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INCOMPLETENESS VIA PARADOX AND COMPLETENESS

Published online by Cambridge University Press:  23 May 2019

WALTER DEAN
WALTER DEAN*
Affiliation:
Department of Philosophy, University of Warwick
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF WARWICK COVENTRY CV4 7AL, UK E-mail: W.H.Dean@warwick.ac.uk
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Abstract

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth.

Keywords

03C6203H1503C5700A30Arithmetized Completeness Theoreminterpretabilityset theoretic paradoxessemantic paradoxesHilbert programDavid HilbertPaul BernaysGeorg KreiselHao Wang
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 3 , September 2020 , pp. 541 - 592
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Copyright
Copyright © Association for Symbolic Logic 2019 

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