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.

In this paper we show (cf. Theorem 22) that in a language L* with equality, whose relation symbols denote arbitrary relations over ω* (=rational integers) and whose function symbols denote (= ∃∀ definable in the arithmetic hierarchy) functions over ω*, (i) a positive sentence is true in Λ* (= isolic integers) iff some Horn reduct is true in ω* with Skolem functions. We also show (cf Theorem 20) that (ii) a universally quantified sentence is true in Λ* iff some Horn reduct is true in Λ*.The latter result is nontrivial because our relations are arbitrary and our functions are In order to obtain (i) it was necessary to generalize the frame extensions of [7]. This is done in §2. Our extension procedure agrees with that of [7] for recursive relations (cf. Theorem 12), and is certainly more general for − relations. What happens in the case is still open. In §3 we develop the basic properties of our extension so that in §4 we can prove a metatheorem (cf. Theorems 8 and 10) about Λ (=isols), in a language L with equality whose relation symbols denote arbitrary relations over ω (=nonnegative integers) and whose function symbols denote almost R↑ combinatorial functions. In Theorem 11 this is generalized to infinitary universal sentences. In §5 generic isols are introduced. These are used (cf. Theorems 16–19) to generalize and simplify the “fundamental lemma” of [8]. The basic induction is patterned after Lemma 4.1 of [8], but is stronger in that any sufficiently generic assignment attainable from a frame yields Skolem functions. Finally in §6 these results are applied to Λ*, yielding the titled result (i) of our paper. Immediately following Theorem 15 there is a discussion which attempts to justify the way we extend relations to Λ.

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.

In [10, §0, E), 5)] Shelah states using the proofs of 7.9 and 6.9 in [9] it is possible to prove that if a countable first-order theory T is ℵ0-stable (totally transcendental) and not ℵ1-categorical, then it has at least ∣1 + α∣ models of power ℵα.

In this note we will give a new proof of this theorem using the work of Baldwin and Lachlan [1]. Our original proof used the generalized continuum hypothesis (GCH). We are indebted to G. E. Sacks for suggesting that the notions of ℵ0-stability and ℵ1-categoricity are absolute, and that consequently our use of GCH was eliminable [8]. Routine results from model theory may be found, e.g. in [2].

Proof (with GCH). In the proof of Theorem 3 of [1] Baldwin and Lachlin show of power ℵα such that there is a countable definable subset in . Let B0 be such a subset. Say . We will give by transfinite induction an elementary chain of models of T of power ℵα such that B[i1 … in] has power ℵβ and such that every infinite definable subset of has power ≥ℵβ. This clearly suffices.

We investigate extensions of Heyting's predicate calculus (HPC). We relate geometric properties of the trees of Kripke models (see [2]) with schemas of HPC and thus obtain completeness theorems for several intermediate logics defined by schemas. Our main results are:

(a) ∼(∀x ∼ ∼ϕ(x) Λ ∼∀xϕ(x)) is characterized by all Kripke models with trees T with the property that every point is below an endpoint. (From this we shall deduce Glivenko type theorems for this extension.)

(b) The fragment of HPC without ∨ and ∃ is complete for all Kripke models with constant domains.

We assume familiarity with Kripke [2]. Our notation is different from his since we want to stress properties of the trees. A Kripke model will be denoted by (Aα, ≤ 0), α ∈ T where (T, ≤, 0) is the tree with the least member 0 ∈ T and Aα, α ∈ T, is the model standing at the node α. The truth value at α of a formula ϕ(a1 … an) under the indicated assignment at α is denoted by [ϕ(a1 … an)]α.

A Kripke model is said to be of constant domains if all the models Aα, α ∈ T, have the same domain.

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.

Let {Wi} be a standard enumeration of all recursively enumerable (r.e.) sets, and for any class A of r.e. sets, let θA denote the index set of A = {n ∣ Wn ∈ A}. (Clearly, .) In [1], the index sets of nonempty finite classes of finite sets were classified under one-one reducibility into an increasing sequence {Ym}, 0 ≤ m < ∞. In this paper we examine further properties of this sequence within the partial ordering of one-one degrees of index sets. The main results are as follows: (1) For each m, Ym < Ym + 1 and < Ym + 1; (2) Ym is incomparable to ; (3) Ym + 1 and ; are immediate successors (among index sets) of Ym and m; (4) the pair (Ym + 1, ) is a “least upper bound” for the pair (Ym, ) in the sense that any successor of both Ym and is ≥ Ym + 1or; (5) the pair (Ym, ) is a “greatest lower bound” for the pair (Ym + 1, ) in the sense that any predecessor of both Ym + 1 and is ≤ Ym or . Since and all Ym are in the bounded truth-table degree of K, this yields some local information about the one-one degrees of index sets which are “at the bottom” in the one-one ordering of index sets.

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

Although we believe the results reported below to have direct philosophical import, we shall for the most part confine our remarks to the realm of mathematics. The reader is referred to [4] for a philosophically oriented discussion, comprehensible to mathematicians, of tense logic.

The “minimal” tense logic T0 is the system having connectives ∼, →, F (“at some future time”), and P (“at some past time”); the following axioms:

(where G and H abbreviate ∼F∼ and ∼P∼ respectively); and the following rules:

(8) from α and α → β, infer β,

(9) from α, infer any substitution instance of α,

(10) from α, infer Gα,

(11) from α, infer Hα.

A tense logic is a system T whose language is that of T0 and whose axioms and rules include (1)–(11). The axioms and rules of T other than (1)–(11) are called proper axioms and rules.

We shall investigate three systems of semantics for tense logics, i.e. three notions of structure and three relations ⊧ which stand between structures and formulas. One reads ⊧ α as “α is valid in .” A structure is a model of a tense logic T if every formula provable in T is valid in . A semantics is adequate for T if the set of models of T in the semantics is characteristic for T, i.e. if whenever T ∀ α then there is a model of T in the semantics such that ∀ α. Two structures and , possibly from different semantics, are called equivalent ( ∼ ) if exactly the same formulas are valid in as in .

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.