Natural Numbers

Overview

The standard way of representing the natural numbers is as follows:

• $0=\varnothing$
• $1=\{0\}=\{\varnothing \}$
• $2=\{0,1\}=\{\varnothing ,\{\varnothing \}\}$
• $\dots$

That is, each natural number corresponds to the set of natural numbers smaller than it.

Inductive Sets

For any set $a$, its successor $a^{+}$ is defined as $a^{+}=a\cup \{a\}$. A set $A$ is inductive if and only if $\varnothing \in A$ and $\mathrm{\forall }a\in A,a^{+}\in A$.

A natural number is a set that belongs to every inductive set.

Peano System

A Peano system is a triple $⟨N,S,e⟩$ consisting of a set $N$, a function $S:N\to N$, and a member $e\in N$ such that the following three conditions are met:

• $e\notin \text{ran}S$;
• $S$ is one-to-one;
• Any subset $A$ of $N$ that contains $e$ and is closed under $S$ equals $N$ itself.

Given $\sigma =\{⟨n,n^{+}⟩\mid n\in \omega \}$, $⟨\omega ,\sigma ,0⟩$ is a Peano system.

Ordering

For natural numbers $m$ and $n$, define $m$ to be less than $n$ if and only if $m\in n$. The following biconditionals hold true:

• $m\in n⟺m^{+}\in n^{+}$
• $m\in n⟺m\subset n$
• $m\underset{―}{\in }n⟺m\subseteq n$