Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an eternal self-similar reflecting ascent. - <a href="Aleph_one" class="mw-redirect" title="Aleph one">$\omega_1lt;/a>, the first uncountable ordinal, and the other uncountable cardinals of the [](Middle_attic.md) - [stable](Stable.md "Stable") ordinals - The ordinals of [](Infinite_time_Turing_machines.md), including - [$\Sigma$](Infinite_time_Turing_machines.md#Sigma "Infinite time Turing machines") = the supremum of the accidentally writable ordinals - [$\zeta$](Infinite_time_Turing_machines.md#zeta "Infinite time Turing machines") = the supremum of the eventually writable ordinals - [$\lambda$](Infinite_time_Turing_machines.md#lambda "Infinite time Turing machines") = the supremum of the writable ordinals, - [admissible](Admissible%20ordinal.md "Admissible") ordinals and [](Church-Kleene.md#relativized_Church-Kleene_ordinal) - [](Church-Kleene.md), the supremum of the computable ordinals - the [](Omega_one_chess.md) - [$\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$](Omega_one_chess.md "Omega one chess") = the supremum of the game values for white of all positions in infinite chess - [$\omega_1^{\mathfrak{Ch},c}$](Omega_one_chess.md "Omega one chess") = the supremum of the game values for white of the computable positions in infinite chess - [$\omega_1^{\mathfrak{Ch}}$](Omega_one_chess.md "Omega one chess") = the supremum of the game values for white of the finite positions in infinite chess - the [Takeuti-Feferman-Buchholz](Buchholz's_ψ_functions.md#Takeuti-Feferman-Buchholz_ordinal "Buchholz's ψ functions") ordinal - the [Bachmann-Howard](Madore's_ψ_function.md#Bachmann-Howard_ordinal "Madore's ψ function") ordinal - the [](Madore's_ψ_function.md#Large_Veblen_ordinal) ordinal - the [](Madore's_ψ_function.md#Small_Veblen_ordinal) ordinal - the [](Extended_Veblen_function.md) - the [Feferman-Schütte](Feferman-Schütte.md "Feferman-Schütte") ordinal [$\Gamma_0$](Feferman-Schütte.md "Feferman-Schütte") - [$\epsilon_0$](Epsilon_naught.md "Epsilon naught") and the hierarchy of [](Epsilon_naught.md#epsilon_numbers) - <a href="Indecomposable" class="mw-redirect" title="Indecomposable">indecomposable</a> ordinal - the [](Small_countable_ordinals.md), such as [](Small_countable_ordinals.md) up to [$\epsilon_0$](Epsilon_naught.md "Epsilon naught") - [](Playroom.md#Hilbert.27s_Grand_Hotel) and other toys in the [playroom](Playroom.md "Playroom") - [$\omega$](Omega.md "Omega"), the smallest infinity - down to the [parlour](Parlour.md "Parlour"), where large finite numbers dream