Ordinal number

Definition

A set $X$ is transitive when each element of an element of $X$ is an element of $X$. An ordinal number, in the definition of von Neumann, is a transitive set $\alpha$ that is well-ordered by the relation $\in$, with $\beta < \gamma$ when $\beta \in \gamma$.

Remarks

The first ordinals are $0 = \varnothing$, $1 = \{0\}$, $2 = \{0, 1\}$, and in general $n + 1 = n \cup \{n\}$. The set $\omega = \{0, 1, 2, \dots\}$ of all finite ordinals is the first infinite ordinal. Each well-ordered set is order isomorphic to exactly one ordinal, its order type. Each ordinal $\alpha$ is the set of all ordinals smaller than $\alpha$.

Explanations

L1 human

Ordinals count positions: first, second, third, and then beyond all finite positions. After $0, 1, 2, \dots$ comes $\omega$, then $\omega + 1$, and so on. They let you continue a step-by-step construction past infinity.