Finite Ordinal

Definition 1. We define

\[\N = \bigcap\{X : \empt \in X \,\land\, \text{$X$ is inductive}\}\]

as the set of finite ordinals or natural numbers. Axiom of Infinity guarantees the existence of at least one such $X$, therefore the existence of $\N$.

An ordinal is infinite if it is not finite.

Definition 2. $\omega := \N$ itself is an ordinal. $\omega$ is used when we treat it as an ordinal, $\N$ when we treat it as a set.

Definition 3. We define

\[0 = \empt, \quad 1 = 0 + 1, \quad 2 = 1 + 1, \quad 3 = 2 + 1\]

and so on, where $\alpha + 1$ is the ordinal successor of an ordinal $\alpha$.

Proposition 1. $\omega$ is the least limit ordinal.