Difference between revisions of "Lower attic"
From Cantor's Attic
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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. | 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. | ||
| − | * | + | * [[aleph_1 | $\omega_1$]], the first uncountable ordinal, and the other uncountable cardinals of [[the middle attic]] |
| − | + | ||
* The ordinals of [[infinite time Turing machines]], including | * The ordinals of [[infinite time Turing machines]], including | ||
** [[infinite time Turing machines#Sigma | $\Sigma$]] = the supremum of the accidentally writable ordinals | ** [[infinite time Turing machines#Sigma | $\Sigma$]] = the supremum of the accidentally writable ordinals | ||
Revision as of 07:53, 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
- $\omega$
