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

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

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

Let be arelational system whereby D is a nonempty set andP1 is anm1-ary relation on D. With we associate the (weak) monadic second-order theory consisting of the first-order predicate calculuswith individual variables ranging over D; monadic predicatevariables ranging over (finite) subsets of D; monadicpredicate quantifiers; and constants corresponding to P1,P2, …. We will often use ambiguously to mean also the set of true sentencesof .

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

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

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

An ω-set is a subset of the recursive ordinals whosecomplement with respect to the recursive ordinals is unbounded and has ordertype ω. This concept has proved fruitful in the study of sets in relation to metarecursion theory. We provethat the metadegrees of the setscoincide with those of the meta-r.e. ω-sets. We then show that, given any set, ametacomplete set can befound which is weakly metarecursive in it. It then follows that weakrelative metarecursiveness is not a transitive relation on the sets,extending a result of G. Driscoll [2, Theorem 3.1]. Coincidentally, wediscuss the notions of total and complete regularity. Finally, we solvePost's problem for the transitive closure of weak relativemetarecursiveness. We recommend the reader look at pp. 324–328 of thefundamental article [6] of Kreisel and Sacks before proceeding. He will findthere a proof of the following very basic fact: a subset of the integers is iff it ismetarecursively 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 setof rational numbers which are less than ƒ is a recursiveset if and only if ƒ is representable as the limit of arecursive, recursively convergent sequence of rational numbers. In thispaper we replace the condition that the set of rational numbers less than ƒ is recursive by the condition that this set is atvarious points in the Kleene hierarchy, and we replace the recursive,recursively convergent limit by a variety of other recursive limitingprocesses.

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 thispaper I expand the analysis of the sub-directly irreducible models for GH1which I give in §1 of [1]. In §2 I give a correct proof that GH1 has thefinite model property. In §3 I exhibit these finite models as models oncertain ordered sets.

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

In recent years much effort has gone into the study of languages whichstrengthen 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 theclassical language, but

(II) still simple enough to yield interesting general results. Languagesinvestigated 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 onreduction to conjunctive normal form. Using the following theorem of S3.5: and itsdual for M over a conjunction, any formula can be reducedby standard methods (as in S52) to a conjunction of disjunctionsof 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 nosubset of higher (Turing) degree. Since it is well knownthat there are nonrecursive sets (e.g. sets of minimal degree) containing nononrecursive subset of lower degree, it is natural tosuppose that these arguments may be combined, but this is false. We provethat every infinite set must contain a nonrecursive subset of either higheror lower degree.

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

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

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

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

If A and B are subsets of natural numberswe say that A is recursively equivalent to B (denoted A ≃ B) if there isa 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 whichmaps A onto B and Ā (thecomplement of A) onto B#x00AF;.