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 $X$.
$\omega := \N$ itself is an ordinal. $\omega$ is used with emphasis on an ordinal, $\N$ is used with emphasis on a set.
An ordinal is infinite if it is not finite.
We define
\[0 = \empt, \quad 1 = 0 + 1, \quad 2 = 1 + 1, \quad 3 = 2 + 1\]and so on.
See Also. Natural Numbers
$\omega$ is the least limit ordinal.
Proof. By LIM-ORD > Proposition 1 (I) $\lrimp$ (V).