The Foundations of Mathematics in the Physical Reality

2013/11/08 by D. H. Homan, Homan, D. H.
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #math.lo

paper · pdf · doi:10.48550/arxiv.1311.3168

openalex publication_date 2013/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ω^ω-number. Then we prove a first ω^ω-number notequal 0 belonging to ordinal γ is an impassable barrier for counting down γ to 0 in a finite number of steps.

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