2015•HAL (Le Centre pour la Communication Scientifique Directe)Open access

Axiom of infinity and construction of N

Francisco R. Ruiz del Portal

Abstract

The aim of the present note is to show that it is possible, in the construction of numbers sets, to replace axiom of substitution and the standard Axiom of infinity (there exists an infinite ordinal) by one simple axiom " there exists an infinite set ". Then the Axiom of choice will ensure the existence of a well-ordered set without maximal element with only one hereditary set. We then show that all these sets are isomorphic, and can then be called " naturel integer sets " .

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Key concepts: Axiom of choice, Axiom, Mathematics, Zermelo–Fraenkel set theory, Urelement, Infinity, Simple (philosophy), Set (abstract data type)