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

A successor ordinal, by definition, is an ordinal $\alpha$ that is equal to $\beta + 1$ for some ordinal $\beta$.

The successor function, denoted as $\beta+1$, $\operatorname{suc} \beta$, or $\beta ^+$ is defined on the ordinals as $\beta \cup \{ \beta \}$

There are no ordinals between $\beta$ and $\beta + 1$.

The union $\bigcup \beta$ can be thought of as an inverse successor function, because $\bigcup ( \beta + 1 ) = \beta$. All limit ordinals are equal to their union.