Some Infinitary Paradoxes and Undecidable Sentences in Peano Arithmetic

According to Chaitin, G"odel 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 G"odel did to the Liar paradox. The results partly verify G"odel’s claim.