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Ordinal numbers

Ordinal numbers describe the way a set might be arranged into a well-ordered sequence. Thus, ordinals have to do with the way a set is or can be ordered, rather than its size or cardinality.

An elegant formulation of the ordinal concept in ZFC was provided by von Neumann: an ordinal is simply a transitive set well-ordered by the set membership relation $\in$. Equivalently, an ordinal is a hereditarily transitive set, meaning that it is transitive, and all of its elements are transitive.

The ordinals are ordered by the relation $\alpha\lt\beta$ just in case $\alpha\in\beta$, and one can show that this is a total order, indeed, a well-order. The collection of all ordinals is a transitive proper class. It can be denoted, for example, $\mathrm{Ord}$, $\mathsf{ORD}$, $\mathrm{On}$ or $\mathrm{OR}$.

Successor ordinals

If $\alpha$ is an ordinal, then so is the set $\alpha\cup\{\alpha\}$, and it is easy to prove that $\alpha\cup\{\alpha\}$ is the successor ordinal to $\alpha$, the smallest ordinal above $\alpha$, and is accordingly denoted $\alpha+1$.

Limit ordinals

A limit ordinal is a nonzero ordinal with no immediate predecessor. Every ordinal is either $0$, a successor ordinal or a limit ordinal.

Transfinite induction

Transfinite induction is a method of proving that a statement $\varphi(\alpha)$ holds of all ordinals $\alpha$. Since the ordinals are well-ordered by $\in$, it follows that every nonempty set or class $X$ of ordinals contains a smallest ordinal. Consequently, one can prove that a statement $\varphi(\alpha)$ holds for all ordinals $\alpha$ by proving that it admits of no least counterexample; in other words, one need only prove that whenever $\varphi(\beta)$ holds for all $\beta\lt\alpha$, then $\varphi(\alpha)$ holds. It follows that it holds for all ordinals, since there can be no least failure. It is sometimes convenient to break the transfinite inductive argument into cases, by proving that $\varphi(0)$ holds, that $\varphi(\alpha)\to\varphi(\alpha+1)$ and that $[\forall\beta\lt\lambda\ \varphi(\beta)]\to \varphi(\lambda)$, when $\lambda$ is a limit ordinal.

Transfinite recursion

Transfinite recursion is a method of constructing a well-ordered sequence of objects $a_\alpha$, by specifying how $a_\alpha$ is constructed, assuming one has already constructed $a_\beta$ for $\beta\lt\alpha$.

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