Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.
View the Project on GitHub neugierde/cantors-attic
Quick navigation
The upper attic
The middle attic
The lower attic
The parlour
The playroom
The library
The cellar
Sources
Cantor's Attic (original site)
Joel David Hamkins blog post about the Attic
Latest working snapshot at the wayback machine
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 $\mathrm{OR}$.
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 $\alpha+1$.
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 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 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.