# Difference between revisions of "Lower attic"

From Cantor's Attic

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* [[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]] | ||

− | * | + | * [[small countable ordinals | $\omega+1$]], [[small countable ordinals | $\omega\cdot 2$]], [[small countable ordinals | $\omega^3$]], [[small countable ordinals | $\omega^{\omega^\omega}$]] and the other [[small countable ordinals]] below [[epsilon naught | $\epsilon_0$]] |

* [[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 | * down to the [[subattic]], containing very large finite numbers |

## Revision as of 20:29, 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 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
- $\omega+1$, $\omega\cdot 2$, $\omega^3$, $\omega^{\omega^\omega}$ 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