# Difference between revisions of "User:C7X/Some ordinals"

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Arranged in order of their size in a "reasonable" model. Brackets contain citations | Arranged in order of their size in a "reasonable" model. Brackets contain citations | ||

− | ω₁ | + | ω₁<sup>CK</sup> - admissible - Π₂-rfl. [RichterAczel74] |

− | ω | + | ω<sub>ω</sub><sup>CK</sup> - limit of the admissibles below it, not admissible [Zoo of ordinals] |

CK fixed point | CK fixed point | ||

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Least rec. inacc. - admissible and limit of admissibles | Least rec. inacc. - admissible and limit of admissibles | ||

− | Least rec. 1-inacc. (in Taranovsky' | + | Least rec. 1-inacc. (in Taranovsky's name, Madore calls this rec. hyper-inacc.) |

Least rec. ω-inacc. - rec. n-inacc. for all n<ω | Least rec. ω-inacc. - rec. n-inacc. for all n<ω | ||

− | Least rec. hyper-inacc. (α that is rec. α-inacc.) - | + | Least rec. hyper-inacc. (α that is rec. α-inacc.) - Hyp_cos says C(Z+<i>d</i>^C(Z*2,0),0) in Taranovsky's notation behaves like this in collapse |

− | Least rec. Mahlo - Π₂-rfl. on the set of Π₂-rfl. ordinals below it - Taranovsky says C(Z+ | + | Least rec. Mahlo - Π₂-rfl. on the set of Π₂-rfl. ordinals below it - Taranovsky says C(Z+<i>d</i><sup>2</sup>,0) behaves like this in collapse |

− | Least rec. 1-Mahlo - Π₂-rfl. on the set of rec. Mahlos - Taranovsky says C(Z+ | + | Least rec. 1-Mahlo - Π₂-rfl. on the set of rec. Mahlos - Taranovsky says C(Z+<i>d</i><sup>3</sup>,0) behaves like this in collapse |

− | Least rec. ω-Mahlo - rec. n-Mahlo for all n<ω - Taranovsky says C(Z+ | + | Least rec. ω-Mahlo - rec. n-Mahlo for all n<ω - Taranovsky says C(Z+<i>d</i><sup>ω+1</sup>,0) |

− | Least Π₃-rfl. - Taranovsky says C(Z+ | + | Least Π₃-rfl. - Taranovsky says C(Z+<i>d</i>^<i>d</i>,0) behaves like this in collapse |

− | Least Π₂-rfl. on the set of Π₃-rfl. ordinals below - Taranovsky says C(Z+ | + | Least Π₂-rfl. on the set of Π₃-rfl. ordinals below - Taranovsky says C(Z+<i>d</i><sup><i>d</i>+1</sup>,0) - Madore says to watch out for this type of reflection and not miss its existence [https://bit.ly/3q88SY9] |

− | Least Π₃-rfl. that's Π₂-rfl. on the Π₃-rfl.s - Taranovsky says C(Z+ | + | Least Π₃-rfl. that's Π₂-rfl. on the Π₃-rfl.s - Taranovsky says C(Z+<i>d</i><sup><i>d</i>*2</sup>,0) |

− | Least Π₃-rfl. on Π₃-rfl.s - Taranovsky says C(Z+ | + | Least Π₃-rfl. on Π₃-rfl.s - Taranovsky says C(Z+<i>d</i><sup><i>d</i><sup>2</sup></sup>,0) - Arai has studied this case, it relates to iterating thinning operators for reflection along lexicographic orderings [Arai2010] |

− | Least "Π₃-rfl. on Π₃-rfl. on Π₃-rfl. on ..." (length ω) - Richter and Aczel proved we can iterate Π | + | Least "Π₃-rfl. on Π₃-rfl. on Π₃-rfl. on ..." (length ω) - Richter and Aczel proved we can iterate Π<sub>n</sub>-reflection quite far before reaching Π<sub>n+1</sub>-reflection [RichterAczel74] - Taranovsky says C(Z+<i>d</i>^<i>d</i>^(ω+1)) |

− | Least Π₄-rfl. - Taranovsky says C(Z+ | + | Least Π₄-rfl. - Taranovsky says C(Z+<i>d</i><sup><i>d</i><sup><i>d</i></sup></sup>,0) - Duchhardt analyzed KP+Π₄-rfl. in 2008 and has an OCF collapsing this ordinal, IDK how it works |

− | Least Π₅-rfl. - Taranovsky says C(Z+ | + | Least Π₅-rfl. - Taranovsky says C(Z+<i>d</i><sup><i>d</i><sup><i>d</i><sup><i>d</i></sup></sup></sup>,0) |

− | Limit of "least Π | + | Limit of "least Π<sub>n</sub>-rfl." for n<ω - Some important structure from here to the next ordinal |

− | Least (+1)-stb. - Π | + | Least (+1)-stb. - Π<sub>n</sub>-rfl. for all n<ω |

− | + | Least α that's Π₂-rfl. on {β∈α|β is α-stable} | |

− | Least α that's | + | Least Σ₂-admissible - least α that's Π₃-rfl. on {β∈α|β is α-stable} [https://doi.org/10.1016/0003-4843(82)90022-5] |

− | Least | + | Least Σ₃-admissible - least α that's Π₄-rfl. on {β∈α|β is (α,2)-stable} [https://doi.org/10.1016/0003-4843(82)90022-5] |

− | Least | + | Least Σ₄-admissible - least α that's Π₅-rfl. on {β∈α|β is (α,3)-stable} [https://doi.org/10.1016/0003-4843(82)90022-5] |

− | + | Least gap ordinal - Σ<sub>n</sub>-admissible for all n<ω - least height of β-model of Z₂ - least height of model of ZFC⁻+"V=HC" [Gaps in the constructible universe] | |

− | + | ||

− | Least gap ordinal - Σ | + | |

/<u>!</u>\ I know less about the remaining structure /<u>!</u>\ | /<u>!</u>\ I know less about the remaining structure /<u>!</u>\ | ||

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β that starts gap of length β⁺ - least height of model of KP+"ω₁ exists", this model satisfies "β is uncountable" [https://arxiv.org/abs/1102.0596] | β that starts gap of length β⁺ - least height of model of KP+"ω₁ exists", this model satisfies "β is uncountable" [https://arxiv.org/abs/1102.0596] | ||

− | Least start of third-order gap - least height of model of ZFC⁻+"beth₁ exists"+"V=H | + | Least start of third-order gap - least height of model of ZFC⁻+"beth₁ exists"+"V=H<sub>beth₁</sub>" (we take image of P(ω) under bijection from P(ω) to beth₁ that exists by choice) [https://core.ac.uk/download/pdf/81133582.pdf]. If α is the previous ordinal this is >α⁺ [https://googology.wikia.org/wiki/Gap_ordinal#Relative_sizes] |

− | Least start of fourth-order gap - least height of model of ZFC⁻+"beth₂ exists"+"V=H | + | Least start of fourth-order gap - least height of model of ZFC⁻+"beth₂ exists"+"V=H<sub>beth₂</sub>" (same trick)? [Alluded to in above source] |

− | Least height of model of ZFC⁻+"beth | + | Least height of model of ZFC⁻+"beth<sub>ω</sub> exists" |

− | Least height of model of ZFC⁻+"beth | + | Least height of model of ZFC⁻+"beth<sub>ω₁</sub> exists" |

Least height of model of ZFC⁻+"beth fixed point exists" | Least height of model of ZFC⁻+"beth fixed point exists" | ||

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Least stable [Size informally mentioned by Taranovsky], this is a limit of gap ordinals [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf] | Least stable [Size informally mentioned by Taranovsky], this is a limit of gap ordinals [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf] | ||

− | Least stable that's also during a gap - height of least β₂-model of Z₂ [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf], if this ordinal is α then it's the αth stable ordinal [ | + | Least stable that's also during a gap - height of least β₂-model of Z₂ [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf], if this ordinal is α then it's the αth stable ordinal [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf] |

− | Past here is some stuff that Rathjen wrote a bit about, the ordinals α such that L | + | Past here is some stuff that Rathjen wrote a bit about, the ordinals α such that L<sub>α</sub> is Σ<sub>n</sub>-elementary-substructure of L when 1<n<ω. I hazard a guess that these are related to heights of β<sub>n-1</sub>-models of Z₂ [maybe http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf, by extending theorem 2.2?]. |

− | The least α such that L | + | The least α such that L<sub>α</sub> is power-admissible should also be in this list at least as far down as Σ₂-admissible [http://matwbn.icm.edu.pl/ksiazki/fm/fm98/fm9818.pdf], but IDK where it is exactly |

## Latest revision as of 19:56, 4 December 2021

Arranged in order of their size in a "reasonable" model. Brackets contain citations

ω₁^{CK} - admissible - Π₂-rfl. [RichterAczel74]

ω_{ω}^{CK} - limit of the admissibles below it, not admissible [Zoo of ordinals]

CK fixed point

Least rec. inacc. - admissible and limit of admissibles

Least rec. 1-inacc. (in Taranovsky's name, Madore calls this rec. hyper-inacc.)

Least rec. ω-inacc. - rec. n-inacc. for all n<ω

Least rec. hyper-inacc. (α that is rec. α-inacc.) - Hyp_cos says C(Z+*d*^C(Z*2,0),0) in Taranovsky's notation behaves like this in collapse

Least rec. Mahlo - Π₂-rfl. on the set of Π₂-rfl. ordinals below it - Taranovsky says C(Z+*d*^{2},0) behaves like this in collapse

Least rec. 1-Mahlo - Π₂-rfl. on the set of rec. Mahlos - Taranovsky says C(Z+*d*^{3},0) behaves like this in collapse

Least rec. ω-Mahlo - rec. n-Mahlo for all n<ω - Taranovsky says C(Z+*d*^{ω+1},0)

Least Π₃-rfl. - Taranovsky says C(Z+*d*^*d*,0) behaves like this in collapse

Least Π₂-rfl. on the set of Π₃-rfl. ordinals below - Taranovsky says C(Z+*d*^{d+1},0) - Madore says to watch out for this type of reflection and not miss its existence [1]

Least Π₃-rfl. that's Π₂-rfl. on the Π₃-rfl.s - Taranovsky says C(Z+*d*^{d*2},0)

Least Π₃-rfl. on Π₃-rfl.s - Taranovsky says C(Z+*d*^{d2},0) - Arai has studied this case, it relates to iterating thinning operators for reflection along lexicographic orderings [Arai2010]

Least "Π₃-rfl. on Π₃-rfl. on Π₃-rfl. on ..." (length ω) - Richter and Aczel proved we can iterate Π_{n}-reflection quite far before reaching Π_{n+1}-reflection [RichterAczel74] - Taranovsky says C(Z+*d*^*d*^(ω+1))

Least Π₄-rfl. - Taranovsky says C(Z+*d*^{dd},0) - Duchhardt analyzed KP+Π₄-rfl. in 2008 and has an OCF collapsing this ordinal, IDK how it works

Least Π₅-rfl. - Taranovsky says C(Z+*d*^{ddd},0)

Limit of "least Π_{n}-rfl." for n<ω - Some important structure from here to the next ordinal

Least (+1)-stb. - Π_{n}-rfl. for all n<ω

Least α that's Π₂-rfl. on {β∈α|β is α-stable}

Least Σ₂-admissible - least α that's Π₃-rfl. on {β∈α|β is α-stable} [2]

Least Σ₃-admissible - least α that's Π₄-rfl. on {β∈α|β is (α,2)-stable} [3]

Least Σ₄-admissible - least α that's Π₅-rfl. on {β∈α|β is (α,3)-stable} [4]

Least gap ordinal - Σ_{n}-admissible for all n<ω - least height of β-model of Z₂ - least height of model of ZFC⁻+"V=HC" [Gaps in the constructible universe]

/__!__\ I know less about the remaining structure /__!__\

Least gap of length 2 - this is a limit of gap ordinals [5]

β that starts a gap of length β - exists according to https://core.ac.uk/download/pdf/81133582.pdf

β that starts gap of length β^β - exists and is mentioned in a corollary of https://core.ac.uk/download/pdf/81133582.pdf

β that starts gap of length β⁺ - least height of model of KP+"ω₁ exists", this model satisfies "β is uncountable" [6]

Least start of third-order gap - least height of model of ZFC⁻+"beth₁ exists"+"V=H_{beth₁}" (we take image of P(ω) under bijection from P(ω) to beth₁ that exists by choice) [7]. If α is the previous ordinal this is >α⁺ [8]

Least start of fourth-order gap - least height of model of ZFC⁻+"beth₂ exists"+"V=H_{beth₂}" (same trick)? [Alluded to in above source]

Least height of model of ZFC⁻+"beth_{ω} exists"

Least height of model of ZFC⁻+"beth_{ω₁} exists"

Least height of model of ZFC⁻+"beth fixed point exists"

Least height of model of ZFC

Least height of model of ZFC+"inaccessible exists"

Least height of model of ZFC+"subtle exists" [Size informally mentioned by Taranovsky]

Least stable [Size informally mentioned by Taranovsky], this is a limit of gap ordinals [9]

Least stable that's also during a gap - height of least β₂-model of Z₂ [10], if this ordinal is α then it's the αth stable ordinal [11]

Past here is some stuff that Rathjen wrote a bit about, the ordinals α such that L_{α} is Σ_{n}-elementary-substructure of L when 1<n<ω. I hazard a guess that these are related to heights of β_{n-1}-models of Z₂ [maybe http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf, by extending theorem 2.2?].

The least α such that L_{α} is power-admissible should also be in this list at least as far down as Σ₂-admissible [12], but IDK where it is exactly