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.

中文翻译：

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通过悖论和完整性的不完整性

本文探讨了集合论和语义的传统悖论与形式不完备现象之间的关系。一个核心工具是将算术完备性定理应用于二阶算术和集合论系统，其中可以形式化一阶语言的各种“矛盾概念”。我将首先讨论这个结果最初由 Hilbert & Bernays (1939) 提出的设置，以及后来如何被 Kreisel (1950) 和 Wang (1955) 改编以获得正式的不可判定性结果。然后将介绍这种方法的一般化，由此罗素悖论、米里马诺夫悖论的变体、骗子和格雷林-纳尔逊悖论可以统一转化为不完备性定理。