Difference between revisions of "User:C7X/Some ordinals"
(Created page with "Arranged in order of their size in a "reasonable" model. Brackets contain citations ω₁^CK  admissible  Π₂rfl. [RichterAczel74] ω\_ω^CK  limit of the admissibles...") 
(No difference)

Revision as of 19:49, 4 December 2021
Arranged in order of their size in a "reasonable" model. Brackets contain citations
ω₁^CK  admissible  Π₂rfl. [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. 1inacc. (in Taranovsky's name, Madore calls this rec. hyperinacc.)
Least rec. ωinacc.  rec. ninacc. for all n<ω
Least rec. hyperinacc. (α that is rec. αinacc.)  Hyp\_cos says C(Z+*d*^C(Z\*2,0),0) in Taranovsky's notation behaves like this in collapse
Least rec. Mahlo  Π₂rfl. on the set of Π₂rfl. ordinals below it  Taranovsky says C(Z+*d*^2,0) behaves like this in collapse
Least rec. 1Mahlo  Π₂rfl. on the set of rec. Mahlos  Taranovsky says C(Z+*d*^3,0) behaves like this in collapse
Least rec. ωMahlo  rec. nMahlo for all n<ω  Taranovsky says C(Z+*d*^(ω+1),0)
Least Π₃rfl.  Taranovsky says C(Z+*d*^*d*,0) behaves like this in collapse
Least Π₂rfl. on the set of Π₃rfl. 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 Π₃rfl. that's Π₂rfl. on the Π₃rfl.s  Taranovsky says C(Z+*d*^(*d*\*2),0)
Least Π₃rfl. on Π₃rfl.s  Taranovsky says C(Z+*d*^(*d*^2),0)  Arai has studied this case, it relates to iterating thinning operators for reflection along lexicographic orderings [Arai2010]
Least "Π₃rfl. on Π₃rfl. on Π₃rfl. on ..." (length ω)  Richter and Aczel proved we can iterate Π\_nreflection quite far before reaching Π\_(n+1)reflection [RichterAczel74]  Taranovsky says C(Z+*d*^*d*^(ω+1))
Least Π₄rfl.  Taranovsky says C(Z+*d*^*d*^*d*,0)  Duchhardt analyzed KP+Π₄rfl. in 2008 and has an OCF collapsing this ordinal, IDK how it works
Least Π₅rfl.  Taranovsky says C(Z+*d*^*d*^*d*^*d*,0)
Limit of "least Π\_nrfl." for n<ω  Some important structure from here to the next ordinal
Least (+1)stb.  Π\_nrfl. for all n<ω
Insert important ordinals from User:C7X/Stability list here
Least α that's Π₂rfl. on {β∈αβ is αstable}  for some results about this see User:C7X/Stability
Least Σ₂admissible  least α that's Π₃rfl. on {β∈αβ is αstable} [2]
Least Σ₃admissible  least α that's Π₄rfl. on {β∈αβ is (α,2)stable} [above source]
Least Σ₄admissible  least α that's Π₅rfl. on {β∈αβ is (α,3)stable} [above]
Least gap ordinal  Σ\_nadmissible for all n<ω  least height of βmodel 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 [3]
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" [4]
Least start of thirdorder 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) [5]. If α is the previous ordinal this is >α⁺ [6]
Least start of fourthorder 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 [7]
Least stable that's also during a gap  height of least β₂model of Z₂ [8], if this ordinal is α then it's the αth stable ordinal [same paper]
Past here is some stuff that Rathjen wrote a bit about, the ordinals α such that L\_α is Σ\_nelementarysubstructure of L when 1<n<ω. I hazard a guess that these are related to heights of β\_(n1)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 poweradmissible should also be in this list at least as far down as Σ₂admissible [9], but IDK where it is exactly