Definition:Minimally Inductive Set

Definition
Let $S$ be an infinite successor set.

The minimal infinite successor set $\omega$ is the infinite successor set given by:


 * $\omega := \displaystyle \bigcap \left\{{S' \subseteq S: \text{$S'$ is an infinite successor set}}\right\}$

that is, $\omega$ is the intersection of every infinite successor set which is a subset of $S$.

Natural Numbers
The natural numbers $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$ can be defined as the elements of the minimal infinite successor set $\omega$:


 * $0 := \varnothing = \left\{{}\right\}$
 * $1 := 0^+ = 0 \cup \left\{{0}\right\} = \left\{{0}\right\}$
 * $2 := 1^+ = 1 \cup \left\{{1}\right\} = \left\{{0, 1}\right\}$
 * $3 := 2^+ = 2 \cup \left\{{2}\right\} = \left\{{0, 1, 2}\right\}$
 * $\vdots$

Also see

 * Existence of Minimal Infinite Successor Set, demonstrating from ZF that $\omega$ exists.