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.