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Incompleteness via paradox and completeness
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Dean, Walter (2020) Incompleteness via paradox and completeness. Review of Symbolic Logic, 13 (3). pp. 541-592. doi:10.1017/S1755020319000212 ISSN 1755-0203.
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Official URL: http://dx.doi.org/10.1017/S1755020319000212
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) andWang (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimano’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.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Social Sciences > Philosophy | ||||||||
Library of Congress Subject Headings (LCSH): | Incompleteness theorems, Gödel's theorem | ||||||||
Journal or Publication Title: | Review of Symbolic Logic | ||||||||
Publisher: | Cambridge University Press | ||||||||
ISSN: | 1755-0203 | ||||||||
Official Date: | September 2020 | ||||||||
Dates: |
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Volume: | 13 | ||||||||
Number: | 3 | ||||||||
Page Range: | pp. 541-592 | ||||||||
DOI: | 10.1017/S1755020319000212 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Reuse Statement (publisher, data, author rights): | This article has been published in a revised form in Review of Symbolic Logic http://doi.org/10.1017/S1755020319000212XXX]. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © copyright holder. | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Copyright Holders: | © Association for Symbolic Logic 2019 | ||||||||
Date of first compliant deposit: | 5 June 2019 | ||||||||
Date of first compliant Open Access: | 10 June 2019 |
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