Journal
REVIEW OF SYMBOLIC LOGIC
Volume 13, Issue 3, Pages 541-592Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/S1755020319000212
Keywords
Arithmetized Completeness Theorem; interpretability; set theoretic paradoxes; semantic paradoxes; Hilbert program; David Hilbert; Paul Bernays; Georg Kreisel; Hao Wang
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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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