So-called “preservation theorems” relate the (possible) syntactic form of the axioms of a theory to certain closure conditions on its class of models. Such results are well known for the first-order predicate calculus, Lω, ω, and there are various expositions; e.g., Keisler [14], [15]. For the language , the first results were the theorems of Lopez-Escobar on sentences preserved under homomorphic images and of Malitz on formulas preserved under substructures. More recently, Feferman added a result on formulas preserved under (or persistent for) ∈-extensions. Some of these theorems will be considered in subsequent sections. A more thorough treatment may be found in Makkai [17]. The main new preservation result obtained here characterizes the sentences preserved under ω-unions. This notion and the statement of the theorem will be explained shortly.

It is a familiar experience in mathematical research that concepts which are equivalent in a special case diverge in general. In the case at hand, one must expect to consider different possible statements for , which generalize a known result for Lω, ω. Moreover, diverse proofs may yield the same result in the special case, not all of which can be extended to the general case. Again, since the compactness theorem fails for , one cannot expect to extend the arguments from Lω, ω which use this in an essential way.

In this paper we discuss subsystems of number theory based on restrictions on induction in terms of quantifiers, and we show that all the natural formulations of ‘n-quantifier induction’ are reducible to one of two (for n ≠ 0) nonequivalent normal forms: the axiom of induction restricted to (or, equivalently, ) formulae and the rule of induction restricted to formulae.

Let Z0 be classical elementary number theory with a symbol and defining equations for each Kalmar elementary function, and the rule of induction

restricted to quantifier-free formulae. Given the schema

let IAn be the restriction of IA to formulae of Z0 with ≤n nested quantifiers, IAn′ to formulae with ≤n nested quantifiers, disregarding bounded quantifiers, the restriction to formulae, the restriction to , formulae. IRn, IRn′, , are analogous.

Then, we show that, for every n, , , IAn, and IAn′, are all equivalent modulo Z0. The corresponding statement does not hold for IR. We show that, if n ≠ 0, is reducible to ; evidently IRn is reducible to . On the other hand, IRn′ is obviously equivalent to IAn′ [10, Lemma 2].

In [3] Kemeny made the following conjecture: Suppose *Z is a nonstandard model of the ring of integers Z. Let

and let F be the subgroup of those cosets ā which contain an element of infinite height in *Z. Kemeny then asked if the ring R = {a: ā ∈ F} is also a nonstandard model of Z. If so then Goldbach's conjecture is false because Kemeny also shows in [3] that Goldbach's conjecture fails in R.

The papers [1] and [5] by Gandy and Mendelson show that R is not a nonstandard model of Z but we give here a simpler proof based on Mendelson's paper. Suppose R is a nonstandard model of Z. Then each positive number in R is a sum of four squares. Choose a in R so that a is a positive element of R of infinite height in *Z. Then since a is infinite in *Z, a − 1 is positive. Thus , xi ∈ R for i = 1, …, 4. Now each xi must be of the form ai + ni, where ai has infinite height in *Z and ni, ∈ Z.

In [1] the notions of strongly minimal formula and algebraic closure were applied to the study of ℵ1-categorical theories. In this paper we study a particularly simple class of ℵ1-categorical theories. We characterize this class in terms of the analysis of the Stone space of models of T given by Morley [3].

We assume familiarity with [1] and [3], but for convenience we list the principal results and definitions from those papers which are used here. Our notation is the same as in [1] with the following exceptions.

We deal with a countable first order language L. We may extend the language L in several ways. If is an L-structure, there is a natural extension of L obtained by adjoining to L a constant a for each (the universe of ). For each sentence A(a1, …, an) ∈ L(A) we say satisfies A(a1, …, an) and write if in Shoenfield's notation If is an L-structure and X is a subset of , then L(X) is the language obtained by adjoining to L a name x for each is the natural expansion of to an L(X)-structure. A structure is an inessential expansion [4, p. 141] of an L-structure if for some .

A relational structure of cardinality ℵ0 is called homogeneous by Fraissé [1] if each isomorphism between finite substructures of can be extended to an automorphism of . In §1 of this paper it is shown that there are isomorphism types of such structures for the first order language L0 with a single (binary) relation symbol, answering a question raised by Fraissé. In fact, as is shown in §2, a family of nonisomorphic homogeneous structures for L0 can be constructed, each member of which satisfies the following conditions (where U is the homogeneous, ℵ0-universal graph, the structure of which is considered in [4]):

(i) The relation R of is asymmetric (R ∩ R−1 = ∅);

(ii) If A is the domain of and S is the symmetric relation R ∪ R−1, then (A, S) is isomorphic to U. That is, each may be regarded as the result of assigning a unique direction to each edge of the graph U.

Let T0 be the first order theory of all homogeneous structures for L0 which have cardinality ℵ0. In §3 (which can be read independently of §2) it is shown that T0 has complete extensions (in L0), each of which is ℵ0-categorical. Moreover, among the complete extensions of T0 are theories of arbitrary (preassigned) degree of unsolvability. In particular, there exists an undecidable, ℵ0-categorieal theory in L0, which answers a question raised by Grzegorczyk [2], [3].

It follows from Theorem 6 of [3] that there are ℵ0-categorical theories of partial orderings which have arbitrarily high degrees of unsolvability. This is in sharp contrast to the situation for linear orderings, which were the motivation for Fraissé's early work. Indeed, as is shown in [10], every ℵ0-categorical theory of a linear ordering is finitely axiomatizable. (W. Glassmire [12] has independently shown the existence of theories in L0 which are all ℵ0-categorical, and C. Ash [13] has independently shown that such theories exist with arbitrary degree of unsolvability.)

Ever since Spector's brilliant application of measure theory to recursion theory in 1958 [6] it has been realized that measure theory promotes sweeping simplifications in the theory of degrees. Results previously thought to be pathological were shown by Spector, and later Sacks [4], [5], to hold for almost all degrees (“almost all” in the sense of Lebesgue measure), often with much simpler proofs. Good examples of this phenomenon are Spector's demonstration that almost all pairs of sets are of incomparable degree (as an immediate consequence of Fubini's theorem) and Sacks' exquisitely simple deduction from this result that almost every degree is the join of two incomparable degrees (for if a random sequence is decomposed into its even and odd parts, the result is a random pair).

The present paper attempts to vindicate the feeling that almost all degrees behave in a simple manner by showing that if the quantifier in the theory of degrees with ′(jump), ∪ (join) and ∩ (meet) is taken to be (almost ∀a) instead of (∀a) then the theory is decidable. We are able to use ∩ because it will be shown that if t1, t2 are any terms built from degree variables a1, …, am with ′ and ∪ then t1 ∩ t2 exists for almost all a1, …, am. Thus the “almost all” theory presents a sharp contrast to the standard theory, where ∩ is not always defined (Kleene-Post [1]) and which is known to be undecidable (Lachlan [2]).

Let N, O and S denote the set of nonnegative integers, the graph of the constant 0 function and the graph of the successor function respectively. For sets P, Q, R ⊆ N2 operations of transposition, composition, and bracketing are defined as follows: P∪ = {〈x, y〉 ∣ 〈y, x〉 ∈ P}, PQ = {〈x, z〉 ∣ ∃y〈x, y〉 ∈ P & 〈y, z〉 ∈ Q}, and [P, Q, R] = ⋃n ∈ M(Pn Q Rn).

Theorem. The class of recursively enumerable subsets of N2 is the smallest class of sets with O and S as members and closed under transposition, composition, and bracketing.

This result is derived from a characterization by Julia Robinson of the class of general recursive functions of one variable in terms of function composition and “definition by general recursion.” A key step in the proof is to show that if a function F is defined by general recursion from functions A, M, P and R then F = [P∪, A∪M, R].

The above definitions of the transposition, composition, and bracketing operations on subsets of N2 can be generalized to subsets of X2 for an arbitrary set X. In this abstract setting it is possible to show that the bracket operation can be defined in terms of K, L, transposition, composition, intersection, and reflexive transitive closure where K: X → X and L: X → X are functions for decoding pairs.

The central result of this paper was proved in order to settle a problem arising from B. H. Neumann's paper [10].

In [10] Neumann proved that if a finitely generated group H is recursively absolutely presentable then H is embeddable in all nontrivial algebraically-closed groups. Harry Simmons [14] clarified this by showing that a finitely generated group H is recursively absolutely presentable if and only if H can be recursively presented with solvable word-problem. Therefore, if a finitely generated group H can be recursively presented with solvable word-problem then H is embeddable in all nontrivial algebraically-closed groups.

The problem arises of characterizing those finitely generated groups which are embeddable in all nontrivial algebraically-closed groups. In this paper we prove, by a forcing argument, that if a finitely generated group H is embeddable in all non-trivial algebraically-closed groups then H can be recursively presented with solvable word-problem. Thus Neumann's result is sharp.

Our results are obtained by the method of forcing in model-theory, as developed in [1], [12]. Our method of proof has nothing to do with group-theory. We prove general results, Theorems 1 and 2 below, about constructing generic structures without certain isomorphism-types of finitely generated substructures. The formulation of these results requires the notion of Turing degree. As an application of the central result we prove Theorem 3 which gives information about the number of countable K-generic structures.

We gratefully acknowledge many helpful conversations with Harry Simmons.

Let < and ≼ be two distinguished binary relation symbols. A structure is κ-like iff is a linear ordering of A, card(A) = κ, and every proper initial segment of A has cardinality < κ. A structure is α-ordered iff is a (reflexive) linear ordering of type α with field a subset of A. We define when a cardinal κ is α-inaccessible. (In this paper, inaccessible always means weakly inaccessible.) The 0-inaccessible cardinals are just the inaccessible cardinals; if α > 0, then κ is α-inaccessible iff for each β < α, each closed, cofinal subset of κ contains a β-inaccessible. (The (1 + α)-inaccessibles are just the ρα cardinals of Mahlo.) This paper is concerned with the proof of the following theorem.

Main Theorem. There is an elementary sentence σ with the property that whenever α is an ordinal and κ an infinite cardinal, then σ has an α-ordered κ-like model iff κ is not α-inaccessible.

This theorem gives some additional answers to a question of Mostowski about languages with generalized quantifiers. Fuhrken [1] showed that this question is equivalent to the following one: For which cardinals κ and λ is it true that if an elementary sentence has a κ-like model, then it has a λ-like model? It is actually this question to which the theorem refers. The theorem limits the possible pairs κ, λ of cardinals which answer the question. In fact, if the question is generalized so as to permit sentences from some more extensive language, then the theorem still limits the possible answers. For a more thorough introduction to this problem, the reader is referred to the aforementioned article of Fuhrken as well as Keisler [2] and Vaught [6].

The main result of this paper is the following transfer theorem: If T is an elementary theory which has a κ-like model where κ is hyperinaccessible of type ω, then T has a λ-like model for each λ > card(T). Helling [3] obtained the same conclusion under the stronger hypothesis that κ is weakly compact. Fuhrken conjectured in [1] that the same conclusion would result if κ were merely inaccessible. (He also showed there the connection with a problem about generalized quantifiers.) Thus, our theorem lies properly between Helling's theorem and Fuhrken's conjecture. In [9] it is shown that this theorem is actually the best possible. This theorem and the other results of this paper were announced by the authors in [10].

In §1 we prove as Theorem 1 a slightly stronger form of the above theorem. This theorem is generalized in §2 to Theorem 2 which concerns theories which permit the omitting of types. The methods used in §§1 and 2 are also applicable to problems regarding Hanf numbers as well as to two-cardinal problems.

We prove in this paper that the -system, an axiom system for set theory suggested for investigation by Takeuti in [2], is inconsistent. We also show that this system without the ω-rule is consistent if Zermelo-Fraenkel set theory with the axiom of choice and an axiom due to Reinhardt and Silver is consistent. The -system is an effort to strengthen Bernays-Gödel set theory by adding a reflection principle.

In addition to the standard notation of set theory, we write X″{x) to mean {y∣〈x, y〉 Є X}.

In [2] “recursive frames” were introduced as a means of extending relations R on the nonnegative integers to relations RΛ on the isols. In [1], this extension procedure was generalized by the introduction of “partial recursive frames”; the resulting extended relation on the isols was called RΛ. It was shown in [1] that the two extension procedures agree for recursive relations R, while RΛ ⊇ RΛ if R is . The case when R is , nonrecursive was left open. We show in this note that the extension procedures in fact agree for all relations R.

In the following, the notation and terminology is that of [1] and [2].

Theorem. If R ⊆ XκQ is a recursively enumerable (r.e.) relation, then RΛ = RΛ.

Proof. Clearly RΛ ⊆ RΛ, since every recursive frame is partial recursive. To prove RΛ ⊆ RΛ, we give a uniform effective method for expanding any partial recursiveR-frame F to a recursiveR-frame G such that F ⊆ G, so that So let F be a (nonempty) partial recursive R-frame, with CF(α) generated by . Let Rn denote the result of performing n steps in a fixed recursive enumeration of R. If g(α) is a partial recursive function, “g(α) is defined in n steps” means that in whichever coding of recursive computations is being used, a terminating computation for g with argument α has length ≤ n.

If Σ is the class of all fields and Σ* is the class of all algebraically closed fields, then it is well known that Σ* is characterized by the following properties:

(i) Σ* is the class of models of some first order theory K*.

(ii) If m1m2 are in Σ* and m1 ⊆ m2 then m1 ≺ m2 (m1 is an elementary substructure of m2, i.e. any first order sentence true in m1 is true in m2).

(iii) If m1 is in Σ then there is a structure m2 in Σ* such that m1 ⊆ m2.

If Σ is some other class of models of a first order theory K and a subclass Σ* of Σ exists satisfying (i)–(iii) then Σ* is uniquely determined and K* (which is unique up to logical equivalence) is called the model-companion of K. This notion is a generalization of the fundamental notion of model-completion introduced and extensively studied by A. Robinson [6], When the model-companion exists it provides the basis for a satisfactory treatment of the notion of an algebraically closed model of K.

Recently A. Robinson has developed a more general formulation of the notion of “algebraically closed” structures in Σ, which is applicable to any inductive elementary class Σ of structures (by elementary we always mean ECΔ). Condition (i) must be weakened to

(i′) Σ* is closed under elementary substructure (i.e. if m1 is in Σ* and m2 ≺ m1 then m2 is in Σ*).

There is a fairly simple algebraic property that distinguishes isols from cosimple isols.

Consider the Löwenheim-Skolem theorem in the form: If a theory in a countable first-order language has a model, then it has a countable model. As is well known, this theorem becomes false if one omits the hypothesis that the language be countable, for one then has the following trivial counterexample.

Example 1. Let the language have uncountably many constants, and let the theory say that they are unequal.

To motivate some of our future definitions and to introduce some notation, we present another, less trivial, counterexample.

Example 2. Let L0 be the language whose n-place predicate (resp. function) symbols are all the n-place predicates (resp. functions) on the set ω of natural numbers. Let be the standard model for L0; we use the usual notation Th() for its complete theory. Add to L0 a new constant e, and add to Th() an axiom schema saying that e is infinite. By the compactness theorem, the resulting theory T has models. However, none of its models are countable. Although this fact is well known, we sketch a proof in order to refer to it later.

By [5, p. 81], there is a family {Aα ∣ < α < c} of infinite subsets of ω, the intersection of any two of which is finite.

The method of inner models, used by Gödel to prove the (relative) consistency of the axiom of choice and the generalized continuum hypothesis [2], cannot be used to prove the (relative) consistency of any statement which contradicts the axiom of constructibility (V = L). A more precise statement of this well-known fact is:

(*)For any formula θ(x) of the language of ZF, there is an axiom α of the theory ZF + V ≠ L such that the relativization α(θ) is not a theorem of ZF.

On p. 108 of [1], Cohen gives a proof of (*) in ZF assuming the existence of a standard model of ZF, and he indicates that this assumption can be avoided. However, (*) is not a theorem of ZF (unless ZF is inconsistent), because (*) trivially implies the consistency of ZF. What assumptions are needed to prove (*)? We know that the existence of a standard model implies (*) which, in turn, implies the consistency of ZF. Is either implication reversible?

From our main result, it will follow that, if the converse of the first implication is provable in ZF, then ZF has no standard model, and if the converse of the second implication is provable in ZF, then so is the inconsistency of ZF. Thus, it is quite improbable that either converse is provable in ZF.

A set D of natural numbers is called Diophantine if it can be defined in the form

where P is a polynomial with integer coefficients. Recently, Ju. V. Matijasevič [2], [3] has shown that all recursively enumerable sets are Diophantine. From this, it follows that a bound for n may be given.

We use throughout the logical symbols ∧ (and), ∨ (or), → (if … then …), ↔ (if and only if), ⋀ (for every), and ⋁ (there exists); negation does not occur explicitly. The variables range over the natural numbers 0,1,2,3, …, except as otherwise noted.

It is the purpose of this paper to show that if we do not insist on prenex form, then every Diophantine set can be defined existentially by a formula in which not more than five existential quantifiers are nested. Besides existential quantifiers, only conjunctions are needed. By Matijasevič [2], [3], the representation extends to all recursively enumerable sets. Using this, we can find a bound for the number of conjuncts needed.

Davis [1] proved that every recursively enumerable set of natural numbers can be represented in the form

where P is a polynomial with integer coefficients. I showed in [5] that we can take λ = 4. (A minor error is corrected in an Appendix to this paper.) By the methods of the present paper, we can again obtain this result, and indeed in a stronger form, with the universal quantifier replaced by a conjunction.

Suppose T is a first order intuitionistic theory (more precisely, a theory of Heyting's predicate calculus, e.g., abelian groups, one unary function, dense linear order, etc.) presented to us by a set of axioms (denoted also by) T.

Question. Is T decidable?

One knows that if the classical counterpart of T (i.e., take the same axioms but with the classical predicate calculus as the underlying logic) is not decidable, then T cannot be decidable. The problem remains for theories whose classical counterpart is decidable. In [8], sufficient conditions for undecidability were given, and several intuitionistic theories such as abelian groups and unary functions (both with decidable equality) were shown to be undecidable. In this note we show decidability results (see Theorems 1 and 2 below), and compare these results with the undecidability results previously obtained. The method we use is the reduction-method, described fully in [12] and widely applied in [3], which is applied here roughly as follows:

Let T be a given theory of Heyting's predicate calculus. We know that Heyting's predicate calculus is complete for the Kripke-model type of semantics. We choose a class M of Kripke models for which T is complete, i.e., all axioms of T are valid in any model of the class and whenever φ is not a theorem of T, φ is false in some model of M.

We prove, among other things, that the number mentioned above cannot be shown to exist without using some instance of the axiom of replacement.

In the March, 1971 issue of this Journal [1] a paper of ours was published purporting to prove the hereditary undecidability of the first-order theory of finite planar graphs. The proof presented there contains an error which is unfortunately “unfixable” by the methods of that paper. The theorem however is true and we demonstrate here a generalization to finite cubic (exactly three edges at each vertex) planar graphs. The method involves coding the halting problem for a Turing machine into the theory of these graphs by considering special printouts of computations. Let us first consider a discussion of the aforementioned mistake and see what can be learned from it.

By a graph we will mean a nonempty set V of points together with a set I of unordered pairs of points of V. Each point i = {u, v) ∈ I is an edge of A graph is called finite if ∣V∣ is finite. A graph is said to be planar iff it can be embedded in the plane (i.e., drawn in the plane so that no two edges intersect).

In the earlier paper the method of proof was a semantic embedding of certain binary relations into finite planar graphs. The essential idea was, for a given relation, to let the vertices of the graph interpret the field of the relation and the edges represent the related pairs. This method works for arbitrary graphs, but in the planar case has two main difficulties.