In this paper we present two theorems concerning relationships between degrees of unsolvability of recursively enumerable sets and their complexity properties.

The first theorem asserts that there are nonspeedable recursively enumerable sets in every recursively enumerable Turing degree. This theorem disproves the conjecture that all Turing complete sets are speedable, which arose from the fact that a rather inclusive subclass of the Turing complete sets, namely, the subcreative sets, consists solely of effectively speedable sets [2]. Furthermore, the natural construction to produce a nonspeedable set seems to lower the degree of the resulting set.

The second theorem says that every speedable set has jump strictly above the jump of the recursive sets. This theorem is an expected one in view of the fact that all sets which are known to be speedable jump to the double jump of the recursive sets [4].

After this paper was written, R. Soare [8] found a very useful characterization of the speedable sets which greatly facilitated the proofs of the results presented here. In addition his characterization implies that an r.e. degree a contains a speed-able set iff a′ > 0′.

We determine precisely those locally finite varieties of unary algebras of finite type which, when augmented by a ternary discriminator, generate a variety with a decidable theory.

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of λ-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent).

A standard enumeration of the recursively enumerable (r.e.) sets is an acceptable numbering {Wn}n∈N of the r.e. sets in the sense of Rogers [5, p. 41], together with a 1:1 recursive function f with range In his quest for nonrecursive incomplete r.e. sets Post [4] constructed a hypersimple set Hf, relative to a fixed but unspecified standard enumeration f. Although it was later shown that hyper-simplicity does not guarantee incompleteness, the ironic possibility remained that Post's own particular hypersimple set might be incomplete. We settle the question by proving that H, may be either complete or incomplete depending upon which standard enumeration f is used. In contrast, D. A. Martin has shown [3] that Post's simple set S [4, p. 298] is complete for any standard enumeration. Furthermore, what most modern recursion theorists would regard as the “natural” construction of a hypersimple set (which we give in §1) is also complete for any standard enumeration.

There are two conclusions to be drawn from these results. First, they substantiate the often repeated remark among recursion theorists that Post's hypersimple set construction is a precursor of priority constructions because priorities play a strong role, and because there is a great deal of “restraint” which tends to keep elements out of the set. Secondly, the results warn recursion theorists that more properties than might have been supposed depend upon which standard enumeration is chosen at the beginning of the construction of some r.e. set.

The notion of a variable restricted in range to some set of values, or to values having some property, is widespread in logic and mathematics. To avoid continual explicit mention of such a restriction the common device is to resort to a typographically different set of letters for such variables. But a notation of this kind is clearly inadequate when a general discussion, including more than one kind of restriction, is under consideration. For example, two restrictions may have some inner connection, as in the case of mutually exclusive restrictions, and this the usual symbolism doesn't depict; or, one may wish to study the dependence of such restrictions upon one or more parameters as in the case of bounded quantifiers of recursive function theory; or, one can even conceive of restrictions which may in turn depend upon restricted variables. In each of these cases the currently used notation is deficient.

If x is an individual-variable and Q a formula of the predicate calculus, then for the idea ‘x such that Q’ the notation ‘vxQ’ is proposed as one having the requisite degree of notational multiplicity – the ‘vx’ acting as a binding operator for free occurrences of x in Q. The expression ‘vxQ’ is referred to as a restricted-variable and it may occupy positions in formulas appropriate to individual-variables, in argument places as well as in quantifiers. In § 2 a first-level predicate calculus, called ‘‘, is presented in which the restricted-variable is an integral part.

One of this author's favorite theorems has long been the Zero-One law discovered independently by Glebskii et al. [12] and Ron Fagin [10]. Let A be any first-order property of graphs and let µn(A) be the proportion of labelled graphs on n vertices for which A holds. Then

This result has inspired much work by logicians, generally in the direction of showing (1) for some powerful languages. Thus it is known [5] that (1) holds when A is a sentence in fixed point logic and it is known [13] that (1) does not always hold when A is a sentence in second-order monadic logic. Here, however, we explore recent work in a totally different direction. Let G(n, p) denote the random graph on n vertices with edge probability p. (In §2 we define the random structures we will deal with.) A property A is an event in the probability space and Pr[G(n, p) ╞ A] is well defined. When p = 1/2, each labelled graph on n vertices has equal weight so that (1) may be rewritten

Fagin's proof actually gives that (2) holds for any constant 0 < p < 1.

In this paper we consider classes of finite structures where we have good control over the sizes of the definable sets. The motivating example is the class of finite fields: it was shown in [1] that for any formula in the language of rings, there are finitely many pairs (d, μ) ∈ ω × Q>0 so that in any finite field F and for any ā ∈ Fm the size |ø(Fn,ā)| is “approximately” μ|F|d. Essentially this is a generalisation of the classical Lang-Weil estimates from the category of varieties to that of the first-order-definable sets.

A Liouville function is an analytic function H: ℂ → ℂ with a Taylor series such the an's form a “very fast growing” sequence of integers. In this paper we exhibit the complete first-order theory of the complex field expanded with H.

We answer some problems set by Priest in [11] and [12], in particular refuting Priest's Conjecture that all LP-models of Th(ℕ) essentially arise via congruence relations on classical models of Th(ℕ). We also show that the analogue of Priest's Conjecture for IΔ0 + Exp implies the existence of truth definitions for intervals [0, a] ⊂eM ⊨ IΔ0 + Exp in any cut [0, a] ⊂eK ⊆eM closed under successor and multiplication.

Halldén, in [1], has recently pointed out that it is highly undesirable, in a system of sentential calculus, for there to exist two formulas α and β such that: (i) α and β contain no variable in common; (ii) neither α nor β is provable; (iii) α ∨ β is provable. We shall call a system unreasonable (in the sense of Halldén) if there exists a pair of formulas α and β having properties (i), (ii), and (iii). Halldén shows (in [1]) that the Lewis systems S1 and S3 are unreasonable in this sense; and that the same is true of any system which is between S1 and S3, as well as of every system which is stronger than S3 but weaker than both S4 and S7. In the present note we shall show that this defect does not occur in S4, nor in S5, nor in any “quasi-normal” extension of S5; we give an example, on the other hand, of an unreasonable system which lies between S4 and S5.

When we speak, in what follows, of a system of modal logic, we shall mean a system having the same class of well-formed formulas as have the various Lewis calculi. Thus the well-formed formulas of a system of modal logic, when written in unabbreviated form, are just those formulas which can be built up from sentential variables by use of the binary connective ‘·’ (conjunction sign), and the two unary connectives ‘˜’ (negation sign) and ‘◇’ (possibility sign). We shall, however, also make use of some of the defined signs of Lewis.

We investigate the effect of a variant of Matet forcing on ultrafilters in the ground model and give a characterization of those P–points that survive such forcing, answering a question left open by Blass [4]. We investigate the question of when this variant of Matet forcing can be used to diagonalize small filters without destroying P–points in the ground model. We also deal with the question of generic existence of stable ordered-union ultrafilters.

Une conséquence de la classification des théories complètes d'algèbres de Boole par Tarski [5] est que la théorie élémentaire d'une algèbre de Boole A est déterminée par le type d'isomorphisme du treillis de ses idéaux définissables et, pour chacun de ces idéaux, par le nombre d'atomes du quotient de A par cet idéal lorsque ce nombre est fini. Une remarque analogue peut être faite à propos des cas particuliers d'algèbres de Boole munies d'un idéal distingué étudiés par Ershov [1] et par Jurie et Touraille [3]; dans to us ces cas, c'est la simplicité des treillis possibles qui permet la classification des théories complètes. Le résultat principal de cet article est que, dans le cas général d'une algèbre de Boole munie d'une famille quelconque d'idéaux distingués, la théorie d'un modèle peut encore être caractérisée grâce à une structure algébrique sur l'ensemble de ses idéaux définissables. Il s'agit d'une structure d'algèbre de Heyting munie d'une opération unaire sa définie par sa(K) = {a: a/K est sans atome}, et cette structure s'avère être engendrée par les idéaux distingués du modèle. La méthode utilisée est l'élimination directe des quantificateurs, par réductions successives des formules. Elle nécessite des propriétés algébriques et topologiques qui sont données aux §§1 et 2: on introduit au §1 la notion d'algèbre de Heyting étoilée, c'est-à-dire d'algèbre de Heyting munie d'une opération unaire * vérifiant des égalités qui permettent de rendre compte, d'une certaine façon, de la dérivation de Cantor-Bendixon; le §2 est consacré à des propriétés topologiques qui, dans le cas de l'espace de Stone d'une algèbre de Boole A, permettent d'éclaircir les relations possibles entre les atomes des quotients de A par des idéaux différents.

Graph normal form, GNF, [1], was used in [2, 3] for analyzing paradoxes in propositional discourses, with the semantics—equivalent to the classical one—defined by kernels of digraphs. The paper presents infinitary, resolution-based reasoning with GNF theories, which is refutationally complete for the classical semantics. Used for direct (not refutational) deduction it is not explosive and allows to identify in an inconsistent discourse, a maximal consistent subdiscourse with its classical consequences. Semikernels, generalizing kernels, provide the semantic interpretation.

We investigate two weak fragments of the double negation shift schema, which are motivated, respectively, from Spector’s consistency proof of ACA0 and from the negative translation of RCA0, as well as double negated variants of logical principles. Their interrelations over both intuitionistic arithmetic and analysis are completely solved.

We develop the theory of Krull dimension for S4-algebras and Heyting algebras. This leads to the concept of modal Krull dimension for topological spaces. We compare modal Krull dimension to other well-known dimension functions, and show that it can detect differences between topological spaces that Krull dimension is unable to detect. We prove that for a T1-space to have a finite modal Krull dimension can be described by an appropriate generalization of the well-known concept of a nodec space. This, in turn, can be described by modal formulas zemn which generalize the well-known Zeman formula zem. We show that the modal logic S4.Zn := S4 + zemn is the basic modal logic of T1-spaces of modal Krull dimension ≤ n, and we construct a countable dense-in-itself ω-resolvable Tychonoff space Zn of modal Krull dimension n such that S4.Zn is complete with respect to Zn. This yields a version of the McKinsey-Tarski theorem for S4.Zn. We also show that no logic in the interval [S4n+1S4.Zn) is complete with respect to any class of T1-spaces.

In [16], Peterzil and Steinhorn proved that if a group G definable in an o-minimal structure is not definably compact, then G contains a definable torsion-free subgroup of dimension 1. We prove here a p-adic analogue of the Peterzil–Steinhorn theorem, in the special case of abelian groups. Let G be an abelian group definable in a p-adically closed field M. If G is not definably compact then there is a definable subgroup H of dimension 1 which is not definably compact. In a future paper we will generalize this to non-abelian G.

Cantor’s first set theory paper (1874) establishes the uncountability of ${\mathbb R}$ . We study this most basic mathematical fact formulated in the language of higher-order arithmetic. In particular, we investigate the logical and computational properties of ${\mathsf {NIN}}$ (resp. ${\mathsf {NBI}}$ ), i.e., the third-order statement there is no injection resp. bijection from  $[0,1]$ to ${\mathbb N}$ . Working in Kohlenbach’s higher-order Reverse Mathematics, we show that ${\mathsf {NIN}}$ and ${\mathsf {NBI}}$ are hard to prove in terms of (conventional) comprehension axioms, while many basic theorems, like Arzelà’s convergence theorem for the Riemann integral (1885), are shown to imply ${\mathsf {NIN}}$ and/or ${\mathsf {NBI}}$ . Working in Kleene’s higher-order computability theory based on S1–S9, we show that the following fourth-order process based on ${\mathsf {NIN}}$ is similarly hard to compute: for a given $[0,1]\rightarrow {\mathbb N}$ -function, find reals in the unit interval that map to the same natural number.