Arranged in order of size, comparisons proven using ZFC+some sufficiently strong large cardinal axiom. Some names are changed, such as "recursively hyperinaccessible" -> "recursively 1-inaccessible", and "γ-recursively Mahlo" -> "recursively γ-Mahlo".
| Ordinal | Properties/Notes | Model properties |
|---|---|---|
| Any recursive ordinal (e.g. ψ₀(Ω_ω)) | ||
| ω₁^CK | Least nonrecursive ordinal Admissible and >ω, i.e. Π₂-reflecting [RichterAczel74] |
Min α with L_α |= KPω |
| ω_ω^CK | Least limit of admissibles This ordinal is not admissible [Zoo] |
Min α with L_α ∩ P(ω) |= Π₁^1-CA₀ [Zoo] |
| Fixed point of α |-> ω_α^CK | ||
| Least recursively inaccessible | Least admissible that's also limit of admissibles [Zoo] This α is the αth admissible [Barwise] |
Min α with L_α |= KPi [Zoo] Min α with L_α ∩ P(ω) |= Δ₂^1-CA₀ [Zoo] |
| Least recursively 1-inaccessible | Least recursively inaccessible that's also limit of recursively inaccessibles [Zoo] | |
| Least recursively ω-inaccessible | Recursively n-inaccessible for all n<ω | |
| Least rec. hyper-inaccessible | α that is rec. α-inaccessible | |
| Least rec. Mahlo | Least Π₂-reflecting on Π₂-reflecting ordinals [RichterAczel74, thm. 1.9] Taranovsky claims C(Ω^2,0) in his Degrees of Reflection behaves like this ordinal [Taranovsky, section 4.3] |
Min α with L_α |= KPM [Zoo] |
| Least rec. 1-Mahlo | Least Π₂-reflecting on rec. Mahlos [RichterAczel74, thm. 1.9] Taranovsky claims C(Z+d^3,0) in Degrees of Reflection behaves like this |
|
| Least rec. 2-Mahlo | Least Π₂-reflecting on rec. 1-Mahlos [RichterAczel74] | |
| Least rec. ω-Mahlo | Rec. n-Mahlo for all n<ω Taranovsky claims C(Ω^(ω+1),0) behaves like this |
|
| Least Π₃-reflecting | This α is rec. α-Mahlo, rec. α-hyper-Mahlo, etc. [RichterAczel74] Least 2-admissible (defined using functionals) [RichterAczel74] Taranovsky claims C(Ω^Ω,0) in Degrees of Reflection behaves like this |
Min α with L_α |= KP+Π₃-reflection schema [Rathjen] |
| Least Π₂-reflecting on Π₃-reflectings | Madore says structure here is important [Comment by Madore] Taranovsky claims C(Ω^(Ω+1),0) [Taranovsky, section 4.3] Duchhardt calls these ordinals "schizophrenic" [Stegert] |
|
| Least Π₃-reflecting that's also Π₂-reflecting on Π₃-reflecting ordinals | Taranovsky says C(Ω^(Ω*2),0) [Taranovsky, section 4.3] | |
| Least Π₃-reflecting on Π₃-reflectings | Taranovsky claims C(Ω^Ω^2,0) [Taranovsky, section 4.3] 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 claims C(Ω^Ω^(ω+1)) |
|
| Least Π₄-reflecting | No groundbreaking structure between here and least Π₅-rfl. [Madore blog post] Taranovsky claims C(Ω^Ω^Ω,0) [Taranovsky, section 4.3] |
Min α with L_α |= KP+Π₄-rfl. schema [Rathjen] |
| Least Π₅-reflecting | Taranovsky claims C(Ω^Ω^Ω^Ω,0) [Taranovsky, section 4.3] | Min α with L_α |= KP+Π₅-rfl. schema [Rathjen] |
| Least ordinal that's for all n<ω a limit of the Π_n-rfl. ordinals | (Some structure from here to the next ordinal) | |
| Least (+1)-stb. | Π_n-rfl. for all n<ω [RichterAczel74] | |
| Least (^+)-stb | This ordinal is Π_1^1-rfl. (Levy hierarchy) [RichterAczel74] | |
| Least Π_1^1-rfl on Π_1^1-rfl. [OrderOfReflection] | ||
| Least Σ_1^1-rfl | This ordinal α is the smallest admissible non-Gandy [Zoo] α is (|α-rec.|)-stb. but not (|α-rec.|+1)-stb. [OrderOfReflection] α is Σ_1^1-rfl on Π_1^1-rfl ordinals [OrderOfReflection] |
|
| Least Π_1^1-rfl on Σ_1^1-rfl. [OrderOfReflection] | ||
| Least Σ_1^1-rfl on Σ_1^1-rfl. [OrderOfReflection] | ||
| Least (^++)-stb. | Sup of closure ordinals of [Π_1^1,Π_1^1] inductive defintions (Cenzer, "Ordinal Recursion and Inductive Definitions", 1974) | |
| Insert important ordinals from User:C7X/Stability list here | ||
| Least Σ₂-admissible | Least α that's Π₃-rfl. on {β∈α|β is α-stable} [Kranakis] | Min α with L_α |= KP+Σ₂-collection |
| ITTM ordinal ζ | This is least α for which there exists β>α such that α is (β,2)-stable [any "λζΣ-theorem" paper] | |
| Welch's E₀-ordinals | (β,2)-stable and limit of (γ,2)-stables below [Welch] | |
| Least α where L_α ≺_Σ₃ L_β for some β [Welch] | ||
| Put Welch stuff here | ||
| Least Σ₃-admissible | Least α that's Π₄-rfl. on {β∈α|β is (α,2)-stable} [Kranakis] | Min α with L_α |= KP+Σ₃-collection |
| Least Σ₄-admissible | Least α that's Π₅-rfl. on {β∈α|β is (α,3)-stable} [Kranakis] | Min α with L_α |= KP+Σ₄-collection |
| Least gap ordinal | Σ_n-admissible for all n<ω | Min β with L_β ∩ P(ω) a β-model of Z₂ [Gaps in the constructible universe] Min β with L_β |= ZFC⁻+"V=HC" [Gaps in the constructible universe] |
| Least morass point | α is limit of {β|L_β |= ZF-Powerset} and L_α satisfies "there is exactly one uncountable cardinal" [Simplified Constructibility Theory, p. 21] | NOTE: Uncertain placement, should this be moved down the table? |
| Least start of gap of length 2 | This is a limit of gap ordinals [Gaps in the constructible universe] | |
| β that starts a gap of length β | Exists according to [Gaps in the constructible universe] | |
| β that starts gap of length β^β | Exists and is mentioned in a corollary of [Gaps in the constructible universe] | |
| β that starts gap of length β⁺ | L_(β⁺) satisfies "β is uncountable" [Arai] | Min β with L_(β⁺) |= KP+"ω₁ exists" [Arai] |
| Least start of third-order gap | If α is the previous ordinal this is >α⁺ [Googology Wiki] | Min β with L_β |= ZFC⁻+"beth₁ exists"+"V=H_beth₁" [Gaps in the constructible universe, we take image of P(ω) under bijection from P(ω) to beth₁ that exists by choice] |
| Least start of fourth-order gap | Least height of model of ZFC⁻+"beth₂ exists"+"V=H_beth₂" [Gpas in the constructible universe, same trick as above row?] | |
| 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" | ||
| Least stable | Size placed by [Marek, Rasmussen, "Spectrum of L"] This is a limit of gap ordinals [Marek, Oct 1973] |
|
| Least stable that's also during a gap | If this ordinal is α then it's the αth stable ordinal [Marek, Oct 1973] | Height of least β₂-model of Z₂ [Marek, Oct 1973] |
| The ordinals α such that L_α is Σ_n-elementary-substructure of L_ω₁ when 1<n<ω. | I hazard a guess that these are related to heights of β_(n-1)-models of Z₂ [maybe Marek, Oct 1973, by extending theorem 2.2?]. | |
| The least non-analytical ordinal | This is least α such that L_α ≺ L_ω₁ [Marek, Rasmussen, "Spectrum of L"] |
Past here may have to ask others who know more about indiscernibles than me (Silver indiscernibles are large)
The least α such that L_α is power-admissible should also be in this list at least as far down as Σ₂-admissible [Marek, Aug 1975], but IDK where it is exactly