## 3.4 Ordinals

A set $T$ is *transitive* if $x\in T$ implies $x\subset T$. A set $\alpha $ is an *ordinal* if it is transitive and well-ordered by $\in $. In this case, we define $\alpha + 1 = \alpha \cup \{ \alpha \} $, which is another ordinal called the *successor* of $\alpha $. An ordinal $\alpha $ is called a *successor ordinal* if there exists an ordinal $\beta $ such that $\alpha = \beta + 1$. The smallest ordinal is $\emptyset $ which is also denoted $0$. If $\alpha $ is not $0$, and not a successor ordinal, then $\alpha $ is called a *limit ordinal* and we have

The first limit ordinal is $\omega $ and it is also the first infinite ordinal. The first uncountable ordinal $\omega _1$ is the set of all countable ordinals. The collection of all ordinals is a proper class. It is well-ordered by $\in $ in the following sense: any nonempty set (or even class) of ordinals has a least element. Given a set $A$ of ordinals, we define the *supremum* of $A$ to be $\sup _{\alpha \in A} \alpha = \bigcup _{\alpha \in A} \alpha $. It is the least ordinal bigger or equal to all $\alpha \in A$. Given any well-ordered set $(S, <)$, there is a unique ordinal $\alpha $ such that $(S, <) \cong (\alpha , \in )$; this is called the *order type* of the well-ordered set.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (6)

Comment #971 by Fred Rohrer on

Comment #1005 by Johan on

Comment #3567 by Christian Hildebrandt on

Comment #3691 by Johan on

Comment #5553 by Jichao on

Comment #5737 by Johan on