Definition of the Concept of Natural Numbers and its Exixtence Theorem. Solution of Hilbert's Second Problem

In this paper，we give the new complete solution of the definition of the concept of natural numbers and its existence. This work is the English version of my work in Ito [6], Chapter 2， Section 2.1. It is important to notice that we do not use Gödel's Incompletcness Theorem.This gives the consideration and the solution of Hilbert's second problem "the proof of consistency of the axioms of arithmetic" in the other angle by changing the point of view.Namely，giving the definition of the concept of natural numbers means providing the complete system of axioms which determines the set of all natural numbera as an algebraic system. The proof of the existence of the concept of natural numbers means constructing a model of natural numbers as the set of all natural numbers as an algebraic system in the concrete manner on the basis of the ZFC set theory. Thereby，at the same time，we give the new complete solutions of the problem of the foundation of analysis and the problem of the foundation of geometry