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
