# Lower attic

From Cantor's Attic

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