This list is a living map. In the infinite, there is always room between the rungs. Explore at your own risk.

I. The Countable Abyss and Proof-Theoretic Ordinals

Before reaching the first uncountable infinity, mathematicians explore the limits of what can be constructed, proven, or computed. This tier maps the precise boundaries where specific proof systems and computable functions reach their absolute limits.

• ω (omega), the smallest infinity
• Hilbert's hotel and other toys in the playroom
• The small countable ordinals (ω+1, ..., ω·2, ..., ω², ..., ω^ω, ..., ω^ω^ω, ... up to ε₀)
• Indecposable ordinal
• ε₀ and the hierarchy of ε_α numbers
• The Feferman-Schütte ordinal Γ₀
• The Extended Veblen function
• The small Veblen ordinal
• The large Veblen ordinal
• The Bachmann-Howard ordinal
• Buchholz’s ψ (psi) functions
• The Takeuti-Feferman-Buchholz ordinal
• Limits of Subsystems of Second-Order Arithmetic (e.g., the limit of Π¹₁-Comprehension)
• The Kripke-Platek Limits (KPi and KPM)
• Game values and ordinals of infinite chess (ω_1^ck, etc.)
• Church-Kleene ω_1^ck, the supremum of the computable ordinals
• Admissible ordinals and relativized Church-Kleene ω_1^ck
• λ = the supremum of the writable ordinals
• ζ = the supremum of the eventually writable ordinals
• Σ = the supremum of the accidentally writable ordinals
• The ordinals of infinite time Turing machines
• Stable ordinals

See: Section I - The Limits of Logic and Computation

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II. The Continuum and The Uncountable

Transitioning into the genuinely uncountable realm, this section charts the Aleph and Beth hierarchies. It also maps the dense, topological characteristics of the real number line, expanding beyond simple bounding numbers to capture the full spectrum of the continuum's structure.

• ω₁ (omega-one), the first uncountable ordinal / ℵ₁ (aleph-one), the first uncountable cardinal
• Cardinals, infinite cardinals, regular and successor cardinals
• ℵ₂ (aleph-two), the second uncountable cardinal
• ℵ_ω (aleph-omega) and singular cardinals
• The aleph numbers and the ℵ_α hierarchy
• ℵ-fixed point
• The descriptive set-theoretic cardinals
• The bounding number b, the dominating number d, the covering numbers, additivity numbers
• The Complete Spectrum of Cichoń’s Diagram (Splitting, Reaping, Tower numbers)
• Cardinal characteristics of the continuum
• The continuum (2^ℵ₀)
• Θ (Theta)
• ℶ_ω (beth-omega) and the strong limit cardinals
• The beth numbers and the ℶ_α hierarchy
• ℶ-fixed point

See: Section II - The Continuum and The Uncountable (The Breaking of the List)

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III. The Lower Large Cardinals: Reflection and Combinatorics

Here, standard mathematics (ZFC) is transcended. This section tracks the combinatorial pressures, infinite graph colorings, and reflection properties that force massive, inaccessible universes to exist.

• Zermelo-Fraenkel set theory (ZFC) as a baseline
• Con(ZFC) and Con₀(ZFC), the iterated consistency hierarchy
• The minimal transitive model / Transitive model of ZFC
• The transitive model universe axiom
• Worldly cardinal and the α-worldly hierarchy, hyper-worldly cardinal
• Morse-Kelley set theory
• Weakly inaccessible cardinal, (strongly) inaccessible cardinal
• Grothendieck universe axiom (existence of a proper class of inaccessibles)
• 1-inaccessible, the α-inaccessible hierarchy, hyper-inaccessible, Ω_α-inaccessible cardinals
• Jäger’s collapsing functions and ρ-inaccessible ordinals
• Σ₂-reflecting, Σ_n-reflecting, and reflecting cardinals
• Ord is Mahlo / Mahlo cardinal
• Σ_n-Mahlo cardinals, Σ_ω-Mahlo, weakly Mahlo, (strongly) Mahlo, 1-Mahlo, α-Mahlo hierarchy, hyper-Mahlo
• The positive set theory GPK⁺^∞
• Σ_n-weakly compact cardinal, Σ_ω-weakly compact cardinal
• Weakly compact (= Π¹₁-indescribable = 0-Ramsey) cardinal
• Indestructible weakly compact cardinal
• Σ_n- and Π_n-indescribable, totally indescribable, η-indescribable cardinals
• Ineffable and Π_n^m-Ineffable Hierarchies
• η-shrewd, shrewd, Λ-η-shrewd, Λ-shrewd cardinals
• Unfoldable cardinal, strongly unfoldable cardinal
• Subtle cardinal
• Weakly ineffable (= genuine 0-Ramsey) cardinal, ineffable (= normal 0-Ramsey) cardinal
• n-Ramsey, genuine n-Ramsey, normal n-Ramsey, <ω-Ramsey cardinals
• The n-subtle, n-almost ineffable, n-ineffable cardinals' hierarchy
• Completely ineffable cardinal (= ω-filter property)
• Weakly Ramsey (= 1-iterable) cardinal, super weakly Ramsey, ω-Ramsey cardinal
• α-Erdős cardinal, weakly remarkable cardinal that is not remarkable
• The α-Erdős, α-iterable, and (ω,α)-Ramsey hierarchy for countable infinite α
• Silver cardinal / 0# (zero-sharp), existence of Silver indiscernibles
• ω1-iterable cardinal, (ω,ω1)-Ramsey cardinal
• ω1-Erdős cardinal and γ-Erdős cardinals for uncountable γ, Chang's conjecture
• Almost Ramsey cardinal
• α-weakly Erdős cardinals, greatly Erdős cardinal
• Virtually Ramsey cardinal, Jónsson cardinal, Rowbottom cardinal, Ramsey cardinal
• Ramsey M-rank
• α-Mahlo-Ramsey hierarchy
• Π_α-Ramsey, completely Ramsey (= ω-very Ramsey), α-hyper completely Ramsey, super completely Ramsey
• α-filter property, α-Ramsey cardinal (for ω < α < κ), almost fully Ramsey (= <κ-Ramsey)
• Strongly Ramsey cardinal, strongly Ramsey M-rank, super Ramsey cardinal, super Ramsey M-rank
• κ-filter property, fully Ramsey (= κ-Ramsey) cardinal
• κ+-filter property, strategic (ω+1)-Ramsey cardinal, strategic fully Ramsey, ω1-very Ramsey, κ-very Ramsey
• Singular Jónsson cardinal

See: Section III - The Lower Large Cardinals (The Rise of Reflection)

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IV. The Core Large Cardinals: Embeddings and Structure

Housing the heavyweights of modern set theory, this tier charts the immense jumps in consistency strength mediated by elementary embeddings and the structural determinacy of projective sets.

• Weakly measurable cardinal, measurable cardinal
• 0† (zero-dagger)
• Nontrivial Mitchell rank, o(κ)=1, o(κ)=κ++
• The θ-strong, hypermeasurability, and θ-tall hierarchies, tall and strong cardinals
• Strong cardinal
• Strongly tall cardinal
• Weakly superstrong cardinal
• Superstrong cardinal, C(n)-superstrong hierarchy
• Mutually Large Cardinals
• Woodin cardinal
• The Axiom of Determinacy and its projective counterpart
• Shelah cardinal
• The Stationary Tower (P_<δ)
• Subcompact cardinal
• The proper forcing axiom and Martin's maximum
• Nearly supercompact and nearly strongly compact cardinals
• λ-strongly compact cardinals, strongly compact cardinal
• λ-supercompact cardinals, supercompact cardinal, C(n)-supercompact cardinals
• Enhanced λ-supercompact, enhanced supercompact, λ-hypercompact, hypercompact cardinal
• Woodin for strong compactness
• α-extendible hierarchy, extendible cardinal, C(n)-extendible hierarchy
• 0-extendible cardinal
• Σ_n-extendible cardinal
• Σ_n correct and Σ_n-correct cardinals / Correct cardinals, V_δ ≺ V and the Feferman theory
• Vopěnka scheme, Vopěnka principle, Vopěnka scheme cardinal, Vopěnka (= Woodin for supercompactness) cardinal
• Shelah for supercompactness
• High-jump cardinal, almost high-jump cardinal, super high-jump cardinal, high-jump with unbounded excess closure cardinal
• (1)-remarkable (= virtually supercompact), virtually measurable, strategic ω-Ramsey, weak Proper Forcing Axiom
• Virtually extendible (= 2-remarkable), virtually C(n)-extendible (= n+1-remarkable), completely remarkable, Generic Vopěnka's Principle
• Virtually Shelah for supercompactness cardinal
• The n-iterable and virtually n-huge hierarchy*
• Virtually rank-into-rank cardinal
• Almost huge, huge, huge, super almost huge, superhuge, ultrahuge, 2-superstrong cardinal*
• n-fold variants of hugeness (plus C(n) variants), extendibility, supercompactness, strongness, etc.

See: Section IV - The Core Large Cardinals (The Heavyweights of Structure)

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V. The Upper Limits and the Choiceless Abyss

The apex of the catalogue approaches the absolute limits of mathematical consistency, eventually demanding the abandonment of the Axiom of Choice, before culminating in the formal collapse of logical triviality.

• The Reflection Principle (Global)
• The wholeness axioms, axioms I₄
• Rank into rank axioms (I3=E0, IEω, IE, I2=E1, Ei, I1=Eω plus m-C(n)-Ei), ω-fold variants, I0 axiom and strengthenings
• The Kunen inconsistency
• Weakly Reinhardt, Reinhardt, super Reinhardt, A-Super Reinhardt, totally Reinhardt cardinal
• Berkeley cardinal, club Berkeley, limit club Berkeley cardinal, X-closed rank Berkeley
• The Bagaria-Koellner Axioms (V_λ <Σn Vκ)
• Inconsistent Multiplicities
• The Burali-Forti Limit (δ > δ)
• Ω (The Absolute Infinite)
• The Inconsistent Triviality (0=1)

See: Section V - The Upper Limits and the Choiceless Abyss (The Edge of Reality)