# Difference between revisions of "Lower attic"

From Cantor's Attic

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[[File:SagradaSpiralByDavidNikonvscanon.jpg | right | Sagrada Spiral photo by David Nikonvscanon]] | [[File:SagradaSpiralByDavidNikonvscanon.jpg | right | Sagrada Spiral photo by David Nikonvscanon]] | ||

− | Welcome to the lower attic, where | + | Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an endlessly self-similar reflecting ascent, whose life-giving everlasting beauty enraptures us. |

− | + | ||

* [[aleph_1 | $\omega_1$]], the first uncountable ordinal, and the other uncountable cardinals of the [[middle attic]] | * [[aleph_1 | $\omega_1$]], the first uncountable ordinal, and the other uncountable cardinals of the [[middle attic]] | ||

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* [[admissible#relativized_admissible | $\omega_1^x$]] | * [[admissible#relativized_admissible | $\omega_1^x$]] | ||

* [[admissible]] ordinals | * [[admissible]] ordinals | ||

+ | * Church-Kleene [[Church-Kleene omega_1 | $\omega_1^{ck}$]], the supremum of the computable ordinals | ||

* [[Gamma | $\Gamma$]] | * [[Gamma | $\Gamma$]] | ||

− | |||

* [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]] | * [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]] | ||

* the [[small countable ordinals]], those below [[epsilon naught | $\epsilon_0$]] | * the [[small countable ordinals]], those below [[epsilon naught | $\epsilon_0$]] |

## Revision as of 20:24, 28 December 2011

Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an endlessly self-similar reflecting ascent, whose life-giving everlasting beauty enraptures us.

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