursive presentation.

References lIARJuNGTON, L. A., 1971, Structures with recursive presentations, Notices, American Mathematical Society, vol. 18, p. 826 KumNE, S. C., 1952, Introduction to Metamathematics (van Nostrand, New York) MAKKAI, M., 1964, On PC" classes in the theory of models, Magyar Tud. Akad. Kututo Int. Kozl., vol. 9, pp. 159-194 MARSH, W., 1966, On w!"categorical but not os-categorical theories, Doctoral dissertation, Dartmouth College MoRLEY, M. D., 1970, The Lilwenheim-Skolem theorem for models with standard part, Symposia Mathematica, vol. 5, pp. 43-52 VAUGHT, R. L., 1961, Denumerable models of complete theories, in: Infinitistic Methods, Proceedings of the Symposium on the Foundations of Mathematics, Warsaw, 1959 (pergamon Press, New York), pp. 303-321