The Foundations of Mathematics in the Physical Reality

Abstract

In this article we present an axiomatic definition of sets with individuals and a definition of natural numbers and ordinals. We use the axioms pairs, union, power, regularity and separation. We define the equality of sets and of individuals. There is no empty set. A single shoe is not a singleton set but an individual, a pair of shoes is a set. We call limit ordinals first numbers, that is a first number of the Peano axioms. An axiom of infinity is postulated and we prove the Peano axioms for ordinals with a first number up to a first $\omega^\omega$-number. Then we prove a first $\omega^\omega$-number notequal $0$ belonging to ordinal $\gamma$ is an impassable barrier for counting down $\gamma$ to $0$ in a finite number of steps.