Results 1  10
of
12
The Value of Information
 in Monotone Decision Problems,” MIT Working Paper
, 2001
"... To the memory of Andrei Kolmogorov, In the 100th year since his birth. There appears to be a gap between usual interpretations of Godel Theorem and what is actually proven. Closing this gap does not seem obvious and involves complexity theory. (This is unrelated to, well studied before, complexity q ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
To the memory of Andrei Kolmogorov, In the 100th year since his birth. There appears to be a gap between usual interpretations of Godel Theorem and what is actually proven. Closing this gap does not seem obvious and involves complexity theory. (This is unrelated to, well studied before, complexity quantifications of the usual Godel effects.) Similar problems and answers apply to other unsolvability results for tasks where required solutions are not unique, such as, e.g., nonrecursive tilings. 1 Introduction. D.Hilbert asked if formal arithmetic can be consistently extended to a complete theory. The question was somewhat vague since an obvious answer was “yes”: just add to the axioms of Peano Arithmetic (PA) 1 a maximal consistent set, clearly existing albeit hard to find. K.Godel formalized this question as existence among such extensions of recursively enumerable ones and gave it a
The Six Semiosic Predicates
"... © This paper is not for reproduction without the express permission of the author. Information is understood as a mediated construction, the Sign, which is a cohesion of meaning, both material and conceptual, derived from the measurement within spatial and temporal parameters of energy/matter by six ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
© This paper is not for reproduction without the express permission of the author. Information is understood as a mediated construction, the Sign, which is a cohesion of meaning, both material and conceptual, derived from the measurement within spatial and temporal parameters of energy/matter by six predicate relations. Three predicate relations enforce states; three predicate relations enforce dynamics. The Sign is understood as a triadic cohesion of three predicates. 1
The scope of logic: deduction, abduction, analogy
"... The present form of mathematical logic originated in the twenties and early thirties from the partial merging of two different traditions, the algebra of logic and the logicist tradition (see [27], [41]). This resulted in a new form of logic in which several features of the two earlier traditions co ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The present form of mathematical logic originated in the twenties and early thirties from the partial merging of two different traditions, the algebra of logic and the logicist tradition (see [27], [41]). This resulted in a new form of logic in which several features of the two earlier traditions coexist. Clearly neither the algebra of logic nor the logicist’s logic is identical to the present form of mathematical logic, yet some of their basic ideas can be distinctly recognized within it. One of such ideas is Boole’s view that logic is the study of the laws of thought. This is not to be meant in a psychologistic way. Frege himself states that the task of logic can be represented “as the investigation of the mind; [though] of the mind, not of minds” [17, p. 369]. Moreover Frege never charges Boole with being psychologistic and in a letter to Peano even distinguishes between the followers of Boole and “the psychological logicians ” [16, p. 108]. In fact for Boole the laws of thought which are the object of logic belong “to the domain of what is termed necessary truth ” [2, p. 404]. For him logic does not depend on psychology, on the contrary psychology depends on logic insofar as it is only through an investigation of logical operations that we could obtain “some probable
GÖDELIAN PLATONISM
, 2009
"... The members of the Comittee appointed to examine the thesis of TEPPEI HAYASHI ..."
Abstract
 Add to MetaCart
The members of the Comittee appointed to examine the thesis of TEPPEI HAYASHI
Gödel on Intuition and on Hilbert’s finitism
"... 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 con ..."
Abstract
 Add to MetaCart
(Show Context)
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 doublereversal, 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 firstorder number theory, P A; but starting in the Dialectica paper
An Ordinal Analysis of parameter free ...Comprehension: Part I
, 1995
"... The objective of this paper is to present an ordinal analysis for the fragment of second order arithmetic with \Delta 1 2 comprehension, bar induction and \Pi 1 2 comprehension for formulae without set parameters. ..."
Abstract
 Add to MetaCart
The objective of this paper is to present an ordinal analysis for the fragment of second order arithmetic with \Delta 1 2 comprehension, bar induction and \Pi 1 2 comprehension for formulae without set parameters.
An Ordinal Representation System for ...Comprehension and Related Systems
, 1995
"... The objective of this paper is to introduce an ordinal representation system which has been employed in the determination of the prooftheoretic strength of \Pi 1 2 comprehension and related systems. ..."
Abstract
 Add to MetaCart
The objective of this paper is to introduce an ordinal representation system which has been employed in the determination of the prooftheoretic strength of \Pi 1 2 comprehension and related systems.
IOS Press Proving as a Computable Procedure
, 2004
"... Abstract. Gödel’s incompleteness theorem states that every finitelypresented, consistent, sound theory which is strong enough to include arithmetic is incomplete. In this paper we present elementary proofs for three axiomatic variants of Gödel’s incompleteness theorem and we use them (a) to illustr ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Gödel’s incompleteness theorem states that every finitelypresented, consistent, sound theory which is strong enough to include arithmetic is incomplete. In this paper we present elementary proofs for three axiomatic variants of Gödel’s incompleteness theorem and we use them (a) to illustrate the idea that there is more than “complete vs. incomplete”, there are degrees of incompleteness, and (b) to discuss the implications of incompleteness and computerassisted proofs for Hilbert’s Programme. We argue that the impossibility of carrying out Hilbert’s Programme is a thesis and has a similar status to the ChurchTuring thesis. 1.