This paper is a sequel to [1]. Our purpose is to exhibit certain basic relationships between (ordered dyadic) trees and regular nest structures. We introduce the notion of a tree being isomorphic to a nest structure, and we study some necessary and sufficient conditions for the existence of such isomorphisms. By virtue of some of our results, Kónig's lemma on infinite trees, and our fundamental lemma of [1] concerning infinite regular nest structures become intimately related — either yields an alternative proof of the other.

Our terminology and notation is the same as that of [1], of which this note is a sequel.

We wish to show that if we take the natural deduction system (N) described in [1], and delete the rules for the quantifiers, we obtain a complete system for propositional logic. [Of course we now construe “X”, “Y”, “Z” as syntactic variables ranging over formulas of propositional logic, rather than sentences of quantification theory.] Moreover the system serves as a neat decision procedure.

The following is a self-contained proof theoretic treatment of two of the principal axiom schemata of current intuitionistic analysis: the axiom of bar induction (Brouwer's bar theorem) and the axiom of continuity. The results are formulated in terms of formal derivability in elementary intuitionistic analysis H(§ 1), so the positive (i.e., derivability) results also apply to elementary classical analysis Z1 (Appendix 1). Both schemata contain the combination of quantifiers νfΛn, where f, g, … are intended to range over free choice sequences of suitable kinds of objects x, y, …; for example, natural numbers or sequences of natural numbers, and n, m, p, r, … over natural numbers (non-negative integers).

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.

Let Ω be an infinite cardinal larger than ω. By Lω we mean a language with infinitely long expressions having no individual constants, and such that the only predicates are < and =, and the length of ν or ∃ in a formula in Lω is smaller than Ω (cf. §2).

The word problem for groups was first formulated by M. Dehn [1], who gave a solution for the fundamental groups of a closed orientable surface of genus g ≧ 2. In the following years solutions were given, for example, for groups with one defining relator [2], free groups, free products of groups with a solvable word problem and, in certain cases, free products of groups with amalgamated subgroups [3], [4], [5]. During the period 1953–1957, it was shown independently by Novikov and Boone that the word problem for groups is recursively undecidable [6], [7]; granting Church's Thesis [8], their work implies that the word problem for groups is effectively undecidable.

In [5] it was shown that to complete Herbrand's argument for his Fundamental Theorem (see [6]) a weak analyzing function for certain applications of the rules of passage is needed. The following theorem describes such a function. (We use the terminology and notation of [5] except that, for each schema S and each p ≧ 1, we shall write D(S,p) rather than DSp for the domain of order p generated by S. In addition, we shall say that an element of D(S, p) is of order k, 1 ≦ k ≦ p, if it belongs to D(S, k) but not to D(S,k—1).)

The matrix defining Łukasiewicz's three-valued logic, constructed in 1920 and described at length in [1], is the following:

This matrix was axiomatized in 1931 by Wajsberg (see [6]), who showed that the following axioms together with the rules of substitution and modus ponens were sufficient:

It is well known that the modal calculus S4 has infinitely many non-equivalent formulae in a single proposition letter (in standard terminology, infinitely many modal functions), whilst S5 has only finitely many. However, the situation regarding the intermediate modal calculi S4.2, S4.3, and Prior's Diodorean tense-logic D does not seem to have been settled. In this note we show that each of these systems, together with a certain proper supersystem D* of D, has infinitely many modal functions.

This is in contrast with the fact that in the intermediate propositional logics KC and LC, which correspond under the McKinsey-Tarski translations to S4.2 and S4.3, there are only finitely many non-equivalent formulae in a single proposition letter.2

In this paper, a simple inductive characterization of the ordinal numbers is stated and developed. The characterization forms the basis for a set of axioms for ordinal theory and also for several short explicit definitions of the ordinals. The axioms are shown to be sufficient for ordinal theory, and, subject to suitable existence assumptions, each of the definitions is shown to imply the axioms.

The present results apply to the familiar von Neumann version of the ordinals, but the methods used are easily adapted to other versions.

This paper contains a rigorous treatment of the species of implication described in [8] and [9], where it was given the name of connexive implication. A brief historical survey will lay bare its roots in antiquity, and it will be shown that none of the well-known systems of propositional logic serves to formalize it.1 In this paper a new system of ‘connexive’ logic will be presented, the system being shown to be (a) consistent, (b) independent of two-valued logic, (c) Post-complete.

J. R. Shoenfield conjectured in a talk at the Berkeley Model Theory Symposium (1963) that, if b and d are non-zero recursively enumerable (r.e.) degrees such that b < d then there exists an r.e. degree c such that c < d and b U c = d. G. E. Sacks echoed this conjecture at the end of [3]. In this paper the conjecture is disproved. We construct r.e. degrees b, d such that 0 < b < d and such that for no r.e. degree c is it true that c < d and b U c = d. We are grateful to G. E. Sacks for suggesting this problem.

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.

Semantical systems that distinguish between logically true and factually true formulas are well-known from the work of Carnap. The present paper is concerned with extending the formalization of this distinction in two ways. First, we show how to construct syntactical (i.e., logistic) systems that correspond to semantical systems of the type just mentioned. Such a syntactical system for propositional logic is developed in section 3. Similar systems for first-order logic are sketched in section 5. Second, we show how to extend semantical systems that make the logical-factual distinction to languages containing modal connectives. Carnap's work on modal logic conspicuously lacks this feature. Section 4 contains such semantical systems for four well-known modal logics. It also contains a syntactical equivalent of one of these modal semantical systems.

An effective method for testing arguments consisting of singly quantified expressions has recourse to Parry's trapezoid symbolism.1 The procedure entails representing in that symbolism each premise of the argument and the negation of the conclusion. Variables are disregarded. After clearing away any negated quantifiers, the universal statements, including a denied existential conclusion, are first set down and their conjunction (KU) derived. Existential statements, including a denied universal conclusion, are then listed. The argument is valid if and only if some contradiction appears, whether as some premise empty of lines, as a KU empty of lines, or as some existential statement having no lines in common with the KU.