Why the Post-Kleene arithmetical hierarchy of degrees of (recursive) unsolvability was extended into the transfinite is not clear. Perhaps it was thought that if a hierarchy of sufficiently fine structure could be described that would include all sets of integers, some light might be thrown on the Continuum Hypothesis, and its truth or falsity possibly even ascertained. There is also some evidence in the 1955 papers of Kleene (cf. Kleene [2], [3], [4]) that it was once hoped that a theorem for the analytical hierarchy analogous to the result of Post and Kleene

Let E be a set of expressions representing real, single valued, partially defined functions of one real variable. E* will be the set of functions represented by expressions in E.

If A is an expression in E, A(x) is the function denoted by A.

It is assumed that E* contains the identity function and the rational numbers as constant functions and that E* is closed under addition, subtraction, multiplication and composition.

Retraceable sets were denned in Dekker and Myhill [5]. In that paper it was pointed out that if A is a retraceable set, then A is recursive in each of its infinite subsets, i.e. A is introreducible. We would like to know to what extent theorems on retraceable sets also hold for introreducible sets; however, we have made very little progress in this direction. On the other hand, “uniformly introreducible” sets (which will be defined in §2) are more tractable, and it turns out that some results on retraceable sets extend to uniformly introreducible sets while others fail.

The purpose of this paper is to present a very versatile family of logical systems of positional or topological logic. These systems—obtained by generalizing the existing systems of chronological logic—are to have a very general nature, capable of reflecting the characteristics of a wide range of logical systems, including not only chronological (also temporal or tense) logic, but also what we may call locative or place logic, and even a logic of “possible worlds”.

Generally speaking, it appears correct to say that in a formulation of first order logic in which a large number of connectives are taken as primitive (e.g. ∼, Λ, ∨, ⊃, ∀, ∃ as opposed to ∼, ⊃, ∀, or still more economically, ↓, ∀ proofs within the formal system tend to be smoother and more natural, but the metatheory tends to be that much more elaborate. In [1] we introduced a unifying ‘α, β, γ, δ” notation (which we also used in [2]–[7] and which we briefly review in this paper) which allows us to have our cake and eat it too.

The real importance of cut-free proofs is not the elimination of cuts per se, but rather that such proofs obey the subformula principle. In this paper we accomplish this latter objective in a different manner.

In the usual formulations of Gentzen systems, there is only one axiom scheme; all the other postulates are inference rules. By contrast, we consider here some Gentzen type axiom systems for propositional logic and Quantification Theory in which there is only one inference rule; all the other postulates are axiom schemes. This admits of an unusually elegant axiomatization of logic.

In the forthcoming monograph of Crossley [1] the question is raised whether the implication 2 + A = A ⇒ 1 + A = A is true for constructive order types. In this paper a partial answer to this question is given, in that a counterexample is constructed, using, however, not the definition of constructive order type (C.O.T.) given in [1], but an earlier one, namely that in Crossley [2]. The difference is that in [1] only orderings which can be imbedded in a standard dense r.e. ordering R by a partial recursive function are considered. The linear ordering constructed in this paper can be shown not to be such. The problem remains open in the case of orderings imbeddable in R.

Kripke's model structure for D2 is a quadruple (G, K, R, N) where N is the set of normal elements, i.e., the subset of K such that for every H in N, HRH. (See [1, p. 220, p. 211].) In fact, however, this structure provides a model for E2, as the work of Lemmon shows ([2, pp. 58–62]).

To show the inadequacy of Kripke's model structure, we show that the E2 thesis □B ⊃ B, not a thesis of D2, is valid under Kripke's modelling for D2. Suppose, for a reductio argument, that □ B ⊃ B is false for some valuation ϕ in some D2 model structure. Thenyϕ(□B ⊃ B, G) = F; so by truth-functional assignments

In the original formulations of Lewis's S1–S5 in [1], Feys's S1°, S2° in [2], and Sobociński's S3°, S4° in [3], a primitive rule is that of replacement for strict equivalents which it is enough to express formally as:

RI: where γ′ is like γ except for having β at one place where γ has α.

This paper is an appendix to Keisler's papers [3], [4]. We assume that the reader is fully acquainted with these papers and use their terminology without further explanation. One of the main tools in a number of proofs in [4] is the following result which is all but stated explicitly in [4].

The well-developed theory of recursion has been applied to certain algebraic structures with countable universes to obtain interesting and natural results on the question of effective procedures in such structures ([2], [9], [10], [13]). It has also been applied to structures related to the real numbers, via approximations; we cite [3] as but one example. We are interested, however, in the problem of what operations in an arbitrary algebraic structure, possibly quite large and unnatural, can reasonably be considered “effective” in terms of the underlying operations and relations of the structure.