# Difference between revisions of "Lower attic"

From Cantor's Attic

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− | [[File:SagradaSpiralByDavidNikonvscanon.jpg | | + | {{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 | + | 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 | ||

+ | * [[Heights of models]] <!--(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 | * 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#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#zeta | $\zeta$]] = the supremum of the eventually writable ordinals, |

− | ** [[infinite time Turing machines#lambda | $\lambda$]] = the supremum of the writable ordinals | + | ** [[infinite time Turing machines#lambda | $\lambda$]] = the supremum of the writable ordinals |

− | * [[admissible# | + | * [[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]] | * [[epsilon naught | $\epsilon_0$]] and the hierarchy of [[epsilon naught#epsilon_numbers | $\epsilon_\alpha$ numbers]] | ||

− | * the [[small countable ordinals]], | + | * [[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 | * [[omega | $\omega$]], the smallest infinity | ||

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

## Revision as of 17:22, 22 September 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
- Heights of models
- The ordinals of infinite time Turing machines, including
- bad ordinals
- reflecting ordinals
- 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