Lower 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
  • $\Sigma$ = the supremum of the accidentally writable ordinals
  • $\zeta$ = the supremum of the eventually writable ordinals
  • $\lambda$ = the supremum of the writable ordinals,
• $\omega_1^x$
• 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