Lower attic
From Cantor's Attic
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
- $\omega$, the smallest infinity
- the sub-attic, containing very large finite numbers