Incompleteness Via Paradox and Completeness

Review of Symbolic Logic 13 (3):541-592 (2020)
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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.

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Walter Dean
University of Warwick

Citations of this work

Current Research on Gödel’s Incompleteness Theorems.Yong Cheng - 2021 - Bulletin of Symbolic Logic 27 (2):113-167.
Formal Notes on the Substitutional Analysis of Logical Consequence.Volker Halbach - 2020 - Notre Dame Journal of Formal Logic 61 (2):317-339.
XV—On Consistency and Existence in Mathematics.Walter Dean - 2021 - Proceedings of the Aristotelian Society 120 (3):349-393.
On the Depth of Gödel’s Incompleteness Theorems.Yong Cheng - forthcoming - Philosophia Mathematica.

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References found in this work

Outline of a theory of truth.Saul Kripke - 1975 - Journal of Philosophy 72 (19):690-716.
The Principles of Mathematics.Bertrand Russell - 1903 - Revue de Métaphysique et de Morale 11 (4):11-12.
Truth and Other Enigmas.Michael Dummett - 1980 - Revue Philosophique de la France Et de l'Etranger 170 (1):62-65.

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