Ordinal numbers describe the way a set might be arranged into a [well-ordered](Ordering_Relations.md "Ordering Relations") sequence. Thus, ordinals have to do with the way a set is or can be ordered, rather than its size or [cardinality](Cardinal.md "Cardinal"). An elegant formulation of the ordinal concept in ZFC was provided by von Neumann: an *ordinal* is simply a [transitive](Transitive.md "Transitive") set [well-ordered](Ordering_Relations.md "Ordering Relations") 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 $\beta\lt\alpha$. This article is a stub. Please help us to improve Cantor's Attic by adding information.