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G\"odel's first and second incompleteness theorems are corner stones ofmodern mathematics. In this article we present a new proof of these theoremsfor ZFC and theories containing ZFC, using Chaitin's incompleteness theorem anda very basic numbers extension. As opposed to the usual proofs, these proofsdon't use any fixed-point theorem and rely solely on sets structure. Unlike inthe original proof, the statements which can be shown to be unprovable by ourtechnique exceed by far one specific statement constructed from the axiom set.Our goal is to draw attention to the technique of number extensions, which webelieve can be used to prove more theorems regarding the provability andunprovability of different assertions regarding natural numbers.

A proof of G\"odel's incompleteness theorems using Chaitin's incompleteness theorem