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

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.