Abstract

A contradiction is obtained, considering the axiom of infinity, then ℕ and Peano axioms, together a list of ℕ subsets and with inclusion relation and union operation. Natural numbers constitute an infinite set, ℕ, but we show the union of its proper subsets, with a specific form, isn’t an infinite set. Also we get a simpler explanation and a symbolic representation. Lastly, inconsistency of Peano successor axiom is a consequence of rejecting infinity.