| _version_ | 1866901179062026240 |
|---|---|
| author | Revista, Zen MFC, 10 |
| author_facet | Revista, Zen MFC, 10 |
| contents | <div>Mathematical Applications of Science Fiction</div> <div> </div> <p>We establish a canonical correspondence between the con-<br>sistency strength of the Proper Forcing Axiom (PFA) and<br>the existence of Woodin cardinals within the core model<br>K. Specifically, we demonstrate that if PFA holds, then for<br>every set X, the inner model K(X) admits a proper class<br>of Woodin cardinals, or exhibits a failure of covering at a<br>measurable cardinal. We construct a fine-structural analysis<br>of the hierarchy of extender models L[⃗E] to show that the<br>failure of □κ principles, implied by PFA, necessitates the ex-<br>istence of δ-strong cardinals in the core model which reflect<br>to Woodin cardinals under generic absoluteness. This work<br>synthesizes the Core Model Induction with Steel’s analysis<br>of the PFA conjecture, providing a transfinite lower bound<br>for the consistency strength of proper forcing in terms of<br>inner model theory.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18110552 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Woodin Cardinals and the Consistency Strength of Proper Forcing Axioms in Inner Models Revista, Zen MFC, 10 <div>Mathematical Applications of Science Fiction</div> <div> </div> <p>We establish a canonical correspondence between the con-<br>sistency strength of the Proper Forcing Axiom (PFA) and<br>the existence of Woodin cardinals within the core model<br>K. Specifically, we demonstrate that if PFA holds, then for<br>every set X, the inner model K(X) admits a proper class<br>of Woodin cardinals, or exhibits a failure of covering at a<br>measurable cardinal. We construct a fine-structural analysis<br>of the hierarchy of extender models L[⃗E] to show that the<br>failure of □κ principles, implied by PFA, necessitates the ex-<br>istence of δ-strong cardinals in the core model which reflect<br>to Woodin cardinals under generic absoluteness. This work<br>synthesizes the Core Model Induction with Steel’s analysis<br>of the PFA conjecture, providing a transfinite lower bound<br>for the consistency strength of proper forcing in terms of<br>inner model theory.</p> |
| title | Woodin Cardinals and the Consistency Strength of Proper Forcing Axioms in Inner Models |
| url | https://doi.org/10.5281/zenodo.18110552 |