Large Cardinals

Origin. Hausdorff described weakly inaccessible cardinals (1908); Mahlo added his hierarchy (1911); Ulam and Tarski 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
κ	U+03BA	Kappa	The large cardinal
j	—	Embedding	Elementary j : V → M
crit(j)	—	Critical point	First ordinal moved
V_κ	—	Rank	The κ-th level of the cumulative hierarchy
M	—	Inner model	Transitive target of the embedding
U	—	Ultrafilter	κ-complete, non-principal
L	—	Constructible universe	The model measurables exclude

Metatheory. Large cardinals decide statements ZFC leaves open: Woodin cardinals give projective determinacy, and a proper class of them 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.