# Difference between revisions of "Lower attic"

From Cantor's Attic

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Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an eternal self-similar reflecting ascent. | 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 | * [[stable]] ordinals | ||

* models of [[ZFC-P]] (ZFC without powerset axiom)<!--much above $\Sigma_n$-admissible, much below ZFC (stable ordinals as part of ZFC have no consistency strength)--> | * models of [[ZFC-P]] (ZFC without powerset axiom)<!--much above $\Sigma_n$-admissible, much below ZFC (stable ordinals as part of ZFC have no consistency strength)--> | ||

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* 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 | + | * [[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 | $\epsilon_0$]] | * 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]] | * [[Playroom#Hilbert's Grand Hotel | Hilbert's hotel]] and other toys in the [[playroom]] | ||

* [[omega | $\omega$]], the smallest infinity | * [[omega | $\omega$]], the smallest infinity | ||

* down to the [[parlour]], where large finite numbers dream | * down to the [[parlour]], where large finite numbers dream |

## Revision as of 16:57, 30 July 2021

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
- models of ZFC-P (ZFC without powerset axiom)
- 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 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