The uniform halting problem (UH) can be stated as follows:

Give a decision procedure which for any given Turing machine (TM) will decide whether or not it has an immortal instantaneous description (ID).

An ID is called immortal if it has no terminal successor. As it is generally the case in the literature (see e.g. Minsky [4, p. 118]) we assume that in an ID the tape must be blank except for some finite number of squares. If we remove this restriction the UH becomes the immortality problem (IP).

In this paper, I present a modified and extended version of combinatory logic. Schönfinkel originated the study of combinatory logic (in [2]), but its development is primarily due to H. B. Curry. In the present paper, I will make use of both the symbolism (with some modification) and the results of Curry, as found in [1].

What is novel about my version of combinatory logic is a kind of combinators which I call discriminators. These combinators discriminate between different symbols, and yield values which are determined by the symbols which are their arguments. If the normal combinators investigated by Curry are considered as functions taking (combinations of) symbols as arguments and yielding symbols as values, these combinators can rearrange and cancel their arguments or introduce new symbols.

According to Sobociński's recollection, the consistency of Mereology was proved by Leśniewski by means of an appropriate interpretation within the framework of the theory of real numbers. His proof was never published, but in a recent paper R. E. Clay has succeeded in reconstructing a version of it.1 Clay's result amounts to showing that if Lesniewski's Ontology expanded by the addition of the axioms for the real numbers is consistent then Mereology is consistent. Without casting any doubts on the validity of the proof one can hardly fail to note that here we have a case where the consistency of a conceptually simple theory is made to depend on the consistency of a theory which from the point of view of intuition is far from being obvious.

A formula α of modal LPC is a de re modality iff it contains a free individual variable within the scope of a modal operator; otherwise α is de dicto.1 We shall say that de re modalities are eliminable in a system S of modal LPC iff there is a transformation R on wffs of S such that R(α) is de dicto and ├sα ≡ R(α). We shall show that where S is LPC + S5 + Pr2 then de re modalities are not eliminable in S.

In the literature of symbolic logic there are many examples of systems having free variables ranging over truth-values, individuals, or predicates. But, though many such systems are equipped with universal and existential quantifiers (and though many other quantifiers, e.g., for exactly one, are thereby definable), the problem of free variables corresponding to such constants has been neglected.

Let be a relational system whereby D is a nonempty set and P1 is an m1-ary relation on D. With we associate the (weak) monadic second-order theory consisting of the first-order predicate calculus with individual variables ranging over D; monadic predicate variables ranging over (finite) subsets of D; monadic predicate quantifiers; and constants corresponding to P1, P2, …. We will often use ambiguously to mean also the set of true sentences of .

One of the basic results in the theory of λ-conversion is the Church-Rosser Theorem, which says that, using certain rules for conversion and reduction of λ-formulae, any two interconvertible formulae can both be reduced to one formula. (I will not explain this in detail, as λ-conversion is described fully in Church's [2], where the Church-Rosser Theorem is Theorem 7 XXVII; see also Chapter 4 of Curry and Feys' [3].) The first part of the present paper contains an abstract form of this theorem.

Let G be a separated (Hausdorff) topological group and let *G be an enlargement of G (see [8]). Thus, *G (i) possesses the same formal properties as G in the sense explained in [8], and (ii) every set of subsets {Aν } of G with the finite intersection property—i.e. such that every nonempty finite subset of {Aν } has a nonempty intersection—satisfies ∩*Aν ≠ ø, where the *Aν are the extensions of the Aν in *G, respectively.

A partially ordered (p.o.) set (P, ≼) is represented by the recursively enumerable (r.e.) class C if (P, ≼) is order isomorphic to (C, ⊆), that is to the p.o. set consisting of C ordered by the inclusion relation. (P, ≼.) is representable if it is represented by some r.e. class. N will denote the set of natural numbers.

Cylindric algebras were introduced by Alfred Tarski about 1952 to provide an algebraic analysis of (first-order) predicate logic. With each cylindric algebra one can, in fact, associate a certain, in general infinitary, predicate logic; for locally finite cylindric algebras of infinite dimension the associated predicate logics are finitary. As with Boolean algebras and sentential logic, the algebraic counterpart of completeness is representability. Tarski proved the fundamental result that every locally finite cylindric algebra of infinite dimension is representable.

An elementary theory T in a language L is (strongly) finitely inseparable if the set of logically valid sentences of L and the set of T-finitely refutable sentences are recursively inseparable. In §1 we establish a sufficient condition for the elementary theory of a class of BA's with operators to be finitely inseparable. This is done using the methods developed independently by M. Rabin and D. Scott (see [6]) on the one hand and by Ershov on the other (see [2]).

Let ε and Λ denote the sets of all nonnegative integers and isols respectively. Let σ be any subset of ε. Following [6], we shall denote the recursive equivalence type of a by Req(σ) and write σ≃τ when Req(σ) = Req(τ).

The notion of polyadic algebra was introduced by Halmos to reflect algebraically the predicate logic without equality. Later Halmos enriched the study with the introduction of the notion of equality. These algebras are very closely related to the cylindric algebras of Tarski. The notion of diagonal free cylindric algebra predates that of cylindric algebra and is also due to Tarski. The theory of diagonal free algebras forms an important fragment of the theories of polyadic and cylindric algebras.

Shoenfield [8] has shown that a hierarchy for the functions recursive in a type-2 object can be set up whenever E2 (the type-2 object that introduces numerical quantification) is recursive in that type-2 object. With a restriction that we will discuss in the next paragraph, Moschovakis [4, pp. 254–259] has solved the analogous problem for type-3 objects. His method seems to generalize for any type-n object, where n ≥ 2. We will solve this same problem of finding hierarchies based on type-n objects by a different method. Instead of using ordinal notations for indexing stages of hierarchies, as do Shoenfield and Moschovakis, we will define notation-independent stages.

In this paper we consider some axiomatic systems of set theory related to the system NF (New Foundations) of Quine. In particular we discuss the possible relations of cardinality between a finite set x and its subset class SC(x) = {y | y ∩ x} and also between x and its unit set class USC(x) = {{y} | y ε x}. Specker [5] has shown that in NF the cardinal of a finite set x can never be the same as the cardinal of SC(x).

This paper is a sequel to the joint publication of Scott and Krauss [8] in which the first aspects of a mathematical theory are developed which might be called “First Order Probability Logic”. No attempt will be made to present this additional material in a self-contained form. We will use the same notation and terminology as introduced and explained in Scott and Krauss [8], and we will frequently refer to the theorems stated and proved in the preceding paper.

Let L be a first order predicate language with two sorts of variables and a single dyadic predicate letter whose first place is to be filled by variables of one sort and whose second place is to be filled by variables of the other sort. In answer to a question of M. H. Löb we show that there is no decision procedure for determining whether or not a sentence of L is universally valid.

This paper compares the subsystem of analysis based on the Π11-comprehension axiom (Π11-CA) with classical Bar Induction (BI). An extension of Bar Induction using infinitary formulae is also considered, which was suggested to us by G. Kreisel (oral communication).

To describe these formal systems, we first describe the language, L, appropriate for their formalization. L contains the language of ENT (1st order arithmetic) plus function variables ƒi, gi, hi, etc. Number-theoretic terms are built up from +, × , ′ , 0, and application.

In this paper a new property of theories, called effective retractability is introduced and used to obtain a characterization for the degrees of subtheories of arithmetic and set theory. By theory we understand theory in standard formalization as defined by Tarski [10]. The word degree refers to the Kleene-Post notion of degree of recursive unsolvability [2]. By the degree of a theory we mean, of course, the degree associated with its decision problem via Gödel numbering.