The Foundations of Mathematics in the Physical Reality

In this article we present an axiomatic definition of sets with individuals and a definition of natural numbers (finite ordinal numbers). We use the axioms pairs, union, regularity and separation of the standard set theory ZF. The equality of sets should be defined thus the axiom of extensionality is not used. And there are individuals thus there is no empty set. The principle of mathematical induction is proved for natural numbers. Then we define ordinal numbers and postulate the union set of all natural numbers and define transfinite ordinal numbers.

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