This is a comprehensive map of the infinite, ranked by size and "consistency strength," going from the smallest infinite sets to the absolute largest limits of mathematical thought.

• ω, the smallest infinity
• Hilbert's hotel and other toys in the playroom
• The small countable ordinals, such as ω, ω+1, …, ω·2, …, ω², …, ω^ω, …, ω^ω^ω, … up to ε₀
• Indecomposable 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
• The Takeuti-Feferman-Buchholz ordinal
• ω₁ᶜʰ = the supremum of the game values for white of the finite positions in infinite chess
• ω₁ᶜʰ,ᶜ = the supremum of the game values for white of the computable positions in infinite chess
• ω₁ᶜʰ ≅ the supremum of the game values for white of all positions in infinite chess
• The omega one of chess
• Church-Kleene ω₁ᶜᵏ, the supremum of the computable ordinals
• Admissible ordinals and relativized Church-Kleene ω₁ˣ
• 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
• ω₁, the first uncountable ordinal

• ℵ₀ and everything mentioned above ^
• Cardinals, infinite cardinals
• ℵ₁, the first uncountable cardinal
• Uncountable, regular and successor cardinals
• ℵ₂, the second uncountable cardinal
• ℵ_ω and singular cardinals
• Buchholz's ψ functions
• 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 and many more
• Cardinal characteristics of the continuum
• The continuum
• Θ
• ℶ_ω and the strong limit cardinals
• The beth numbers and the ℶ_α hierarchy
• ℶ-fixed point
• Σ_n-extendible cardinal
• 0-extendible cardinal
• Σ₂ correct and Σ_n-correct cardinals
• Correct cardinals, V_δ ≺ V and the Feferman theory

• Zermelo-Fraenkel set theory
• Con(ZFC) and Conα(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 (the existence of a proper class of inaccessible cardinals)
• 1-inaccessible, the α-inaccessible hierarchy, hyper-inaccessible cardinals, Ωα-inaccessible cardinals
• Jäger's collapsing functions and ρ-inaccessible ordinals
• Σ2-reflecting, Σn-reflecting and reflecting cardinals
• Ord is Mahlo
• Pseudo uplifting cardinal, uplifting cardinal
• Σn-Mahlo cardinals, Σω-Mahlo cardinal, weakly Mahlo cardinal, (strongly) Mahlo cardinal, 1-Mahlo, the α-Mahlo hierarchy, hyper-Mahlo cardinals, Ωα-Mahlo cardinals
• The positive set theory GPK∞+
• Σn-weakly compact cardinals, Σω-weakly compact cardinal, weakly compact (=Π11-indescribable=0-Ramsey) cardinal
• Σnm- and Πnm-indescribable, totally indescribable, η-indescribable cardinals
• η-shrewd, shrewd, A-η-shrewd, A-shrewd cardinals
• Unfoldable cardinal, strongly unfoldable cardinal
• Weakly superstrong cardinal
• Strongly uplifting (=superstrongly unfoldable) cardinal
• Ethereal cardinal
• Subtle cardinal
• Weakly ineffable (=almost 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 cardinals, ω-Ramsey cardinal
• (1-)remarkable (=virtually supercompact), virtually measurable, strategic ω-Ramsey cardinals, weak Proper Forcing Axiom
• Virtually extendible (=2-remarkable), virtually C(n)-extendible (=n+1-remarkable) cardinals, completely remarkable cardinal, Generic Vopěnka's Principle
• Virtually Shelah for supercompactness cardinal
• The n-iterable and virtually n-huge* hierarchy
• Virtually rank-into-rank cardinal
• ω-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
• Πα-Romsey, completely Romsey (=ω-very Ramsey), α-hyper completely Romsey, super completely Romsey cardinals
• α-filter property, α-Ramsey cardinal (for ω<α<κ), almost fully Ramsey (=<κ-Ramsey) cardinal
• 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 cardinal, ω1-very Ramsey cardinal, κ-very Ramsey cardinal
• Singular Jónsson cardinal
• 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
• Strongly tall cardinal
• Woodin cardinal
• The axiom of determinacy and its projective counterpart
• Shelah cardinal
• Superstrong cardinal, C(n)-superstrong hierarchy
• Subcompact cardinal
• The proper forcing axiom and Martin's maximum
• Indestructible weakly compact cardinal
• 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 cardinals, enhanced supercompact cardinal, λ-hypercompact cardinals, hypercompact cardinal
• Woodin for strong compactness
• α-extendible hierarchy, extendible cardinal, C(n)-extendible hierarchy
• 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
• 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…
• 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

Also see: Kunen Inconsistency, Berkeley Cardinal Hierarchy, The Expanded Large Cardinal Catalogue,