# 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 the countably infinite ordinals climb ever higher, one upon another, in eternal self-similar reflecting ascent | + | Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in eternal self-similar reflecting ascent with endless enrapturing beauty. |

− | + | ||

* [[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]] |

## Revision as of 20:47, 28 December 2011

Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in eternal self-similar reflecting ascent with endless enrapturing beauty.

- $\omega_1$, 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
- $\omega+1$, $\omega\cdot 2$, $\omega^3$, $\omega^{\omega^\omega}$ and the other small countable ordinals below $\epsilon_0$
- Hilbert's hotel
- $\omega$, the smallest infinity
- down to the subattic, containing very large finite numbers