Set Theory: Constructing The Integers

Introduction

Taking the finite ordinals gives a model for N. We can prove all of the standard properties of N.
Natural numbers are good for indicating the number of times you want to iterate a function. But what if you want to allow iterations of the inverse function.
How do we introduce the negative numbers?

**********

Step 1: Introducing equivalence relations

 In general an equivalence relation results when we wish to “identify” two elements of a set that share a common attribute. The definition is motivated by observing that any process of “identification” must behave somewhat like the equality relation, and the equality relation satisfies the reflexive (x = x for all x), symmetric (x = y implies y = x), and transitive (x = y and y = z implies x = z) properties.

Equivalence Relations by R.C.Lacher: pdf

**********

Step 2: Constructing integers from natural numbers

The natural numbers are designed for measuring the size of finite sets, but what if you want to compare the sizes of two sets? For example, you might want to compare the number of chairs in a classroom with the number of students to determine the number of free chairs. If there are more students than chairs, you would use negative integers to indicate the absence of free chairs.

The Integers by Wayne Aitken: pdf

 **********

More understanding:

Constructing the Integers by Bernd Schröder: pdf