## 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.

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 $\geq$ to $\lt$ to keep the analogy with $\in$ .

Comment #5553 by Jichao on

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

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