The Integers

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $P=\mathbb{N}\times\mathbb{N}$, $\sim$ and $[a,b]$ be as in Construction of the Integers: Pairs of Natural Numbers up to Equal Differences, with Sum, Product, Negation and Order §equivalence.

¶ The set of integers is the quotient $\mathbb{Z}=P/{\sim}$, which is a set whose elements are the classes $[a,b]$ with $a,b\in\mathbb{N}$, by Construction of the Integers: Pairs of Natural Numbers up to Equal Differences, with Sum, Product, Negation and Order §equal.

¶ The sum $+$, the product $\cdot$, the negation $-$ and the order $\le$ on $\mathbb{Z}$ are those of Construction of the Integers: Pairs of Natural Numbers up to Equal Differences, with Sum, Product, Negation and Order §operations. For $x,y\in\mathbb{Z}$, $x-y$ denotes $x+(-y)$, and $x<y$ means $x\le y$ and $x\neq y$.

¶ $0_{\mathbb{Z}}=[1,1]$ and $1_{\mathbb{Z}}=[2,1]$.

¶ $\iota:\mathbb{N}\to\mathbb{Z}$ is the map with $\iota(n)=[n+1,1]$, given by Maps and Relations Given by Formulas §map.

Here $2$ and $n+1$ are natural numbers by The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §background, so the classes above are integers.