NOTE: Of course this page does not list all infinite numbers, because then it would be, well, infinitely long.

RED (Small countable ordinals) - ω to ψ(Ω^Ω)

ω - The first transfinite ordinal, the first limit ordinal, the first additively decomposable ordinal and the first admissible ordinal. It has a cardinality of א‎₀. Using ZFC, ω = N (the natural numbers). Occasionally referred to as γ₀.

ω + 1 - The first infinite successor ordinal. Equal to, of course, ω + 1! Although, confusingly, it still has a cardinality of א‎₀. In fact, all countably ordinals do. Using ZFC, ω + 1 = N ∪ {ω}.

2ω - The second limit ordinal. Equal to ω + ω. Using ZFC, 2ω = ⋃ (N ∪ {ω + (α - 1)}) ∀α ≤ ω.

ω² - The second additively indecomposable ordinal. Equal to ωω. Using ZFC, ω² = ⋃ (⋃ (N ∪ ω + (α - 1))) ∀α ≤ ω? Occasionally referred to as γ₁.

ω^ω - The first multiplicatively indecomposable ordinal. I'm not going to and probably couldn't if I tried to convert this into a ZFC set 😂. Occasionally referred to as δ₀.

ω^ω↑ω - The ωth multiplicatively indecomposable ordinal (not exponentially indecomposable though).

ε₀ - The first exponentially indecomposable ordinal. Equal to ω ↑↑ ω. The smallest number not definable using the one-argument Veblen function; the limit of the sequence ω, ω^ω, ω^ω↑ω, ...; the first fixed point of α → ω^α and the proof-theoretic ordinal of Peano arithmetic and ACA₀.

ε₁ - The second exponentially indecomposable ordinal. Equal to ε₀ ↑↑ ω. The second fixed point of α → ω^α.

ε_ω - The (ω + 1)th exponentially indecomposable ordinal. Equal to ε_{ω - 1} ↑↑ ω (although, because ω is a limit ordinal, ω-1 is undefined). The (ω + 1)th fixed point of α → ω^α.

ε_ε0 - The (ε₀)th exponentially indecomposable ordinal. The (ε₀ + 1)th fixed point of α → ω^α.

ζ₀ - The first tetrationally indecomposable ordinal. Equal to ε_εε...ε0 with ε₀ ε's. The limit of the sequence ε₀, ε_ε0, ε_εε0, ... and first fixed point of α → ε_α.

φ(ω, 0) - The first ↑^{(ω + 2)} indecomposable ordinal. The limit of the sequence φ(ω-1, 0), φ(ω-1, φ(ω-1, 0)), φ(ω-1, φ(ω-1, φ(ω-1, 0))), ... This is the smallest ordinal > ω closed under primitive recursive ordinal functions.

Feferman-Schütte ordinal (Γ₀) - The first fixed point of α → φ(α, 0). It is written as φ(1, 0, 0) using the Veblen function or ψ(Ω^Ω) using Buchholz's psi function. The smallest number not definable using the two-argument Veblen function; the limit of the sequence ε₀, φ(ε₀, 0), φ(φ(ε₀, 0), 0), ...; the proof-theoretic ordinal of arithmetic transfinite recursion and the last (largest) red-class ordinal using this scheme I created.

ORANGE (Large countable ordinals and first uncountable ordinal) - ψ(Ω^Ω + Ω) to Ω

Ackermann ordinal - φ(1, 0, 0, 0) in using the Veblen function or ψ(Ω^Ω↑2) using Buchholz's psi function. The smallest number not definable using the three-argument Veblen function.

"Small" Veblen ordinal - ψ(Ω^Ω↑ω) using Buchholz's psi function. The limit of φ(1, 0), φ(1, 0, 0), φ(1, 0, 0, 0), ...

"Large" Veblen ordinal - ψ(Ω^Ω↑Ω) using Buchholz's psi function.

Bachmann-Howard ordinal - ψ(ε_{Ω + 1}) using Buchholz's psi function. The proof-theoretic ordinal of Kripke-Platek set theory (KP).

Takeuti-Feferman-Buchholz ordinal - ψ(ε_{Ωω + 1}) using Buchholz's psi function. The proof-theoretic ordinal of Π¹₁ - comprehension + transfinite induction and Π¹₁ - comprehension + bar induction.

Madore's ordinal (this is the name I have dubbed it, it does not have an official name) - collapse of ε_{I + 1} using an unknown function. I is the first inaccessible cardinal. The proof-theoretic ordinal of Kripke-Platek set theory augmented by the recursive inaccessibility of the class of ordinals (KPi), and Δ¹₂ - comprehension + bar induction.

"Small" Rathjen ordinal (this is the name I have dubbed it, it does not have an official name), aka ϑ (the designation given to it by Michael Rathjen, but I mean, what kind of a name is that?) - collapse of ε_{M + 1} using Buchholz's psi function from "a note on the ordinal analysis of KPM", not to be confused with the regular, more well-known. M is the first Mahlo cardinal. The proof-theoretic ordinal of KPM, an extension of Kripke–Platek set theory based on a Mahlo cardinal.

"Large" Rathjen ordinal (this is the name I have dubbed it, it does not have an official name) - collapse of ε_{K + 1} using Rathjen's Psi function. K is the first weakly compact cardinal. The proof-theoretic ordinal of KP + Π₃ - Ref.

"Small" Stegert ordinal (this is the name I have dubbed it, it does not have an official name) - Ψ^{εΞ + 1}_X using Stegert's Psi function. Ξ is the first Π²₀-indescribable cardinal and X = (ω⁺; P₀; ϵ; ϵ; 0) The proof-theoretic ordinal of KP + Π_ω - Ref.

"Large" Stegert ordinal (this is the name I have dubbed it, it does not have an official name) - Ψ^{εY + 1}_X using Stegert's Psi function. X = (ω⁺; P₀; ϵ; ϵ; 0). The proof-theoretic ordinal of Stability, an extension of KP.

Church-Kleene ordinal - The first non-recursive ordinal, the first non-hyperarithmetical ordinal and the second admissible ordinal. Its order type is equal to all ordinals below it (the recursive ordinals).

Great Church-Kleene ordinal (this is the name I have dubbed it, it does not have an official name) - The smallest limit of admissible ordinals; yet it isn't admissible itself. The smallest α such that L_α ∩ P(ω) is a model of Π¹₁ - comprehension.

Recursively inaccessible ordinals - Recursively inaccessible ordinals are admissible ordinals which are also a limit of admissibles. The first recursively inaccessible ordinal is the smallest ordinal α such that L_α ⊨ KPi, or, on the arithmetical side, such that L_α ∩ P(ω) is a model of ∆¹₂ - comprehension.

Recursively hyperinaccessible ordinals - Recursively hyperinaccessible ordinals are recursively inaccessible ordinals which are also a limit of recursively inaccessibles.

Recursively Mahlo ordinals - The first recursively Mahlo ordinal is the smallest admissible ordinal α such that or any α-recursive function f : α → α there is an admissible β < α which is closed under f. Alternatively, this is the smallest ordinal α such that L_α ⊨ KPM.

(+1)-stable ordinals - The first (+1)-stable ordinal is the smallest α such that L_α ⪯₁ L_{α + 1}. This is also the smallest Π¹₀-reflecting ordinal (i.e., Π_n-reflecting for every n ∈ ω).

(⁺)-stable ordinals - The first (⁺)-stable ordinal is the smallest α such that L_α ⪯₁ L_α +, where α⁺ is the first admissible ordinal > α. This is also the smallest Π¹₁-reflecting ordinal.

Σ¹₁-reflecting ordinals - A limit ordinal α is said to be Σ¹₁-reflecting if for any Σ₁ formula φ(x) in the Levy hierarchy and any b ∈ L_α, if (L_α, ∈) ⊨ φ(b) there exists a β ∈ On ∩ α such that b ∈ L_β and (L_β, ∈) ⊨ φ(b) and if for every Σ¹₁ sentence Φ in the Levy hierarchy, L_α ⊨ Φ implies that there exists a γ ∈ A ∩ α(L_γ ⊨ Φ). (⁺)-stable ordinals are the same as Σ¹₁-reflecting ordinals, apparently.

(⁺⁺)-stable ordinals - The first (⁺⁺)-stable ordinal is the smallest α such that L_α ⪯₁ L_α++, where α⁺ and α⁺⁺ are the first admissible ordinals > α. (⁺⁺)-stable ordinals are Σ¹₁-reflecting, although the first (⁺⁺)-stable ordinal is greater than the first Σ¹₁-reflecting ordinal.

Inaccessibly-stable ordinals - The first inaccessibly-stable ordinal is the smallest α such that L_α ⪯₁ L_β, where β is the smallest recursively inaccessible ordinal > α.

Mahlo-stable ordinals - The first Mahlo-stable ordinal is the smallest α such that L_α ⪯₁ L_β, where β is the smallest recursively Mahlo ordinal > α.

Doubly (+1)-stable ordinals - The first doubly (+1)-stable ordinal is the smallest α such that L_α ⪯₁ L_β ⪯₁ L_{β + 1}.

Nonprojectible ordinals - The first nonprojectible ordinal is the smallest β such that β is a limit of β-stable ordinals. In other words, the smallest β such that L_β ⊨ KPi + “the stable ordinals are unbounded”.

Σ₂-admissible ordinals - The first Σ₂-admissible ordinal is the smallest β such that L_β ⊨ KPω + ∆₂ - Sep, or, in other words, such that L_β ∩ P(ω) is a model of ∆¹₃ - comprehension.

The ordinal of ramified analysis (β₀) - The smallest β such that L_β ⊨ ∧_n Σ_n - Sep (the full separation scheme), or, in other words, such that L_β ∩ P(ω) is a model of full second-order analysis (second-order comprehension).

Devlin-Jech ordinal - The smallest ordinal α such that L_α ⊨ KP + “ω₁ exists”, in other words, the smallest admissible α which is not locally countable, or equivalently, the smallest α such that L_α ⊨ KP + “P(ω) exists”.

Stable ordinals - The smallest stable ordinal is the smallest σ such that L_σ ⪯₁ L, or equivalently: L_σ ⪯₁ L_ω1. This is also the smallest Σ¹₂-reflecting ordinal.

ω₁ - The smallest uncountably infinite ordinal and the first infinite ordinal to not have a cardinality of א‎₀ (ω₁ has a cardinality of א‎₁). Also known as Ω. ω₁ ≠ R (the real numbers), although |R| = ω₁. According to the continuum hypothesis, א‎₁ = P(א‎₀) = 2^א‎0.

YELLOW (Small uncountable ordinals) - ω₂ to ω_ω

Originally, this section would instead have been Ω + 1 to 3Ω, but such numbers don't exist according to Cantor's Attic. This section is a WIP.

ω₂ - The second uncountable ordinal. Has a cardinality of א‎₂ = P(א‎₁) = 2^א‎1. Not that significant of a number.

ω_ω - The ωth uncountable ordinal. Has a cardinality of א‎_ω = P(א‎_ω-1) = 2^א‎ω-1, which is the first uncountable singular cardinal. It is the supremum of the set {ω, ω₁, ..., ω_n, ... | n < ω}, or, equivalently, the limit of the sequence ω, ω₁, ..., ω_n, ...

LIGHT GREEN (Medium uncountable ordinals) - 1st ω fixed point to Σ_n-extendible cardinals

This section is a WIP.

ω_ω...ω - The 1st omega-fixed point, defined as the smallest α such that ω_α = α. It has a cardinality of _א..._אא. It is the limit of the sequence ω, ω_ω, ω_ωω, ...

Σ_n-extendible cardinals - A cardinal κ is Σ_n-extendible if, for some ordinal θ, there exists a Σ_n-elementary embedding j: V_κ → V_θ with critical point κ, or alternatively, if V_κ ⪯_Σn V_θ.

LIGHT BLUE (Large uncountable ordinals and small large cardinals) - 0-extendible to Ω^α-inaccessible cardinals

0-extendible cardinals - A cardinal κ is 0-extendible if, for some ordinal θ, there exists an elementary embedding j: V_κ → V_θ with critical point κ.

Worldly cardinals - A cardinal κ is worldly if V_κ ⊨ ZF. All worldly cardinals are strong limits and beth fixed points.

α-Worldly cardinals - A cardinal κ is α-worldly if it is worldly and for every β < α, the β-worldly cardinals are unbounded in κ.

Hyper-worldly cardinals - A cardinal κ is hyper-worldly if it is κ-worldly (it is worldly and for every β < κ, the β-worldly cardinals are unbounded in κ).

Weakly inaccessible cardinals - A cardinal κ is weakly inaccessible if it is an uncountable, regular and is neither a successor cardinal nor zero.

Strongly inaccessible cardinals - A cardinal κ is strongly inaccessible if it is an uncountable, regular and for all λ < κ, 2^λ < κ.

α-Inaccessible cardinals - A cardinal κ is α-inaccessible if it is inaccessible and for all β < α, κ is a limit of β-inaccessible cardinals.

Hyper-inaccessible cardinals - A cardinal κ is hyper-inaccessible if it is κ-inaccessible (it is inaccessible and for all β < κ, κ is a limit of β-inaccessible cardinals).

Ω^α-inaccessible cardinals - A cardinal κ is Ω^α-inaccessible if it is hyper^{κ↑↑(α-1)}-inaccessible (it is hyper-inaccessible and for all β < κ↑↑(α-1), κ is hyper^β-inaccessible).

DARK GREEN (Medium large cardinals) - ρ-inaccessible to Super weakly Ramsey

ρ-Inaccessible cardinals - A cardinal κ is ρ-inaccessible if it is a regular cardinal and for all α < ρ, κ is a limit of α-inaccessible cardinals. Not sure how this differs from the α-Inaccessible cardinals.

Pseudo-uplifting cardinals - A cardinal κ is pseudo-uplifting if for every ordinal θ there is a cardinal γ > θ such that V_κ ≺ V_γ is a proper elementary extension.

Uplifting cardinals - A cardinal κ is uplifting if for every ordinal θ there is an inaccessible cardinal γ > θ such that V_κ ≺ V_γ is a proper elementary extension.

Σ_n-Mahlo cardinals - A cardinal κ is Σ_n-Mahlo if it is regular and every club in κ that is Σ_n-definable in H(κ) contains an inaccessible cardinal.

Σ_ω-Mahlo cardinals - A cardinal κ is Σ_ω-Mahlo if it is regular and every club subset of κ that is definable (with parameters) in H(κ) contains an inaccessible cardinal.

Weakly Mahlo cardinals - A cardinal κ is weakly Mahlo if it is regular and the set of regular cardinals below κ is stationary in κ.

Mahlo cardinals - A cardinal κ is Mahlo if it is inaccessible and the regular cardinals below κ form a stationary subset of κ.

α-Mahlo cardinals - A cardinal κ is α-Mahlo if it is Mahlo and for each β < α, the class of β-Mahlo cardinals is stationary in κ.

Hyper-Mahlo cardinals - A cardinal κ is hyper-Mahlo if it is κ-Mahlo (it is Mahlo and for each β < κ, the class of β-Mahlo cardinals is stationary in κ).

Ω^α-Mahlo cardinals - A cardinal κ is Ω^α-Mahlo if it is hyper^{κ↑↑(α-1)}-Mahlo (it is hyper-Mahlo and for all β < κ↑↑(α-1), κ is hyper^β-Mahlo).

Σ_n-weakly compact cardinals - A cardinal κ is Σ_n-weakly compact if for every R ⊆ V_κ which is definable by a Σ_n formula (with parameters) over V_κ and every Π¹₁ sentence Φ, if ⟨V_κ, ∈, R⟩ ⊨ Φ then there is α < κ (equivalently, unboundedly-many α < κ) such that ⟨V_α, ∈, R ∩ Vα⟩ ⊨ Φ.

Weakly compact cardinals - A cardinal κ is weakly compact if for every transitive set M of size κ with κ ∈ M there is a transitive set N and an embedding j: M → N with critical point κ. All weakly compact cardinals are Π¹₁-indescribable.

Σⁿ_m-, Δⁿ_m or Πⁿ_m-indescribable - A cardinal κ is Πⁿ_m-indescribable (resp. Σⁿ_m-indescribable or Δⁿ_m-indescribable) if for every Π_m (resp. Σ_m and Δ_m) first-order sentence ϕ and all S ⊆ V_κ(⟨V_{κ + n}; ∈, S⟩ ⊨ ϕ → ∃α < κ(⟨V_{α + n}; ∈, S ∩ V_α⟩ ⊨ ϕ))

Totally indescribable - A cardinal κ is totally indescribable if it is Πⁿ_m-, Σⁿ_m- and Δⁿ_m- indescribable for all natural n and m.

β-indescribable - A cardinal κ is β-indescribable if for every first-order sentence ϕ and all S ⊆ V_κ(⟨V_{κ + β}; ∈, S ⊨ ϕ → ∃α < κ(⟨V_{α + β}; ∈, S ∩ V_α⟩ ⊨ ϕ)).

η-Shrewd - A cardinal κ is η-shrewd if for all X ⊆ V_κ and for every formula ϕ(x₁, x₂), if V_{κ + η} ⊨ ϕ(X, κ), then ∃_{0 < κ0, η0 < κ} V_{κ0 + η0} ⊨ ϕ(X ∩ V_κ0, κ₀).

Shrewd - A cardinal κ is shrewd if it is η-shrewd for all η > 0.

A-η-shrewd - A cardinal κ is A-η-shrewd if for all X ⊆ V_κ and every formula ϕ(x₁, x₂), if ⟨V_{κ + η}, A ∩ V_{κ + η}⟩ ⊨ ϕ(X, κ), then ∃_{0 < κ0, η0 < κ} ⟨V_{κ0 + η0}, A ∩ V_{κ0 + η0}⟩ ⊨ ϕ(X ∩ Vκ₀, κ₀).

A-shrewd - A cardinal κ is A-shrewd if, for all η > 0, κ is A-η-shrewd.

Unfoldable - A cardinal κ is unfoldable if, for every θ, and every A ⊆ κ, there is some transitive M with A ∈ M ⊨ ZFC and some j: M → N with critical point κ such that j(κ) ≥ θ.

Strongly unfoldable - A cardinal κ is strongly unfoldable if for every θ, and every A ⊆ κ , there is some transitive M with A ∈ M ⊨ ZFC and some j: M → N with critical point κ such that j(κ) ≥ θ and V_θ ⊆ N.

Weakly superstrong - A cardinal κ is weakly superstrong if for every transitive set M of size κ with κ ∈ M and M < κ ⊆ M, a transitive set N and an elementary embedding j: M → N with critical point κ, for which V_j(κ) ⊆ N.

Strongly uplifting - A cardinal κ is strongly uplifting if for every ordinal θ and every A ⊆ V_κ, there exists some inaccessible λ > θ and an A ⊆ V_λ such that (V_κ; ∈, A) ≺ (V_λ; ∈, A) is a proper elementary extension.

Ethereal - A cardinal κ is ethereal if for every club C in κ and sequence (S_α | α < κ) of sets such that for α < κ, |S_α| = |α| and S_α ⊆ α, there are elements α, β ∈ C such that α < β and |S_α ∩ S_β| = |α|.

Subtle - A cardinal κ is subtle if for every ⟨A_α ∣ α < κ⟩ with A_α ⊆ α and every club C ⊆ κ there are α < β in C such that A_β ∩ α= A_α.

Weakly ineffable - A cardinal κ is weakly ineffable if for every sequence ⟨A_α ∣ α < κ⟩ with A_α ⊆ α there is A ⊆ κ such that the set S={α < κ ∣ A ∩ α = A_α} has size κ.

n-Ramsey - A cardinal κ is n-Ramsey if, for every function F:[κ]^<ω → 2 there is H ⊆ κ with |H| = κ such that F ↾ [H]ⁿ is constant for every n. Here [X]^y is the set of all y-elements subsets of X.

Completely ineffable - Define that a collection R ⊆ P(κ) is a stationary class if R≠∅; for all A ∈ R, A is stationary in κ; and if A ∈ R and B ⊇ A, then B ∈ R. A cardinal κ is completely ineffable if there is a stationary class R such that for every A ∈ R and F: [A]² → 2, there is H ∈ R such that F ↾ [H]² is constant.

Weakly Ramsey - A cardinal κ is weakly Ramsey if every A ⊆ κ is contained in a weak κ-model M for which there exists a well-founded, weakly amenable M-ultrafilter on κ. Weakly Ramsey cardinals are 1-iterable (discussed later).

Super weakly Ramsey - A cardinal κ is super weakly Ramsey if every A ⊆ κ is contained, as an element, in a weak κ-model M ≺ H(κ⁺) for which there exists a κ-powerset preserving the elementary embedding j∶ M → N.

DARK BLUE (Large large cardinals) - Remarkable to almost fully Ramsey

Remarkable - A cardinal κ is remarkable if for every η > κ, there is α < κ such that in a set-forcing extension there is an elementary embedding j: V_α → V_η with j(crit(j))=κ.

Virtually measurable - A cardinal κ is virtually measurable if for every regular ν > κ there exists a transitive M and a forcing P such that, in V^P, there is an elementary embedding j: H^V_ν → M with critical point κ.

Virtually extendible - A cardinal κ is virtually extendible if for every α > κ, in a set-forcing extension there is an elementary embedding j: V_α → V_β with critical point κ and j(κ) > α.

Virtually C⁽ⁿ⁾ extendible - A cardinal κ is virtually C⁽ⁿ⁾ extendible if for every α > κ, in a set-forcing extension there is an elementary embedding j: V_α → V_β with critical point κ and j(κ) > α and j(κ) is Σ_n-correct.

Completely remarkable - If 1-remarkable is remarkable and virtually C⁽ⁿ⁾ extendible is (n+1)-remarkable, then a cardinal κ is completely remarkable if it is n-remarkable for all n > 0.

Virtually Shelah for supercompactness - A cardinal κ is virtually Shelah for supercompactness if for every function f: κ → κ there are λ > κ and λ¯ < κ such that in a set-forcing extension there is an elementary embedding j: V_λ¯ → V_λ with j(crit(j)) = κ, λ¯ ≥ f(crit(j)) and f ∈ ran(j).

Virtually α-huge - A cardinal κ is virtually α-huge for some α > κ, in a set-forcing extension, κ is the critical point of an elementary embedding j: V_α ≺ V_β such that jⁿ(κ) < α.

Virtually rank-into-rank - A cardinal κ is virtually rank-into-rank if, in a set-forcing extension, it is the critical point of an elementary embedding j: V_λ → V_λ for some λ > κ.

Weakly remarkable - A cardinal κ is weakly remarkable if for every η > κ, there is α such that in a set-forcing extension there is an elementary embedding j:V_α → V_η with j(crit(j))=κ.

α-Erdős - A cardinal κ is α-Erdős for an infinite limit ordinal α if it is the least cardinal κ such that for every function F: [κ]^<ω → 2 there is H ⊆ κ with |H| = α such that F ↾ [H]ⁿ is constant for every n. Here [X]^y is the set of all y-elements subsets of X.

α-Iterable - Suppose M is a weak κ-model and U is an M-ultrafilter on κ. U is 0-good if it is well-founded, 1-good if it is 0-good and weakly amenable, and β-good for an ordinal β > 1 it it produces at least β-many well-founded iterated ultrapowers. A cardinal κ is α-iterable if every A ⊆ κ is contained in a weak κ-model M for which there exists an α-good M-ultrafilter on κ.

(ω, α)-Ramsey - For X ⊆ κ and ordinal β, define G_R(X, β) is a certain game for two players with finitely many moves. A cardinal κ is (ω, α)-Ramsey for countably infinite α if player 1 has no winning strategy in G^θ_ω(κ, α) for all regular θ > κ.

Silver - A cardinal κ is silver if, in a set-forcing extension, there is a club in κ of generating indiscernibles for V_κ of order-type κ.

Almost Ramsey - A cardinal κ is almost Ramsey if for every function F:[κ]^<ω → γ there is H ⊆ κ with |H| = λ such that F↾[H]ⁿ is constant for all n, and all λ, γ < κ. Here, [X]^y is the set of all y-elements subsets of X.

Weakly α-Erdős - Suppose that κ has uncountable cofinality, A is a κ-structure, with X ⊆ κ, and t_A(X) = {limit ordinal α ∈ κ | there exists a set I ⊆ α ∩ X of good indiscernibles for A cofinal in α}. Using this one can define a hierarchy of normal filters F_α potentially for all α < κ⁺; these are generated by suprema of sets of nested indiscernibles for structures A on κ using the above basic t_A(X) operation. A cardinal κ is weakly α-Erdős when F_α is non-trivial.

Greatly Erdős - A cardinal κ is greatly Erdős if it is weakly α-Erdős for all α < κ⁺.

Virtually Ramsey - For A ⊆ κ, define that I = {α < κ ∣ there is an unbounded good set of indiscernibles I_α ⊆ α for ⟨L_κ[A], A⟩}. A cardinal κ is virtually Ramsey if for every A ⊆ κ, the set I contains a club subset of κ.

Jónsson - A cardinal κ is Jónsson if for every θ > κ there exists a transitive set M with κ ∈ M and an elementary embedding j: M → H_θ such that j(κ) = κ and critical point < κ, if for every θ > κ there exists a transitive set M with κ ∈ M and an elementary embedding j: M → V_θ such that j(κ) = κ and critical point < κ.

Rowbottom - For all λ < κ, a cardinal κ is Rowbottom if for any function f: [κ]^<ω → λ, there is some set of ordinals H ⊆ κ such that (H, <) has order type κ and |f "[H]^<ω| < ω₁. Here, [X]^y is the set of all y-elements subsets of X.

Ramsey - A cardinal κ is Ramsey if for every function F: [κ]^<ω → 2 there is H ⊆ κ with |H| = κ such that F↾[H]ⁿ is constant for every n. Here, [X]^y is the set of all y-elements subsets of X.

Π_α-Ramsey - For a regular, uncountable cardinal κ, we define I^κ₋₂ = P_<κ(κ); I^κ₋₁ = the set of non-stationary subsets of κ; for n < ω, I^κ_n = R(I^κ_n−2); for α ≥ ω, I^κ_α+1 = R(I^κ_α) and for a limit ordinal γ, I^κ_γ = ⋃R(I^κ_β) for all β < γ. A cardinal κ is Π_α-Ramsey if it is regular, uncountable and κ ∉ I^κ_α.

Completely Ramsey - A cardinal κ is completely Ramsey if it is Π_α-Ramsey for all α.

α-Hyper completely Ramsey - A sequence ⟨f_α: α < κ⁺⟩ of elements of ^κκ is a canonical sequence on κ if both: for all α, β ∈ κ, α < β implies f_α < f_β; and for any other sequence ⟨g_α: α < κ⁺⟩ of elements of κ^κ such that ∀_{α < β < κκ} g_α < g_β, we have ∀_{α < κ+} f_α < g_α. For a regular uncountable cardinal κ, let f^→ = ⟨f_α: α < κ⁺⟩ be the canonical sequence on κ. κ is 0-hyper completely Ramsey if κ is completely Ramsey. For α < κ⁺, κ is α-hyper completely Ramsey if κ is (α-1)-hyper completely Ramsey and there is a completely Ramsey subset X such that for all λ ∈ X, λ is f_α(λ)-hyper completely Ramsey.

Super completely Ramsey - A cardinal κ is super completely Ramsey if κ is κ⁺-hyper completely Ramsey.

γ-filter property - For X ⊆ κ and ordinal α, define GR(X, α) as a certain game for two players with finitely many moves. A cardinal κ has the γ-filter property if player 1 does not have a winning strategy in G^θ_γ(κ) for all regular θ > κ.

α-Ramsey - For α ≤ κ, κ is α-Ramsey if for arbitrarily large regular cardinals θ, every A ⊆ κ is contained, as an element, in some weak κ-model M ≺ H(θ) which is closed under < α-sequences, and for which there exists a κ-powerset preserving elementary embedding j∶ M → N. Note that, in the case α = κ, a weak κ-model closed under <κ-sequences is exactly a κ-model.

Almost fully Ramsey - A cardinal κ is almost fully Ramsey if it is <κ-Ramsey.

INDIGO (Larger large cardinals) - Strongly Ramsey to hypercompact cardinals

Strongly Ramsey - A cardinal κ is strongly Ramsey if every A ⊆ κ is contained in a κ-model M for which there exists a weakly amenable M-ultrafilter on κ. An M-ultrafilter for a κ-model M is automatically countably complete since ⟨M, U⟩ satisfies that it is κ-complete and it must be correct about this since M is closed under sequences of length less than κ.

Super Ramsey - A weak κ-model M is a κ-model if additionally M^<κ ⊆ M. A cardinal κ is super Ramsey if and only if for every A ⊆ κ, there is some κ-model M with A ⊆ M ≺ H_κ+ such that there is some N and some κ-powerset preserving nontrivial elementary embedding j: M ≺ N.

Fully Ramsey - A cardinal κ is fully Ramsey if it is κ-Ramsey. Fully Ramsey cardinals are super Ramsey limits of super Ramsey cardinals.

ω₁-Very Ramsey - For X ⊆ κ and ordinal α, define GR(X, α) as a certain game for two players with finitely many moves. X is ω₁-very Ramsey if player 2 has a winning strategy in GR(X, ω₁).

Measurable - For every transitive M ⊨ ZFC⁻ of size κ⁺ closed under <κ sequences with κ ∈ M, there exists a transitive N of size κ⁺ closed under <κ sequences and a cofinal elementary embedding j: M ⟶ N with critical point κ such that N = {j(f)(κ) | f ∈ M; f: κ ⟶ M}.

γ-Strong - A cardinal κ is γ-strong if it is the critical point of some elementary embedding j: V → M for some transitive class M such that V_γ ⊂ M.

Hypermeasurable - A cardinal κ is x-hypermeasurable for a set x if it is the critical point of some elementary embedding j: V → M for some transitive class M such that x ∈ M. A cardinal κ is λ-hypermeasurable if it is H_λ-hypermeasurable (where Hλ is the set of all sets of hereditary cardinality less than λ). A cardinal κ is hypermeasurable if it is λ-hypermeasurable for all λ.

θ-tall - A cardinal κ is θ-tall if there is an elementary embedding j: V → M into a transitive class M with critical point κ such that j(κ) > θ and M^κ ⊂ M.

Tall - A cardinal κ is tall if it is θ-tall for every θ.

Strong - A cardinal κ is strong if it is γ-strong for each γ.

Strongly tall - A cardinal κ is strongly tall if there is some measure U on a set S witnessing κ's θ-tallness in the ultrapower of V by U. More precisely, the ultrapower embedding j: V ≺ M has critical point κ, M_κ ⊂ M, and j(κ) > θ. κ is strongly tall if it is strongly θ-tall for every θ.

Woodin - A cardinal δ is Woodin if it is inaccessible and for every function f: δ → δ there exists κ < δ such that κ is closed under f and there exists a nontrivial elementary embedding j: V → M with critical point κ such that V_j(f)(κ) ⊆ M.

Weakly hyper-Woodin - A cardinal δ is weakly hyper-Woodin if for every set A, there is a normal measure U on δ such that {κ < δ | κ is <δ-A-strong} ∈ U.

Shelah - A cardinal δ is Shelah if for every function f: δ → δ there exists a nontrivial elementary embedding j: V → M with critical point δ such that V_j(f)(δ) ⊆ M.

Hyper-Woodin - A cardinal δ is hyper-Woodin if U is a normal measure on δ and for every set A, {κ < δ | κ is <δ-A-strong} ∈ U.

Superstrong - A cardinal κ is superstrong if it is the critical point of some elementary embedding j: V → M such that M is a transitive class and V_j(κ) ⊂ M.

C⁽ⁿ⁾-Superstrong - A cardinal κ is C⁽ⁿ⁾-superstrong if there exists an elementary embedding j: V → M for transitive M, with critical point κ, V_j(κ) ⊆ M and j(κ) ∈ C(n).

Subcompact - A cardinal κ is subcompact if for every A ⊂ H(κ⁺) there is a non-trivial elementary embedding j: (H(μ⁺), B) → (H(κ⁺), A) (where H(κ⁺) is the set of all sets of cardinality hereditarily less than κ⁺) with critical point μ and j(μ) = κ.

Nearly supercompact - A cardinal κ is nearly θ-supercompact if and only if for every A ⊆ θ, there exists a transitive M ⊨ ZFC⁻ closed under <κ sequences with A, κ, θ ∈ M, a transitive N, and an elementary embedding j: M → N with critical point κ such that j(κ) > θ and j′′(θ) ∈ N. A cardinal is nearly supercompact if it is nearly θ-supercompact for all θ.

θ-strongly compact - A cardinal κ is θ-strongly compact if there is an elementary embedding j: V → M for a transitive class M with critical point κ, such that j′′(θ) ⊂ s ∈ M for some set s ∈ M with |s|M < j(κ).

Strongly compact - A cardinal κ is strongly compact if it is θ-strongly compact for all θ.

θ-supercompact - A cardinal κ is θ-supercompact if there is an elementary embedding j: V → M with M for a transitive class M with critical point κ and M is closed under arbitrary sequences of length θ.

Supercompact - A cardinal κ is supercompact if it is θ-supercompact for all θ.

C⁽ⁿ⁾-Supercompact - A cardinal κ is θ-C⁽ⁿ⁾-supercompact if there is an elementary embedding j: V → M with transitive M and critical point κ, j(κ) > λ, M is closed under arbitrary sequences of length θ and j(κ) ∈ C(n). A cardinal κ is C⁽ⁿ⁾-supercompact if it is θ-C(n)-supercompact for all θ > κ.

Enhanced supercompact - A cardinal κ is enhanced supercompact if there is a strong cardinal θ > κ such that for every cardinal λ > θ, there is a λ-supercompactness embedding j: V → M such that θ is strong in M.

α-Hypercompact - A cardinal κ is α-hypercompact if for every ordinal β < α and for every cardinal λ ≥ κ, there exists a cardinal λ′ ≥ λ and an elementary embedding j: V → M generated by a normal fine ultrafilter on P_κλ such that κ is β-hypercompact in M.

Hypercompact - A cardinal κ is hypercompact if it is β-hypercompact for every ordinal β.

VIOLET (Largest large cardinals) - Woodinised strongly compact cardinals to limit club Berkeley cardinals

Woodinised strongly compact - A cardinal δ is Woodinised strongly compact if for every A ⊆ δ there is κ < δ which is <δ-strongly compact for A.

η-extendible - A cardinal κ is η-extendible if there is an elementary embedding j: V_{κ + η} → V_θ, with critical point κ, for some ordinal θ.

Extendible - A cardinal κ is extendible if it is η-extendible for all η.

C⁽ⁿ⁾-Extendible - A cardinal κ is C⁽ⁿ⁾-extendible if there is an ordinal µ and an elementary embedding j: V_λ → V_µ, with critical point κ, j(κ) > λ and j(κ) ∈ C(n).

A-Extendible - A cardinal κ is A-extendible, for a class A, if for every ordinal λ > κ there is an ordinal θ such that there is an elementary embedding j: ⟨V_λ, ∈, A ∩ V_λ⟩ → ⟨V_θ, ∈, A ∩ V_θ⟩ with critical point κ such that λ < j(κ).

Vopěnka cardinal - A cardinal κ is Vopěnka if it is inaccessible and V_κ satisfies Vopěnka's principle (for any language L and any proper class C of L-structures, there are distinct structures M, N ∈ C and an elementary embedding j: M → N).

Shelah for supercompactness - A cardinal δ is Shelah for supercompactness if for every function f: δ → δ there exists a nontrivial elementary embedding j: V → M with critical point δ such that M^j(f)(δ) ⊆ M.

High-jump - A cardinal κ is high-jump if it is the critical point of an elementary embedding j: V → M such that M is closed under sequences of length sup{j(f)(κ) | f: κ → κ}.

Almost high-jump - A cardinal κ is almost high-jump if it is the critical point of an elementary embedding j: V → M such that M is closed under sequences of length <θ.

Super high-jump - A cardinal κ is super high-jump if there are high-jump embeddings with arbitrarily large clearance, i.e. it is "high-jump order Ord".

Almost huge - A cardinal κ is almost huge if λ = j(κ) and M is closed under all of its sequences of length less than λ.

Huge - A cardinal κ is huge if λ = j(κ) and M is closed under all of its sequences of length λ.

Super almost huge - A cardinal κ is super almost huge if for every γ, there is some λ > γ for which κ is almost huge.

Superhuge - A cardinal κ is superhuge if for every γ, there is some λ > γ for which κ is huge.

Ultrahuge - A cardinal κ is λ-ultrahuge for λ > κ if there exists a nontrivial elementary embedding j: V → M for some transitive class M such that j(κ) > λ, M^j(κ) ⊆ M and V_j(λ) ⊆ M. A cardinal is ultrahuge if it is λ-ultrahuge for all λ ≥ κ.

Weakly Reinhardt - A cardinal κ is weakly Reinhardt if there exists a nontrivial elementary embedding j: V_λ+2 → V_λ+2 with critical point κ such that V_κ ≺ V_λ ≺ V_γ (for some γ > λ > κ).

Reinhardt - A cardinal κ is Reinhardt if there exists a nontrivial elementary embedding j: V → V with critical point κ.

Super Reinhardt - A cardinal κ is super Reinhardt if there exists a nontrivial elementary embedding(s) j: V → V with critical point κ, where j(κ) can be as large as desired.

Totally Reinhardt - For a proper class A, cardinal κ is called A-super Reinhardt if for all ordinals λ there is a non-trivial elementary embedding j: V → V with critical point κ, j(κ) >λ and j⁺(A) = A. (where j⁺(A): = ∪_{α ∈ Ord}j(A ∩ Vα)). A cardinal κ is totally Reinhardt if for each A ∈ V_{κ + 1}, (V_κ, V_{κ + 1}) ⊨ ZF₂ + “There is an A-super Reinhardt cardinal”.

Berkeley - A cardinal κ is Berkeley if for any transitive set M with κ ∈ M and any ordinal α < κ there is an elementary embedding j: M ≺ M with α < critical point < κ.

Club Berkeley - A cardinal κ is club Berkeley if κ is regular and for all clubs C ⊆ κ and all transitive sets M with κ ∈ M there is j ∈ E(M) with critical point ∈ C.

Limit club Berkeley - A cardinal κ is limit club Berkeley if it is a club Berkeley cardinal and a limit of Berkeley cardinals.

MY INFINITE NUMBERS

Megacompact - A cardinal κ is megacompact if there is an elementary embedding j: V → M with M for a transitive class M with critical point κ and M is closed under arbitrary sequences of length j^ω(θ) for all θ.

n-Megacompact - A cardinal κ is n-megacompact if, for all λ < j^nω(κ), j^nω(κ) is a limit of λ-megacompact cardinals.

Megastrong - A cardinal κ is megastrong if there exists some elementary embedding j: V → M for some transitive class M with critical point κ and V_jω(κ) ⊂ M.

n-Megastrong - A cardinal κ is n-megastrong if, for all λ < j^nω(κ), j^nω(κ) is a limit of λ-megastrong cardinals.