Woodin Cardinals and the Consistency Strength of Proper Forcing Axioms in Inner Models
Year: 2025
Type: article
Abstract: Mathematical Applications of Science Fiction We establish a canonical correspondence between the con-sistency strength of the Proper Forcing Axiom (PFA) andthe existence of Woodin cardinals within the core modelK. Specifically, we demonstrate that if PFA holds, then forevery set X, the inner model K(X) admits a proper classof Woodin cardinals, or exhibits a failure of covering at ameasurable cardinal. We construct a fine-structural analysisof the hierarchy of extender models L[⃗E] to show that thefailure of □κ principles, implied by PFA, necessitates the ex-istence of δ-strong cardinals in the core model which reflectto Woodin cardinals under generic absoluteness. This worksynthesizes the Core Model Induction with Steel’s analysisof the PFA conjecture, providing a transfinite lower boundfor the consistency strength of proper forcing in terms ofinner model theory.
Authors: Revista, Zen, MFC, 10
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7117935911
FWCI:
Cites:
Cited by:
Related to: