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.

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)).

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.

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.

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]).

Friedman has posed (see [F, p. 117]) the following problem: “35. Define the set E of expressions by (i) Con is an expression; (ii) if A, B are expressions so are (~ A), (A&B), and Con(A). Each expression ϕ in E determines a sentence ϕ in PA [classical first-order arithmetic] by taking Con* to be “PA is consistent,” ( ~ A) * to be ~ (A*), (A&B)* to be A*&B*, and Con(A)* to be “PA + ‘A*’ is consistent.” The set of expressions ϕ ∈ E such that ϕ* is true is recursive.

The formalized second incompleteness theorem reads

In order to simplify notation, we will reformulate Friedman's problem slightly. Let Con be the usual sentence of PA expressing the consistency of PA, ~ A the negation of A, (A&B) the conjunction of A and B, etc., and Bew(A) the result of substituting the numeral for the Gödel number of A for the free variable in the usual provability predicate for PA. Let Con(A) = ~ Bew(~ A). (Con(A) is equivalent in PA to the usual sentence expressing the consistency of PA + A.) And let the class of F-sentences be the smallest class which contains Con and which also contains ~ A, (A&B) and Con(A) whenever it contains A and B. Since ⊦PA Bew(A) ↔ Bew(B) if ⊦PAA ↔ B, Friedman's problem is then the question whether the class of true F-sentences is recursive.

The answer is that it is recursive. To see why, we need a definition and a theorem.

Definition. An atom is a sentence Conn for some n ≥ 1, where Cont = Con and Conn + 1 = Con(Conn).