$\mathsf {Sealing}$ from iterability
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- by Grigor Sargsyan and Nam Trang;
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 229-248
- DOI: https://doi.org/10.1090/btran/65
- Published electronically: March 2, 2021
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Abstract:
We show that if $V$ has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy (as defined in the paper) then $\mathsf {Sealing}$ holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to other work by the authors where it is shown that $\mathsf {Sealing}$ holds in a generic extension of a certain minimal universe. The current theorem is more general in that no minimality assumption is needed. A corollary of the main theorem is that $\mathsf {Sealing}$ is consistent relative to the existence of a Woodin cardinal which is a limit of Woodin cardinals. This improves significantly on the first consistency of $\mathsf {Sealing}$ obtained by W.H. Woodin.
The $\mathsf {Largest\ Suslin\ Axiom}$ ($\mathsf {LSA}$) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let $\mathsf {LSA\text {-}over\text {-}uB}$ be the statement that in all (set) generic extensions there is a model of $\mathsf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. The other main result of the paper shows that assuming $V$ has a proper class of inaccessible cardinals which are limit of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, in the universe $V[g]$, where $g$ is $V$-generic for the collapse of the successor of the least strong cardinal to be countable, the theory $\mathsf {LSA}$-$\mathsf {over}$-$\mathsf {UB}$ fails; this implies that $\mathsf {LSA\text {-}over\text {-}UB}$ is not equivalent to $\mathsf {Sealing}$ (over the base theory of $V[g]$). This is interesting and somewhat unexpected, in light of other work by the authors. Compare this result with Steel’s well-known theorem that “$\mathsf {AD}^{L(\mathbb {R})}$ holds in all generic extensions” is equivalent to “the theory of $L(\mathbb {R})$ is sealed” in the presence of a proper class of measurable cardinals.
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Bibliographic Information
- Grigor Sargsyan
- Affiliation: Department of Mathematics, Rutgers University, New Jersey
- MR Author ID: 677243
- Email: gs481@math.rutgers.edu
- Nam Trang
- Affiliation: Department of Mathematics, University of North Texas, Denton, Texas
- MR Author ID: 1067824
- ORCID: 0000-0002-7528-682X
- Email: Nam.Trang@unt.edu
- Received by editor(s): October 9, 2019
- Received by editor(s) in revised form: December 15, 2020
- Published electronically: March 2, 2021
- Additional Notes: The authors would like to thank the NSF for its generous support. The first author was supported by NSF Career Award DMS-1352034. The second author was supported by NSF Grants DMS-1565808, DMS-1849295, and NSF Career Grant DMS-1945592.
- © Copyright 2021 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 229-248
- MSC (2020): Primary 03E15, 03E45, 03E60
- DOI: https://doi.org/10.1090/btran/65
- MathSciNet review: 4223043