The lower attic

Sagrada Spiral photo by David Nikonvscanon

Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an eternal self-similar reflecting ascent.

• $\omega_1$, the first uncountable ordinal, and the other uncountable cardinals of the middle attic
• stable ordinals
• The ordinals of infinite time Turing machines, including
  • $\Sigma$ = the supremum of the accidentally writable ordinals
  • $\zeta$ = the supremum of the eventually writable ordinals
  • $\lambda$ = the supremum of the writable ordinals,
• the Bachmann-Howard ordinal
• admissible ordinals and relativized Church-Kleene $\omega_1^x$
• Church-Kleene $\omega_1^{ck}$, the supremum of the computable ordinals
• the omega one of chess $\omega_1^{\rm chess}]] * the [[Feferman-Schütte]] ordinal [[Feferman-Schütte | $\Gamma_0$]] * [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]] * the [[small countable ordinals]], such as [[small countable ordinals | $\omega,\omega+1,\ldots,\omega\cdot 2,\ldots,\omega^2,\ldots,\omega^\omega,\ldots,\omega^{\omega^\omega},\ldots$]] up to [[epsilon naught | $\epsilon_0$]] * [[Hilberts hotel | Hilbert's hotel]] * [[omega | $\omega$, the smallest infinity
• down to the parlour, where large finite numbers dream