In this paper we discuss an interpretation of intuitionistic type theory in many-sorted arithmetic with so-called conditional application. Via the formulas-as-types correspondence the arithmetical system in turn can be embedded in ML, resulting in a characterization of strong Σ-elimination by an axiom of conditional choice.

We use the spaces and n of complete types and of existential types to investigate various notions which appear in the theory of the algebraic structure of models.

Lewis, in the presentation of his calculus of strict implication, contends that this calculus accords with the usual meaning of “implies” such that “p strictly implies q” is synonymous with “q is deducible from p” (pp. 126–127, 235–262). It is the purpose of this paper to present, in 1, certain considerations in the light of which this statement does not hold, and in 2, a new implicative relation upon which a calculus of propositions can be based such that it will accord with the relation of deducibility and from which it is possible to derive the calculi of strict and material implications.

1. Strict implication regarded as synonymous with deducibility. In the presentation of his calculus of strict implication Lewis observes that despite inclusion of both intensional and extensional propositions in this calculus, the meaning of its elements, p, q, r, etc., “remain fixed, whether it is their extensional or their intensional relations which are in question” (p. 120). But if the meaning or content of the elements of this calculus is irrelevant to the logical truth which it contains, then, as Weiss has already remarked, the propositions of this calculus “are really extensionally treated, involving multiple values,” e.g., possibility, on the analogy of truth and falsity. The importance of this fact is that, in consequence, the relation of strict implication is subject to certain paradoxes, such as (19.74) “a proposition which is impossible strictly implies any proposition” or (19.75) “a proposition which is necessarily true is strictly implied by any proposition” (p. 174) or (19.84) “any two necessarily true propositions are strictly equivalent” (p. 176).

Assuming determinacy, the model L[T2] does not depend on the choice of T2.

We consider the infinite game where player ONE chooses terms of a strictly increasing sequence of first category subsets of a space and TWO chooses nowhere dense sets. If after ω innings TWO's nowhere dense sets cover ONE's first category sets, then TWO wins. We prove a theorem which implies for the real line: If TWO has a winning strategy which depends on the most recent n moves of ONE only, then TWO has a winning strategy depending on the most recent 3 moves of ONE (Corollary 3). Our results give some new information concerning Problem 1 of [S1] and clarifies some of the results in [B-J-S] and in [S1].

The purpose of this note is to demonstrate that a weak form of club guessing on ω1 implies the existence of an Aronszajn line with no Countryman suborders. An immediate consequence is that the existence of a five element basis for the uncountable linear orders does not follow from the forcing axiom for ω-proper forcings.

We make certain notational conventions. These are referred to as CL (Computational Logic). The relation of interchangeability is introduced as the basic connection between logical formulas. This approach lends perspicuity to the results of sentential calculus. With its technical devices CL is able to rephrase logical theorems in rather succinct manner. Our exposition tries to steer a middle course between informality and strict rigor.

It is felt that the method of CL offers advantages for the teaching of logic. Proofs are algorithmic and resemble those of elementary algebra. All inferences are reversible and practically non-tentative.

Sign is a primitive term of CL. Its denning property is a capacity for entering into binary combination with other signs, or with itself, according to this convention:

x and y are signs if, and only if, (x y) is a sign. Bracketing is looked upon as an operation on signs in terms of which other operations are definable. The practice of some authors in classifying brackets under the heading of symbols, seems to us questionable; for unlike symbols or signs brackets are never used to denote anything. Brackets enter into the composition of signs, not to denote a grouping, but rather to exhibit it, in the manner of a diagram.

An unending list of letters, with or without subscripts,

serve to denote arbitrary signs. An arbitrary sign may or may not have other signs as parts. The numeral “2” is used as a constant. Bracketing abbreviation is as follows:

and so on. Outermost brackets will ordinarily be omitted. This kind of bracketing may be termed left-associative. For convenience, these bracketing conventions are crystallized into a rule.

Following [1] we write {n} for the nth recursively enumerable (re) set; that is, {n} = {x|VyT(n, x, y)}. By a “pair (T, α)” we mean a consistent re extension T of Peano arithmetic P and an RE-formula α which numerates the non-logical axioms of T in P [4]. Given a pair (T, α) and a particular formula which binumerates the Kleene T predicate in P, there can be defined a primitive recursive function Nα such that and which has the additional property that {Nα(Nα(n))} = ø for all n.

The purpose of the present paper is to introduce a generalisation of the concept of productive set introduced by Post (2) and studied by Dekker (3) and others. Throughout the paper we shall use small Latin letters to denote both non-negative integers (referred to as numbers) and functions (both partial and total) from numbers to numbers. Sets of numbers will be denoted by small Greek letters and classes of such sets by capital Latin letters. ωη is the range of the ηth partial recursive function.