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

The *Church-Kleene* ordinal $\omega_1^{ck}$ is the supremum of the
computable ordinals, where an ordinal $\alpha$ is *computable* if there
is a computable relation $\lhd$ on $\mathbb{N}$ of order type
$\alpha$, that is, such that
$\langle\alpha,\lt\rangle\cong\langle\mathbb{N},\lhd\rangle$.

This ordinal is closed under all of the elementary ordinal arithmetic operations, such as successor, addition, multiplication and exponentiation.

The Church-Kleene ordinal is the least admissible ordinal.

The Church-Kleene idea easily relativizes to oracles, where for any real $x$, we define $\omega_1^x$ to be the supremum of the $x$-computable ordinals. This is also the least admissible ordinal relative to $x$, and every countable successor admissible ordinal is $\omega_1^x$ for some $x$.

```
This article is a stub. Please help us to improve Cantor's Attic by adding information.
```