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{{DISPLAYTITLE: The lower attic}}
[[File:SagradaSpiralByDavidNikonvscanon.jpg | thumb | Sagrada Spiral photo by David Nikonvscanon]]
[[Category: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.

*  [[Aleph#Aleph_one| $\omega_1$]], the first uncountable ordinal, and the other uncountable cardinals of the [[middle attic]]
* [[stable]] ordinals
* [[Heights of models]], [[Skolem's paradox]] <!--(ZFC without powerset axiom) is much above $\Sigma_n$-admissible, much below ZFC (stable ordinals as part of ZFC have no consistency strength)-->
* The ordinals of [[infinite time Turing machines]], including  
** [[infinite time Turing machines#Sigma | $\Sigma$]] = the supremum of the accidentally writable ordinals,
** [[infinite time Turing machines#zeta | $\zeta$]] = the supremum of the eventually writable ordinals, 
** [[infinite time Turing machines#lambda | $\lambda$]] = the supremum of the writable ordinals
* [[bad]] ordinals
* [[reflecting ordinal|reflecting]] ordinals
* [[admissible]] ordinals and [[Church-Kleene#relativized Church-Kleene ordinal | relativized Church-Kleene $\omega_1^x$]]
* [[Church-Kleene | Church-Kleene $\omega_1^{ck}$]], the supremum of the computable ordinals
* the [[omega one chess | omega one of chess]]
** [[omega one chess|$\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$]] = the supremum of the game values for white of all positions in infinite chess
** [[omega one chess| $\omega_1^{\mathfrak{Ch},c}$]] = the supremum of the game values for white of the computable positions in infinite chess 
** [[omega one chess| $\omega_1^{\mathfrak{Ch}}$]] = the supremum of the game values for white of the finite positions in infinite chess
* the [[Buchholz's ψ functions#Takeuti-Feferman-Buchholz ordinal|Takeuti-Feferman-Buchholz]] ordinal
* the [[Madore's ψ function#Bachmann-Howard ordinal|Bachmann-Howard]] ordinal
* the [[Madore's ψ function#Large Veblen ordinal|large Veblen]] ordinal
* the [[Madore's ψ function#Small Veblen ordinal|small Veblen]] ordinal
* the [[Extended Veblen function]]
* the [[Feferman-Schütte]] ordinal [[Feferman-Schütte | $\Gamma_0$]]
* [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]]
* [[Limit_ordinal#Types_of_Limits|indecomposable]] ordinal
* 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 | $\epsilon_0$]] 
* [[Playroom#Hilbert's Grand Hotel | Hilbert's hotel]] and other toys in the [[playroom]]
* [[omega | $\omega$]], the smallest infinity
* down to the [[parlour]], where large finite numbers dream