The Stacks project

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

\[ \alpha = \bigcup \nolimits _{\gamma \in \alpha } \gamma . \]

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.

Comments (6)

Comment #971 by Fred Rohrer on

I suggest changing "well ordered" to "well-ordered" (or at least use the same spelling throughout). The other occurrences without a hyphen are in 00YP, 065T, 03C3, 09E0, 06RF and 06RG.

Comment #3567 by Christian Hildebrandt on

I would suggest changing to to keep the analogy with .

Comment #5553 by Jichao on

Why don't we have the conclusion by the definition? And, in another way, it feels strange that there appears an in the definition of itself. Won't that makes not well-defined?

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05N1. Beware of the difference between the letter 'O' and the digit '0'.