Large Cardinals

Origin. Hausdorff described weakly inaccessible cardinals (1908); Mahlo added his hierarchy (1911); Ulam introduced measurables (1930). Scott (1961) proved that a measurable cardinal implies V ≠ L, which turned the subject from curiosity into method. Solovay, Martin, Woodin, and Magidor built the modern hierarchy and its connection to determinacy.

Models. Axioms asserting cardinals so large that ZFC cannot prove they exist. Each is a strengthening of "there are many ordinals" iterated past the point where the iteration is itself definable, and each is characterized—above the smallest ones—by an elementary embedding of the universe into an inner model.

Formalism.

Inaccessible: κ uncountable, regular, and a strong limit (λ < κ ⟹ 2^λ < κ). V_κ ⊨ ZFC, so Con(ZFC) follows — hence unprovable in ZFC by Gödel II.

Mahlo: κ inaccessible and {λ < κ : λ inaccessible} is stationary in κ.

Measurable — the embedding characterization: κ carries a κ-complete non-principal ultrafilter U. Equivalently: ∃ elementary j : V → M, M transitive, with crit(j) = κ. The critical point is the first ordinal moved: j(α) = α for α < κ, j(κ) > κ.

The hierarchy as closure of M: Strong: V_λ ⊆ M for each λ Woodin: a closure condition on the whole embedding family Supercompact: M^λ ⊆ M for all λ < j(κ) Huge: M^{j(κ)} ⊆ M Rank-into-rank: j : V_λ → V_λ

Kunen inconsistency (1971): There is no nontrivial elementary j : V → V. This bounds the hierarchy from above; every axiom must stop short of it.

Consistency strength: LC₁ ≤ LC₂ iff Con(ZFC + LC₂) ⟹ Con(ZFC + LC₁). Empirically almost linear, with no known reason why.

Symbols.

Symbol	Unicode	Name	Meaning
¹	U+00B9	Superscript one	Second-order level marker in Π¹_{n+1}
Π	U+03A0	Pi	Universal level of the projective hierarchy
α	U+03B1	Alpha	An ordinal below the critical point
κ	U+03BA	Kappa	The large cardinal
λ	U+03BB	Lambda	A cardinal below κ, or the closure index of an embedding
₁	U+2081	Subscript one	Index in LC₁
₂	U+2082	Subscript two	Index in LC₂
ℝ	U+211D	Double-struck R	The reals, in L(ℝ)
→	U+2192	Rightwards arrow	The embedding arrow, j : V → M
∃	U+2203	There exists	Assertion that an embedding exists
≠	U+2260	Not equal to	Scott's conclusion V ≠ L
≤	U+2264	Less than or equal	The consistency-strength ordering
⊆	U+2286	Subset of or equal to	Closure conditions on the target model
⊨	U+22A8	Satisfaction	V_κ ⊨ ZFC for κ inaccessible
⟹	U+27F9	Implies	Relative-consistency implication
j	—	Embedding	Elementary j : V → M
crit(j)	—	Critical point	The first ordinal moved by j
V, V_κ	—	Universe and rank	The cumulative hierarchy, and its κ-th level
M	—	Inner model	Transitive target of the embedding
M^λ	—	Closure	Every λ-sequence from M lies in M
U	—	Ultrafilter	κ-complete, non-principal, witnessing measurability
L	—	Constructible universe	The model measurables exclude
L(ℝ)	—	Constructible closure of the reals	Where a proper class of Woodins gives AD
Con	—	Consistency	Con(ZFC) follows from an inaccessible
LC₁, LC₂	—	Large cardinal axioms	Compared by consistency strength
AD	—	Axiom of Determinacy	Every infinite game on the integers has a winning strategy
ZFC	—	Zermelo–Fraenkel with Choice	The theory the axioms extend
stationary	—	Stationary set	Meets every club subset of κ

Metatheory. Large cardinals decide statements ZFC leaves open: Martin and Steel (1989) proved that n Woodin cardinals with a measurable above them give determinacy for Π¹_{n+1} sets of reals, and Woodin proved that a proper class of Woodins gives AD in L(ℝ). They calibrate the consistency strength of theories that never mention them — determinacy hypotheses, forcing axioms, and combinatorial principles all land on the same scale. That the scale is almost linearly ordered is the subject's central unexplained fact. Scott's theorem (measurable ⟹ V ≠ L) is the reason the inner model programme exists: for each large cardinal one seeks a canonical L-like model containing it, and the programme has not reached supercompacts.

Applies to. Independence and consistency-strength calibration. Determinacy. Inner model theory. Forcing axioms. Any question about what ZFC cannot settle.

Limitations. Not provable in ZFC — that is what makes them axioms rather than theorems, and Gödel's second theorem makes the situation permanent. Their justification is not deductive: reflection arguments, the empirical linearity of the scale, and the fruitfulness of the consequences are the case, and none of the three is a proof. Whether the axioms are true, as opposed to useful and mutually consistent, is not settled by any of that. The inner model programme's failure to reach supercompacts leaves the upper hierarchy without the canonical models that made the lower one tractable.