There are not many global results on modal logics. One of these is the following theorem by Sahlqvist on completeness and correspondence for a wide class of modal formulae (including many well known logics, e.g. D, T, B, S4, K4, S5, …) (see [S]).

Sahlqvist's Theorem. Let A be any modal formula which is equivalent to a conjunction of formulae of the form □m(A1 → A2), where m ≥ 0, A2 is positive and A1 is obtained from propositional variables and their negations applying ∧, ∨, ♢, and □ in such a way that no positive occurrence of a variable is in a subformula of the form B1 ∨ B2 or ♢ B1 within the scope of some □. Then A corresponds effectively to a first order formula, and L + A is canonical whenever Lis a canonical logic.

A formula A satisfying the above conditions is henceforth called a Sahlqvist formula. Unfortunately, till now, the only complete proof was the original proof of Sahlqvist (a proof of the correspondence half has also been given by van Benthem [vB]). It is so complicated and long that even in an advanced textbook of modal logic [HC] it has not found a place. Here, by considering general frames as topological spaces, an attitude which we developed in [TD], we give a proof of Sahlqvist's theorem simplified to such an extent that one can easily grasp the key idea on which it is based and apply the resulting algorithm to specific modal formulae in a straightforward manner, suitable even for implementation on a personal computer. This key idea also improves on previous preliminary work in the same direction (see [S1], [S2]).

Nonstandard set theory is an attempt to generalise nonstandard analysis to cover the whole of classical mathematics. Existing versions (Nelson, Hrbáček, Kawai) are unsatisfactory in that the unlimited idealisation principle conflicts with the wish to have a full theory of external sets.

I re-analyse the underlying requirements of nonstandard set theory and give a new formal system, stratified nonstandard set theory, which seems to meet them better than the other versions.

We show that a translator ƒ: ω → ω from a Gödelnumbering φ into a direct sum η of a r.e. family of Friedbergnumberings satisfies ƒ ≰T0′. In particular, η cannot be a Gödelnumbering.

In the following we use standard notation (cf.[3]): for i ≥ 1, Pi (respectively, Ri) is the set of partial (total) recursive i-place functions; φ is a Gödel numbering of P1. By φi, s we denote a recursive standard approximation for φi, i.e., φi, s is a finite function, φi, s ⊆ φi, s + 1, φi, s ⊆ φi, φi = ⋃ {φi, s ∣ s ≥ 0}, and a canonical index for φi, s can be computed uniformly in i, s (cf.[3, p. 16 f]).

We call v ∈ P2 a numbering of P1 iff {λx.v(i, x)}i ∈ ω = P1; we denote λx.v(i, x) by vi. Let v and γ be numberings of P1. A total function g: ω → ω is called a translator from v into γ iff ∀i: vi = γg(i).

Church's thesis, that all reasonable definitions of “computability” are equivalent, is not usually thought of in terms of computability by a continuous computer, of which the general-purpose analog computer (GPAC) is a prototype. Here we prove, under a hypothesis of determinism, that the analytic outputs of a C∞ GPAC are computable by a digital computer.

In [POE, Theorems 5, 6, 7, and 8], Pour-El obtained some related results. (The proof there of Theorem 7 depends on her Theorem 2, for which the proof in [POE] is incorrect, but for which a correct proof is given in [LIR]. Also, the proof in [POE] of Theorem 8 depends on the unproved assertion that a solution of an algebraic differential equation must be analytic on an open subset of its domain. However, this assertion was later proved in [BRR].) As in [POE], we reduce the problem to a problem about solutions of certain systems of algebraic differential equations (ADE's). If such a system is nonsingular (i.e. if the “separant” does not vanish along the given solution), then the argument is very easy (see [VSD] for an even simpler situation), so that the essential difficulties arise from singular systems. Our main tools in handling these difficulties are drawn from the excellent (and difficult) paper [DEL] by Denef and Lipshitz. The author especially wants to thank Leonard Lipshitz for his kind help in the preparation of the present paper.

We emphasize here that our proof of the simulation result applies only to the GPAC as described below. The GPAC's form a natural subclass of the class of all analog computers, and are based on certain idealized components (“black boxes”), mostly associated with the technology of past decades. One can easily envisage other kinds of black boxes of an input-output character that would lead to different kinds of analog computers. (For example, one could incorporate delays, or spatial integrators in addition to the present temporal integrators, etc.) Whether digital simulation is possible for these “extended” analog computers poses a rich and challenging set of research questions.

A computable groupoid is an algebra ‹N, g› where N is the set of natural numbers and g is a recursive (total) binary operation on N. A set L of natural numbers is a computable list of computable groupoids iff L is recursive, ‹N, ϕe› is a computable groupoid for each e ∈ L, and e ∈ L whenever e codes a primitive recursive description of a binary operation on N.

Theorem 1. Let L be any computable list of computable groupoids. The set {e ∈ L: the equational theory of ‹N, ϕe› is finitely axiomatizable} is not recursive.

Theorem 2. Let S be any recursive set of positive integers. A computable groupoid GS can be constructed so that S is inifinite iff GS has a finitely axiomatizable equational theory.

The problem of deciding which finite algebras have finitely axiomatizable equational theories has remained open since it was first posed by Tarski in the early 1960's. Indeed, it is still not known whether the set of such finite algebras is recursively (or corecursively) enumerable. McKenzie [7] has shown that this question of finite axiomatizability for any (finite) algebra of finite type can be reduced to that for a (finite) groupoid.