In this project we will learn about both primitive recursive and general recursive functions. We will also learn about Turing computable functions, and will discuss why the class of general recursive functions coincides with the class of Turing computable functions. We will introduce the effectively calculable functions, and the ideas behind Alonzo Church’s (1903–1995) proposal to identify the

Since the cognitive revolution, it’s become commonplace that cognition involves both computation and information processing. Is this one claim or two? Is computation the same as information processing? The two terms are often used interchangeably, but this usage masks important differences. In this paper, we distinguish information processing from computation and examine some of their mutual relations, shedding light on the role each can play in a theory of cognition. We recommend that theorists of cognition be explicit and careful in choosing 1 notions of computation and information and connecting them together. Much confusion can be avoided by doing so.

This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to know the dynamical properties of a c.e. set We , not merely whether an element x is enumerated in We but when, relative to its appearance in other c.e. sets. We present here

Computation and information processing are among the most fundamental notions in cognitive science. They are also among the most imprecisely discussed. Many cognitive scientists take it for granted that cognition involves computation, information processing, or both – although others disagree vehemently. Yet different cognitive scientists use ‘computation ’ and ‘information processing ’ to mean different things, sometimes without realizing that they do. In addition, computation and information processing are surrounded by several myths; first and foremost, that they are the same thing. In this paper, we address this unsatisfactory state of affairs by presenting a general and theory-neutral account of computation and information processing. We also apply our framework by analyzing the relations between computation and information processing on one hand and classicism and connectionism/computational neuroscience on the other. We defend the relevance to cognitive science of both computation, at least in a generic sense, and information processing, in three important senses of the term. Our account advances several foundational debates in cognitive science by untangling some of their conceptual knots in a theory-neutral way. By leveling the playing field, we pave the way for the future resolution of the debates ’ empirical aspects.

There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of first-order number theory, P A; but starting in the Dialectica paper

Abstract. This article demonstrates the invalidity of Theorem VI in Gödel’s monograph of 1931, by showing that (15) xBκ(17Gen r) − → Bewκ[Sb(r17 Z(x))], (16) xBκ(17Gen r) − → Bewκ[Neg(Sb(r17 Z(x))], (derived by means of definition (8.1) Q(x, y) ≡ xBκ[Sb(y19)] respectively

Abstract. This article demonstrates the invalidity of the so-called Gödel’s first incompleteness theorem, Theorem VI in Gödel’s 1931 article, showing that propositions (15) and (16), derived from the definition 8.1, in its proof, are false in PA. Introduction. Developed as a consequence of the crisis of the foundation of mathematics due to the discovery of the antinomies, Hilbert’s formalism planned as criterions of adequacy for the axiomatic systems, the achievement of their coherence and completeness [8][9]. The result of incompleteness for any system embodying the arithmetic of the positive integers, obtained by Gödel in 1931, grafted on Hilbert’s

Abstract. This article demonstrates the invalidity of Theorem VI of Gödel’s monograph of 1931, by showing that (15) xBκ(17Gen r) − → Bewκ[Sb(r17 Z(x))], (16) xBκ(17Gen r) − → Bewκ[Neg(Sb(r17 Z(x))], (derived by means of definition (8.1) Q(x, y) ≡ xBκ[Sb(y19)] respectively

Abstract. This article demonstrates the invalidity of Theorem VI in Gödel’s monograph of 1931, by showing that (15) xBκ(17Gen r) − → Bewκ[Sb(r17 Z(x))], (16) xBκ(17Gen r) − → Bewκ[Neg(Sb(r17 Z(x))], (derived by means of definition (8.1) Q(x, y) ≡ xBκ[Sb(y19)] respectively

Abstract. This article demonstrates the invalidity of Theorem VI of Gödel’s monograph of 1931, by showing that (15) xBκ(17Gen r) − → Bewκ[Sb(r17 Z(x))], (16) xBκ(17Gen r) − → Bewκ[Neg(Sb(r17 Z(x))], (derived by means of definition (8.1) Q(x, y) ≡ xBκ[Sb(y19)] respectively