# Difference between revisions of "Lower attic"

From Cantor's Attic

m |
|||

Line 17: | Line 17: | ||

** [[omega one chess| $\omega_1^{\mathfrak{Ch}}$]] = the supremum of the game values for white of the finite 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 [[Madore's ψ function#Bachmann-Howard ordinal|Bachmann-Howard]] ordinal | * the [[Madore's ψ function#Bachmann-Howard ordinal|Bachmann-Howard]] ordinal | ||

− | * the [[large Veblen]] ordinal | + | * the [[Madore's ψ function#Large Veblen ordinal|large Veblen]] ordinal |

− | * the [[small Veblen]] ordinal | + | * the [[Madore's ψ function#Small Veblen ordinal|small Veblen]] ordinal |

* the [[Extended Veblen function]] | * the [[Extended Veblen function]] | ||

* the [[Feferman-Schütte]] ordinal [[Feferman-Schütte | $\Gamma_0$]] | * the [[Feferman-Schütte]] ordinal [[Feferman-Schütte | $\Gamma_0$]] |

## Latest revision as of 04:14, 15 April 2017

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
- 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 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