# Difference between revisions of "Lower attic"

From Cantor's Attic

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* [[Hilberts hotel | Hilbert's hotel]] | * [[Hilberts hotel | Hilbert's hotel]] | ||

* [[omega | $\omega$]], the smallest infinity | * [[omega | $\omega$]], the smallest infinity | ||

− | * down to the [[subattic]], containing very large finite numbers | + | * $\searrow$ down to the [[subattic]], containing very large finite numbers |

## Revision as of 09:24, 28 December 2011

Welcome to the lower attic, where we store the comparatively smaller notions of infinity. Roughly speaking, this is the realm of countable ordinals and their friends.

- $\omega_1$, the first uncountable ordinal, and the other uncountable cardinals of the middle attic
- The ordinals of infinite time Turing machines, including
- $\omega_1^x$
- admissible ordinals
- $\Gamma$
- Church-Kleene $\omega_1^{ck}$, the supremum of the computable ordinals
- $\epsilon_0$ and the hierarchy of $\epsilon_\alpha$ numbers
- the small countable ordinals, those below $\epsilon_0$
- Hilbert's hotel
- $\omega$, the smallest infinity
- $\searrow$ down to the subattic, containing very large finite numbers