Some syntactically simple fragments of intuitionistic logic possess considerable expressive power compared with their classical counterparts.

In particular, we consider in this paper intuitionistic second order propositional logic (ISPL) a formalisation of which may be obtained by adding to the intuitionistic propositional calculus quantifiers binding propositional variables together with the usual quantifier rules and the axiom scheme (Ex), where is a formula not containing x.

The main purpose of this paper is to show that the classical first order predicate calculus with identity can be (isomorphically) embedded in ISPL.

It turns out an immediate consequence of this that the classical first order predicate calculus with identity can also be embedded in the fragment (PLA) of the intuitionistic first order predicate calculus whose only logical symbols are → and (.) (universal quantifier) and the only nonlogical symbol (apart from individual variables and parentheses) a single monadic predicate letter.

Another consequence is that the classical first order predicate calculus can be embedded in the theory of Heyting algebras.

The undecidability of the formal systems under consideration evidently follows immediately from the present results.

We shall indicate how the methods employed may be extended to show also that the intuitionistic first order predicate calculus with identity can be embedded in both ISPL and PLA.

For the purpose of the present paper it will be convenient to use the following formalisation (S) of ISPL based on [3], rather than the one given above.

In this paper we show that any countable upper semilattice with zero can be embedded as an initial segment of the degrees of unsolvability. This provides a characterization of the order types of the countable initial segments of the degrees since any such initial segment is trivially an initial segment of a countable upper semilattice. Let a segment S of the degrees be a set of degrees such that, if a, b Є S and a < c < b, then c Є S. One may readily observe that our result also characterizes the order types of all countable segments of the degrees of unsolvability.

Mostowski [11] shows that if a structure has a decidable theory, then its weak direct power has one as well; his proof however never produces decision procedures which are elementary recursive.

Some very general results are obtained here about the nature of the weak direct power of a structure, which in most cases lead to elementary recursive decision procedures for weak direct powers of structures which themselves have elementary recursive procedures.

In particular, it is shown that 〈N*, +〉, the weak direct power of 〈N, +〉, can be decided in space

for some constant c. As corollaries, the same upper bound is obtained for the theory of the structure 〈N +, ·〉 of positive integers under multiplication, and for the theory of finite abelian groups. Fischer and Rabin [7] have shown that the theory of 〈N,* +〉 requires time

even on nondeterministic Turing machines.

Let α be an admissible ordinal. In this paper we study the structure of the upper semilattice of α-recursively enumerable degrees. Various results about the structure which are of fundamental importance had been obtained during the past two years (Sacks-Simpson [7], Lerman [4], Shore [9]). In particular, the method of finite priority argument of Friedberg and Muchnik was successfully generalized in [7] to an α-finite priority argument to give a solution of Post's problem for all admissible ordinals. We refer the reader to [7] for background material, and we also follow closely the notations used there.

Whereas [7] and [4] study priority arguments in which the number of injuries inflicted on a proper initial segment of requirements can be effectively bounded (Lemma 2.3 of [7]), we tackle here priority arguments in which no such bounds exist. To this end, we focus our attention on the fine structure of Lα , much in the fashion of Jensen [2], and show that we can still use a priority argument on an indexing set of requirements just short enough to give us the necessary bounds we seek.

In [2] we described an arithmetic theory of constructions (ATC) and showed that first-order intuitionistic arithmetic (HA) could be interpreted in it. In [3] we went on to show that the interpretation of HA in ATC is faithful. The purpose of the present paper is to apply these ideas to intuitionistic arithmetic in all finite types. Tait has shown [6] that a conservative extension of HA is obtained by adding the Gödel functionals with intuitionistic logic and intensional identity in all finite types. Below we show that this extension remains conservative on the addition of certain axioms of choice which are evident on the intended interpretation of the intuitionistic logical connectives. This theorem (Corollary 6.2 below) was first obtained by a more complicated argument in our dissertation [1]. Some of its implications are discussed in Goodman and Myhill [4].

We assume that the reader is familiar with [2] and [3].

Modal reduction principles (MRPs) are modal formulas of the following form: Mp → Np, where M, N are (possibly empty) sequences of modal operators (i.e. □ or ◊). The notation M, N will be used to abbreviate such an MRP. We study a certain semantic correspondence between modal formulas and relational properties and obtain two main results. (1) On transitive semantic structures every MRP corresponds to a first-order relational property. (2) For the general case a syntactic criterion exists for distinguishing modal formulas with corresponding first-order properties from the others.

Several results in the theory of recursion in higher types indicate that the effect of a higher type functional on the lower types does not reflect the high type, i.e. the same effect could be obtained by functionals of relatively low type. The two main results here are:

Plus -1 - Theorem (G. Sacks [6] for k = 1, [7] for k > 1). Let H be a normal functional of type ≥ k + 1. Then there exists a normal functional F of type k + 1 such that k-sc(F) = k-sc(H), i.e. the same subsets of tp(k − 1) are recursive in F and H.

Plus - 2 - Theorem (L. Harrington [1]). Let H be a normal functional of type ≥ k + 2. Then there exists a normal functional F of type k +2 such that k-en(H) = k-en(F), i.e. the same subsets of tp(k − 1) are semirecursive in F and H.

The results in this paper also indicate that higher types cannot have too much influence on lower types. The key is the Skolem-Löwenheim theorem. Among the results we mention:

(1) Let n < m. A ⊆ tp(n) × tp(m) be Kleene-semicomputable. Let x ∈ B ⇔ ∀y∈tp(m), ⟨x, y⟩ ∈ A. Then B is . This result may be relativized to a functional of type n + 1.

(2) Let k0 be the type-k-functional that is constant zero. Let F be a functional of type < k. Then, for i ≤ k −2, i-sc(F ,k0) = i-sc(F), i-en(F, k0) = ∀tp(i)(i-en(F)).

The question of continuity of functions defined on Baire space NN or on the reals has been of great interest to constructivists. We have little hope of answering the question without a fundamental philosophical analysis of the notion of constructive function. However, one can ask whether the continuity of functions can be derived in known intuitionistic formal systems, and answer these questions by mathematical means.

The constructivist does not generally wish to restrict himself to reals (or members of NN ) given by a recursive law; therefore the most natural formal systems have variables for reals, sequences and functions. The axioms of these systems turn out to be compatible with the assumption that every real or sequence is given by a recursive law, and every function by an effective operation. An underivability result, therefore, is only strengthened if this assumption is taken as an axiom; and once this is done, we may as well work with (indices of) effective operations, within arithmetic.

By KLS is meant the assertion that each effective operation on Baire space NN is continuous. In [B1] it is shown that this statement cannot be proved in various intuitionistic formal systems. The continuity of functions on the reals has been of even greater interest to constructivists than the continuity of functions on NN ; by KLS(R) is meant the assertion that each effective operation from the real numbers R to R is continuous. It is the purpose of the present note to show that KLS(R) is also underivable in intuitionistic formal systems.

In [7] it is shown that if Σ is a type omitted in the structure = ω, +, ·, < and complete with respect to Th() then Σ is omitted in models of Th() of all infinite powers. The proof given there extends readily to other models of P. In this paper the result is extended to models of ZFC. For pre-tidy models of ZFC, the proof is a straightforward combination of the methods in [7] and in Keisler and Morley ([9], [6]). For other models, the proof involves forcing. In particular, it uses Solovay and Cohen's original forcing proof that GB is a conservative extension of ZFC (see [2, p. 105] and [5, p. 77]).

The method of proof used for pre-tidy models of set theory can be used to obtain an alternate proof of the result for This new proof yields more information. First of all, a condition is obtained which resembles the hypothesis of the “Omitting Types” theorem, and which is sufficient for a theory T to have a model omitting a type Σ and containing an infinite set of indiscernibles. The proof that this condition is sufficient is essentially contained in Morley's proof [9] that the Hanf number for omitting types is so the condition will be called Morley's condition.

If T is a pre-tidy theory, Morley's condition guarantees that T will have models omitting Σ in all infinite powers.

Theorem. Assume that there exists a standard model of ZFC + V = L. Then there is a model of ZFC in which the partial ordering of the degrees of constructibility of reals is isomorphic with a given finite lattice.

The proof of the theorem uses forcing. The definition of the forcing conditions and the proofs of some of the lemmas are connected with Lerman's paper on initial segments of the upper semilattice of the Turing degrees [2]. As an auxiliary notion we shall introduce the notion of a sequential representation of a lattice, which slightly differs from Lerman's representation.

Let K be a given finite lattice. Assume that the universe of K is an integer l. Let ≤ K be the ordering in K. A sequential representation of K is a sequence Ui ⊆ Ui+1 of finite subsets of ω i such that the following holds:

(1) For any s, s′ Є Ui , i Є ω, k, m Є l, k ≤ K m & s(m) = s′(m) → s(k) = s′(k).

(2) For any s Є Ui , i Є ω, s(0) = 0 where 0 is the least element of K.

(3) For any s, s′ Є i Є ω, k,j Є l, if k y Kj = m and s(k) = s′(k) & s(j) = s′(j) → s(m) = s′(m), where v K denotes the join in K.

This paper is concerned with the proof theoretic development of certain infinite languages. These languages contain the usual infinite conjunctions and disjunctions, but in addition to homogeneous quantifiers such as ∀x0x1x2 … and ∃y0y1y2 …, we shall investigate particular subclasses of the dependent quantifiers described by Henkin [1]. The dependent quantifiers of Henkin can be thought of as partially ordered quantifiers defined by a function from one set to the power set of another set. This function assigns to each existentially bound variable, the set of universally bound variables on which it depends.

The natural extension of Gentzen's first order predicate calculus to an infinite language with homogeneous quantifiers results in a system that is both valid and complete, and in which a cut elimination theorem can be proved [2]. The problem then arises of devising, if possible, a logical system dealing with general dependent quantifiers that is valid and complete. In this paper a system is presented that is valid and complete for an infinite language with homogeneous quantifiers and dependent quantifiers that are anti-well-ordered, for example, … ∀x2∃y2∀x1∃y1∀x0∃y0.

Certain notational conventions will be employed in this paper. Greek letters will be used for ordinal numbers. The ordinal ω is the set of all natural numbers, and 2ω is the set of all ω -sequences of elements of 2 = {0,1}. The power set of S is denoted by P(S). μα[A(α)] stands for the smallest ordinal α such that A (α) holds.

The main result of this paper is the fact that for T a complete extension of either P or ZF + V = L, with no new symbols, there is a type which is omitted in a model of T of power ℵ1 but which is realized in all models of higher power. Therefore, the “Hanf number” for omitting types over T is greater than ℵ2.

The present section contains some basic definitions and a discussion of related results. §1 discusses generic relations on models of P and ZFC, and describes a special forcing technique due to Addison. In §2, this technique is used to obtain the main result.

The use of forcing is not essential. Those who do not wish to read a forcing argument may skip §1. At the point in §2 where forcing is used, a proof without forcing is sketched. This proof was obtained only after a careful study of the forcing argument. The forcing argument is given because it is simple, and because the technique of forcing makes intuitively clear why this and other similar results should be true.

All theories in the paper are assumed to be countable and to have infinite models. Let T be a complete theory in a language L. The Hanf number for omitting types over T, denoted by H(T), is the first infinite cardinal κ such that for all L-types, Σ, if T has models omitting Σ in all infinite powers less than κ, then T has models omitting Σ in all infinite powers.

In this article, unpublished methods of Solovay and Kunen are applied to describe conditions under which an uncountable regular κ can satisfy weak (κ, λ)-compactness (see 1.1(3) below), yet lie below 2μ for some μ < κ. The argumentation is in informal ZFC, and general set-theoretic notation is standard. The lower-case Greek letters κ, λ, μ, ν are reserved for cardinals in the sense of some transitive or inner model of a reasonable set theory, φ, Ψ, θ, are (arithmetizations of) formulas in some extension of the first-order language of set theory, and other lower-case Greek letters except Є are metavariables for arbitrary ordinals. If M is transitive, M ⊨ φ abbreviates 〈M, Є 〉 ⊨ φ. [2], [3], [9] and [1] provide more information about large-cardinal theory for those who wish it.

1.1. Definitions. (1) κ is inaccessible iff κ is regular and ℵκ = κ; strongly inaccessible iff κ is regular and ℶκ = κ, i.e., λ < κ for all λ < κ; weakly inaccessible iff κ is inaccessible but not strongly inaccessible.

(2) L κλ is the infinitary language with conjunctions and disjunctions of length < κ and quantification over sequences of length < λ, and PL κ the prepositional language with κ letters and conjunctions and disjunctions of length < κ.

We provide a semantics for Zermelo-Fraenkel set theory. The semantics was essentially given by W. Ackermann [1], and the proof of completeness follows from a lemma of Levy [4]. Furthermore, a close relationship between Ackermann's set theory and Zermelo-Fraenkel set theory has been established by Levy [4] and Reinhardt [6]. Thus, any interest in this paper must consist solely in the explicit formulation of the completeness result and the consequential evidence of the fundamental character of Ackermann's principle and the ad hoc character of the full law of excluded middle and the strong transitivity of Ackermann's unary relation

In this paper we treat the ramified type theory of Russell [6], afterwards adopted by Whitehead and Russell in Principia mathematica [12], so that we may compare Russell's resolution of the semantical antinomies by ramified type theory with the now widely accepted resolution of them by the method of Tarski in [7], [8], [9].

To avoid impredicativity the essential restriction is that quantification over any domain (type) must not be allowed to add new members to the domain, as it is held that adding new members changes the meaning of quantification over the domain in such a way that a vicious circle results. As Whitehead and Russell point out, there is no one particular form of the doctrine of types that is indispensable to accomplishing this restriction, and they have themselves offered two different versions of the ramified hierarchy in the first edition of Principia (see Preface, p. vii). The version in §§58–59 of the writer's [1], which will be followed in this paper, is still slightly different.

To distinguish Russellian types or types in the sense of the ramified hierarchy from types in the sense of the simple theory of types, let us call the former r-types.

There is an r-type i to which the individual variables belong. If β1, β2, …, βm are any given r-types, m ≧ 0, there is an r-type (β1, β2, …, βm)/n to which there belong m-ary functional variables of level n, n ≧ 1. The r-type (α1, α2, …, αm)/k is said to be directly lower than the r-type (β1, β2, …, βm)/n if α1 = β1, α2 = β2, …, αm = βm, k < n.

Though equations are among the simplest sentences available in a first order language, many of the most familiar notions from algebra can be expressed by sets of equations. It is the task of this paper to expose techniques and theorems that can be used to establish that many collections of finite sets of equations characterized by common algebraic or logical properties fail to be recursive. The following theorem is typical.

Theorem. In a language provided with an operation symbol of rank at least two, the collection of finite irredundant sets of equations is not recursive.

Theorems of this kind are part of a pattern of research into decision problems in equational logic. This pattern finds its origins in the works of Markov [8] and Post [20] and in Tarski's development of the theory of relation algebras; see Chin [1], Chin and Tarski [2], and Tarski [23]. The papers of Mal′cev [7] and Perkins [16] are more directly connected with the present paper, which includes generalization of much of Perkins' work as well as extensions of a theorem of D. Smith [22]. V. L. Murskii [14] contains some of the results below discovered independently. Not all known results concerning undecidable properties of finite sets of equations seem to be susceptible to the methods presented here. R. McKenzie, for example, shows in [9] that for a language with an operation symbol of rank at least two, the collection of finite sets of equations with nontrivial finite models is not recursive. D. Pigozzi has extended and elaborated the techniques of this paper in [17], [18], and [19] to obtain new results concerning undecidable properties, particularly those of algebraic character.

A constructive proof is given which shows that every nonrecursive r.e. many-one degree is represented by the family of decision problems for partial implicational propositional calculi whose well-formed formulas contain at most two distinct variable symbols.

The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p-adically closed fields, from the model-theoretical point of view. Cohen [5], from a standpoint less model-theoretic, also contributed much to this analogy.

In this paper we shall point out a feature of all the above treatments which obscures one important resemblance between real and p-adic fields. We shall outline a new treatment of the p-adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and p-adically closed fields.

We want to describe the definable subsets of p-adically closed fields. Tarski [9] in his pioneering work described the first-order definable subsets of real closed fields. Namely, if K is a real-closed field and X is a subset of K first-order definable on K using parameters from K then X is a finite union of nonoverlapping intervals (open, closed, half-open, empty or all of K). In particular, if X is infinite, X has nonempty interior.

Now, there is an analogous question for p-adically closed fields. If K is p-adically closed, what are the definable subsets of K? To the best of our knowledge, this question has not been answered until now.

What is the difference between the two cases? Tarski's analysis rests on elimination of quantifiers for real closed fields. Elimination of quantifiers for p-adically closed fields has been achieved [3], but only when we take a cross-section π as part of our basic data. The problem is that in the presence of π it becomes very difficult to figure out what sort of set is definable by a quantifier free formula. We shall see later that use of the cross-section increases the class of definable sets.

In [B1] and [B2] we have shown that the classical theorems on the continuity of effective operations are not derivable in various constructive formal systems. The purpose of this paper is to establish that these systems are closed under the corresponding rules of inference, which have the form: If e can be proved to be (the index of) an effective operation, then e can be proved to be continuous. In fact we establish closure under apparently stronger rules of the form, if e can be proved to be an effective operation, then e* can be proved to be (the index of) a modulus of continuity for e, where e* is canonically obtained from e.

We again deal with two kinds of effective operations: (i) partial operations on partial arguments, and (ii) total operations on total arguments. The continuity theorem for operations of kind (i) is due to Myhill-Shepherdson [MS] and is called MS; for kind (ii) is due to Kreisel, Lacombe, and Shoenfield [KLS] and is called KLS. We also deal with effective operations on the (recursive) real numbers, which are of kind (ii). In [B1], [B2] it is shown that KLS and MS are derivable in in-tuitionistic arithmetic HA from Markov's principle MP (this follows from examination of the classical proofs); that they do not imply MP in HA; and that they are not derivable in HA. (These theorems remain true when HA is replaced by various stronger systems.) The derived rules to be established in this paper therefore represent nontrivial closure properties of the systems to which they apply.

I first heard the name of Kurt Gödel when, as a graduate student at Princeton in the fall of 1931, I attended a colloquium at which John von Neumann was the speaker, von Neumann could have spoken on work of his own; but instead he gave an exposition of Gödel's results of formally undecidable propositions [1931].

Today I shall begin with Gödel's paper [1930] on The completeness of the axioms of the functional calculus of logic, or of what we now often call “the first-order predicate calculus”, using “predicate” as synonymous with “propositional function”.

Alonzo Church wrote ([1944, p. 62] and [1956, pp. 288–289]), “the first explicit formulation of the functional calculus of first order as an independent logistic system is perhaps in the first edition of Hilbert and Ackermann's Grundzüge der theoretischen Logik (1928).” Clearly, this formalism is not complete in the sense that each closed formula or its negation is provable. (A closed formula, or sentence, is a formula without free occurrences of variables.) But Hilbert and Ackermann observe, “Whether the system of axioms is complete at least in the sense that all the logical formulas which are correct for each domain of individuals can actually be derived from them is still an unsolved question.” [1928, p. 68].

This question Gödel answered in the affirmative in his Ph.D. thesis (Vienna, 1930), of which the paper under discussion is a rewritten version.

I shall not describe Gödel's proof. Perhaps no theorem in modern logic has been proved more often than Gödel's completeness theorem for the first-order predicate calculus. It stands at the focus of a complex of fundamental theorems, which different scholars have approached from various directions (e.g. Kleene [1967, Chapter VI]).