The Gödel Incompleteness Theorems (1931) by the Axiom of Choice

Abstract

Those incompleteness theorems mean the relation of (Peano) arithmeticand (ZFC) set theory, or philosophically, the relation of arithmetical finiteness andactual infinity. The same is managed in the framework of set theory by the axiom ofchoice (respectively, by the equivalent well-ordering "theorem'). One may discuss thatincompleteness form the viewpoint of set theory by the axiom of choice rather thanthe usual viewpoint meant in the proof of theorems. The logical corollaries from that"nonstandard" viewpoint the relation of set theory and arithmetic are demonstrated