Seeing the Number Structures

Abstract

In the previous chapter, we introduced and named an actually infinite set. The set of natural numbers ℕ={0,1,2,…}$${\mathbb {N}}=\{0,1,2,\dots \}$$. What is the structure of this set? We will give a simple answer to this question, and then we will proceed with a reconstruction of the arithmetic structures of the integers and the rational numbers in terms of first-order logic. The reconstruction is technical and rather tedious, but it serves as a good example of how some mathematical structures can bee seen with the eyes of logic inside other structures. This chapter can be skipped on the first reading, but it should not be forgotten.