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).

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 .

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]).

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.

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.

An ω-set is a subset of the recursive ordinals whose complement with respect to the recursive ordinals is unbounded and has order type ω. This concept has proved fruitful in the study of sets in relation to metarecursion theory. We prove that the metadegrees of the sets coincide with those of the meta-r.e. ω-sets. We then show that, given any set, a metacomplete set can be found which is weakly metarecursive in it. It then follows that weak relative metarecursiveness is not a transitive relation on the sets, extending a result of G. Driscoll [2, Theorem 3.1]. Coincidentally, we discuss the notions of total and complete regularity. Finally, we solve Post's problem for the transitive closure of weak relative metarecursiveness. We recommend the reader look at pp. 324–328 of the fundamental article [6] of Kreisel and Sacks before proceeding. He will find there a proof of the following very basic fact: a subset of the integers is iff it is metarecursively enumerable (metafinite).

In [3] F. P. Ramsey proved as a theorem of Zermelo-Fraenkel set theory (ZF) with the Axiom of Choice (AC) the following result:

(1) Theorem. Let A be an infinite class. For each integer n and partition {X, Y} of the size n subsets of A, there exists an infinite subclass of A all of whose size n subsets are contained in only one of X or Y.

Let ƒ be a real number. It is well known [7] that the set of rational numbers which are less than ƒ is a recursive set if and only if ƒ is representable as the limit of a recursive, recursively convergent sequence of rational numbers. In this paper we replace the condition that the set of rational numbers less than ƒ is recursive by the condition that this set is at various points in the Kleene hierarchy, and we replace the recursive, recursively convergent limit by a variety of other recursive limiting processes.

In [1, §4], my ‘proof’ that GH1 has the finite model property is incorrect; there are considerable obscurities towards the end of §1, particularly on p. 33; and I should have exhibited the finite models for GH1. In §1 of this paper I expand the analysis of the sub-directly irreducible models for GH1 which I give in §1 of [1]. In §2 I give a correct proof that GH1 has the finite model property. In §3 I exhibit these finite models as models on certain ordered sets.

In this paper we show how the assumption of the generalized continuum hypothesis (GCH) can be removed or partially removed from proofs in Zermelo-Frankel set theory (ZF) of statements expressible in the simple theory of types. We assume the reader is familiar with the latter language, especially with the classification of formulas and sentences of that language into Σκη and Πκη form (cf. [1]) and with how that language can be relatively interpreted into the language of ZF.

In recent years much effort has gone into the study of languages which strengthen the classical first-order predicate calculus in various ways. This effort has been motivated by the desire to find a language which is

(I) strong enough to express interesting properties not expressible by the classical language, but

(II) still simple enough to yield interesting general results. Languages investigated include second-order logic, weak second-order logic, ω-logic, languages with generalized quantifiers, and infinitary logic.

In this note we sketch a decision procedure for S3.51 based on reduction to conjunctive normal form. Using the following theorem of S3.5: and its dual for M over a conjunction, any formula can be reduced by standard methods (as in S52) to a conjunction of disjunctions of the form where Í is (p ⊃ p), 0 is ∼(p ⊃ p) and α — λ are all PC-wffs (i.e. they contain no modal operators).

In [2] we constructed an infinite set of natural numbers containing no subset of higher (Turing) degree. Since it is well known that there are nonrecursive sets (e.g. sets of minimal degree) containing no nonrecursive subset of lower degree, it is natural to suppose that these arguments may be combined, but this is false. We prove that every infinite set must contain a nonrecursive subset of either higher or lower degree.

I am interested in extending modal calculi by adding propositional quantifiers, given by the rules for quantifier introduction: provided that p does not occur free in A.

In this paper we introduce a system S5U, formed by adding to the modal system S5 a new connective U, Up being read “probably”. A few theorems are derived in S5U, and the system is provided with a decision procedure. Several decidable extensions of S5U are discussed, and probability logic is related to plurality quantification.

In the axiomatic development of the logic of nonrelativistic quantum mechanics it is not difficult to set down certain plausible axioms which ensure that the quantum logic of propositions has the structure of an orthomodular poset. This can be done in a number of ways, for example, as in Gunson [2], Mackey [4], Piron [5], Varadarajan [7] and Zierler [8], and we summarise one of these ways in §2 below.

Rose has shown in [2] that the following axioms are sufficient, with modus ponens, for m-valued Łukasiewiczian implication.

If A and B are subsets of natural numbers we say that A is recursively equivalent to B (denoted A ≃ B) if there is a one-one partial recursive function which maps A onto B, and that A is recursively isomorphic to B (denoted A ≅ B) if there is a one-one total recursive function which maps A onto B and Ā (the complement of A) onto B#x00AF;.