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Category forcings, $\text {\textsf {MM}}^{+++}$, and generic absoluteness for the theory of strong forcing axioms

Author: Matteo Viale
Journal: J. Amer. Math. Soc. 29 (2016), 675-728
MSC (2010): Primary 03E35; Secondary 03E40, 03E57
Published electronically: August 19, 2015
MathSciNet review: 3486170
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Abstract: We analyze certain subfamilies of the category of complete boolean algebras with complete homomorphisms, families which are of particular interest in set theory. In particular we study the category whose objects are stationary set preserving, atomless complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We introduce a maximal forcing axiom $\text {\textsf {MM}}^{+++}$ as a combinatorial property of this category. This forcing axiom strengthens Martin’s maximum and can be seen at the same time as a strenghtening of Baire’s category theorem and of the axiom of choice. Our main results show that $\text {\textsf {MM}}^{+++}$ is consistent relative to large cardinal axioms and that $\text {\textsf {MM}}^{+++}$ makes the theory of the Chang model $L([\text {\textrm {Ord}}]^{\leq \aleph _1})$ with parameters in $P(\omega _1)$ generically invariant for stationary set preserving forcings that preserve this axiom. We also show that our results give a close to optimal extension to the Chang model $L([\text {\textrm {Ord}}]^{\leq \aleph _1})$ of Woodin’s generic absoluteness results for the Chang model $L([\text {\textrm {Ord}}]^{\aleph _0})$ and give an a posteriori explanation of the success forcing axioms have met in set theory.

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Additional Information

Matteo Viale
Affiliation: Department of Mathematics “Giuseppe Peano”, University of Torino, via Carlo Alberto 10, 10125, Torino, Italy
MR Author ID: 719238

Received by editor(s): January 22, 2013
Received by editor(s) in revised form: April 15, 2013, January 15, 2014, and April 5, 2015
Published electronically: August 19, 2015
Additional Notes: The author acknowledges support from PRIN grant 2009 (Modelli e Insiemi), PRIN grant 2012 (Logica, Modelli e Insiemi), Kurt Gödel Research Fellowship 2010, San Paolo Junior PI grant 2012; the Fields Institute in Mathematical Sciences.
Dedicated: To Chiara and to our kids, Pietro and Adele.
Article copyright: © Copyright 2015 American Mathematical Society