  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.
