Woodin cardinals

Woodin cardinals are a class of large cardinal axioms in set theory introduced by W. Hugh Woodin that play a central role in the study of determinacy, inner models, and forcing extensions. They strengthen measurability and extend concepts from Solovay-style large cardinals to provide fine control over the structure of the set-theoretic universe, especially with respect to definable subsets of the real numbers. Woodin cardinals occur in hierarchies that connect with Measurable cardinal, Supercompact cardinal, and hypotheses like Projective Determinacy and the Axiom of Determinacy-related frameworks.

Definition and basic properties

A cardinal κ is called Woodin (in one common formulation) if for every function f: κ → κ there exists a cardinal λ < κ and an elementary embedding j: V → M with critical point less than κ such that j(f)(λ) > κ and V_{j(f)(λ)} ⊂ M; equivalently, κ is strongly inaccessible and for every A ⊂ V_κ there is a <κ-strongness embedding reflecting A. This formulation connects to earlier notions like σ-strongly compact cardinal and Mahlo cardinal through reflection properties. Basic structural properties include that every Woodin cardinal is Inaccessible cardinal and, under mild background hypotheses, is limits of Measurable cardinals or carries a rich collection of extenders analogous to those used for Supercompact cardinals. Woodin cardinals are preserved under canonical inner models that accommodate extender sequences similar to those in Mitchell order analyses.

Equivalent formulations and characterizations

Multiple equivalent characterizations of Woodin cardinals exist: via elementary embeddings with closure properties, via stationary tower combinatorics such as the existence of certain stationary sets in P_κ(λ), and via extender algebra generics producing models with strong closure. One can phrase the Woodin property using the existence of strongness measures or using the failure of the κ-approximation and κ-covering properties in appropriate ultrapowers. The interplay with the Stationary set reflection principles and with the Approachability ideal yields alternative descriptions used in applications to Descriptive set theory and inner models like L[E]. In many proofs, characterizations through the nonexistence of certain counterexamples in H_{θ} or via generic elementary embeddings obtained from the Stationary tower forcing are particularly useful.

Large-cardinal strength and consistency

Woodin cardinals occupy a precise place in the large-cardinal hierarchy: a single Woodin cardinal is weaker than a measurable limit of Woodin cardinals or a supercompact cardinal but stronger than a measurable cardinal in many relative-consistency senses. Finite sequences of Woodin cardinals, ω-sequences, and countable sets of Woodin cardinals produce increasingly strong hypotheses that imply determinacy for more complex pointclasses, linking to consistency strengths calibrated by inner model comparisons with measurable and Strong cardinals. Consistency results often relate Woodin cardinals to statements like Projective Determinacy and determinacy for larger pointclasses; for example, ZFC + "there are n Woodin cardinals with a measurable above" has the consistency strength of certain fragments of projective determinacy. Relative consistency proofs typically use extenders and comparisons with core models such as K or K^c and exploit the fine-structural analysis developed in work by Steel, Mitchell, and Magidor.

Forcing, indestructibility, and preservation

Forcing techniques interact richly with Woodin cardinals. The Stationary tower forcing was devised to exploit the Woodin property to produce elementary embeddings and extend determinacy arguments. Woodin cardinals can be indestructible under certain <κ-closed or strategically closed forcings, but not under arbitrary small forcing; delicate preservation theorems specify which posets maintain the Woodin property. Easton-type forcings and Prikry-type forcings can change combinatorial features near a Woodin without destroying higher instances, while preparations using Laver-like functions for supercompactness have analogues in the Woodin context that yield partial indestructibility. Techniques by Woodin and Shelah show when the Woodin property is preserved by adding reals or by collapsing cardinals, and when generic embeddings obtained from forcing correspond to internal extender sequences in models like M_n# or core models with finitely many Woodins.

Applications in determinacy and inner model theory

Woodin cardinals are pivotal in bridging large-cardinal axioms and determinacy hypotheses: the existence of sufficiently many Woodin cardinals implies strong forms of determinacy such as Determinacy in L(R), Projective Determinacy, and stronger determinacy for higher pointclasses of Borel hierarchy-type sets. Inner model theory uses Woodin cardinals to construct models such as the minimal iterable models with Woodin cardinals, producing objects like M_n and HOD analyses with determinacy consequences. Results by Martin, Steel, Woodin, and Neeman tie Woodin hypotheses to the regularity properties of sets of reals and to absoluteness phenomena like Σ^2_1 absoluteness and Ω-logic formulations. Furthermore, Woodin cardinals enable coding of reals into extender models, support comparison lemmas for hybrid mice, and underpin projective-scale constructions central to descriptive set theory interactions with large-cardinal axioms.

Category:Large cardinals