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

Walter Dean
University of Warwick
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 and also how it was later adapted by Kreisel and Wang 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, and axiomatic theories of truth.
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DOI 10.1017/s1755020319000212
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References found in this work BETA

Outline of a Theory of Truth.Saul Kripke - 1975 - Journal of Philosophy 72 (19):690-716.
Truth and Other Enigmas.Michael Dummett - 1980 - Revue Philosophique de la France Et de l'Etranger 170 (1):62-65.
Reflecting on Incompleteness.Solomon Feferman - 1991 - Journal of Symbolic Logic 56 (1):1-49.
Axiomatizing Kripke’s Theory of Truth.Volker Halbach & Leon Horsten - 2006 - Journal of Symbolic Logic 71 (2):677 - 712.

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Citations of this work BETA

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.

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