Difference between revisions of "Lower attic"
From Cantor's Attic
| Line 12: | Line 12: | ||
* [[admissible]] ordinals and [[Church-Kleene#relativized Church-Kleene ordinal | relativized Church-Kleene $\omega_1^x$]] | * [[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 | * [[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 [[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 [[large Veblen]] ordinal | ||
| Line 17: | Line 21: | ||
* the [[Feferman-Schütte]] ordinal [[Feferman-Schütte | $\Gamma_0$]] | * the [[Feferman-Schütte]] ordinal [[Feferman-Schütte | $\Gamma_0$]] | ||
* [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]] | * [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]] | ||
| − | |||
| − | |||
| − | |||
| − | |||
* [[indecomposable]] ordinal | * [[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 | $\epsilon_0$]] | * 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 | $\epsilon_0$]] | ||
Revision as of 22:03, 28 January 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 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
