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 $\forall a\in A,a^{+}\in A$.

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

Bibliography

• Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).