---
tag: 0RDJK7EQYY
kind: definition
name: Ordinal number
msc: ["03E10"]
defines: 0YSCSX5YZC
uses: [0F1D6PK9RZ, 0EZ758XE73]
sources:
  - url: https://en.wikipedia.org/wiki/Ordinal_number
    locator: lead section
---

## 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|0F1D6PK9RZ]] 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|0Y74KN99A5]] 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.