The lower attic

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,
• 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
  • $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$ = the supremum of the game values for white of all positions in infinite chess
  • $\omega_1^{\mathfrak{Ch},c}$ = the supremum of the game values for white of the computable positions in infinite chess
  • $\omega_1^{\mathfrak{Ch}}$ = the supremum of the game values for white of the finite positions in infinite chess
• the Takeuti-Feferman-Buchholz ordinal
• the Bachmann-Howard ordinal
• the large Veblen ordinal
• the small Veblen ordinal
• the Extended Veblen function
• the Feferman-Schütte ordinal $\Gamma_0$
• $\epsilon_0$ and the hierarchy of $\epsilon_\alpha$ numbers
• indecomposable ordinal
• the small countable ordinals, such as $\omega,\omega+1,\ldots,\omega\cdot 2,\ldots,\omega^2,\ldots,\omega^\omega,\ldots,\omega^{\omega^\omega},\ldots$ up to $\epsilon_0$
• Hilbert’s hotel and other toys in the playroom
• $\omega$, the smallest infinity
• down to the parlour, where large finite numbers dream