Kleene states the theorem with V = N, relative to specific φ, S n , supplied by his Enumeration Theorem, m = 0 (no parameters ~y) and n ≥ 1, i.e., not allowing nullary partial functions. And most of the time, this is all we need; but there are a few important applications where choosing “the right” φ, S n , restricting the values to a proper V ( N or allowing m > 0 or n = 0 simplifies the proof...

We describe a strictly classical dice game, which emulates the main features of the EPR experiment, including violation of Bell’s inequalities. Therefore, the standard interpretation that Bell’s theorem provides necessary conditions for ‘local realism’ is disproved. PACS 03.65.Ud (Entanglement and quantum non locality)

In 1961 G. Higman proved a remarkable theorem establishing a deep connection between the logical notion of recursiveness and questions about finitely presented groups. The basic aim of the present paper is to provide the reader with a rigorous and detailed proof of Higman’s Theorem. All the necessary preliminary material, including elements of group theory and recursive functions theory, is sys...

We generalize Dirichlet’s S-unit theorem from the usual group of S-units of a number field K to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over S. Specifically, we demonstrate that the group of algebraic S-units modulo torsion is a Q-vector space which, when normed by the Weil height, spans a hyperplane determined by the product formula, a...

Given function f holomorphic at infinity, the n-th diagonal Padé approximant to f , say [n/n]f , is a rational function of type (n,n) that has the highest order of contact with f at infinity. Equivalently, [n/n]f is the n-th convergent of the continued fraction representing f at infinity. BernsteinSzegő theorem provides an explicit non-asymptotic formula for [n/n]f and all n large enough in the...

This short and informal article shows that, although Godel's theorem is valid using classical logic, there exists some four-valued logical system that is able to prove that arithmetic is both sound and complete. This article also describes a four-valued Prolog in some informal, brief and intuitive manner.

Abst ract The original statement of Kharitonov's theorem requires that all polynomials in the family have the same degree. It has been shown recently that such an assumption is unnecessary. Here we show that the validity of the stronger statement follows from a classical proof combined with additional elementary considerations. 1 Introdu ction The original statement of Kharitonov's theorem 3] r...

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing, and the stat...

This paper deals with the numerical content of Picard's Thsorem. Two classically equivalent versions of this theorem are proved which are distinct from a computational point of view. The proofs are elementary, and constructive in the sense of Bishop. A Brouwerian counterexample is given to the original version of the theorem.