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2016•arXiv (Cornell University)Open access

Some Infinitary Paradoxes and Undecidable Sentences in Peano Arithmetic

Ka-Yue Cheng

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Abstract

According to Chaitin, Gödel once told him "it doesn't matter which paradox you use [to prove the First Incompleteness Theorem]". In this paper I will present a few infinitary paradoxes and show how to "translate" them to some undecidable sentences in Peano arithmetic, like what Gödel did to the Liar paradox. The results partly verify Gödel's claim.

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According to Chaitin, Gödel once told him "it doesn't matter which paradox you use [to prove the First Incompleteness Theorem]". In this paper I will present a few infinitary paradoxes and show how to "translate" them to some undecidable sentences in Peano arithmetic, like what Gödel did to the Liar paradox. The results partly verify Gödel's claim.

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Available abstract

According to Chaitin, Gödel once told him "it doesn't matter which paradox you use [to prove the First Incompleteness Theorem]". In this paper I will present a few infinitary paradoxes and show how to "translate" them to some undecidable sentences in Peano arithmetic, like what Gödel did to the Liar paradox. The results partly verify Gödel's claim.

Key concepts: Undecidable problem, Peano axioms, Gödel's incompleteness theorems, Mathematics, Arithmetic, Second-order arithmetic, Discrete mathematics, Decidability

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