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

HTML articles powered by AMS MathViewer

- by Matteo Viale
- J. Amer. Math. Soc.
**29**(2016), 675-728 - DOI: https://doi.org/10.1090/jams/844
- Published electronically: August 19, 2015
- PDF | Request permission

## 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.## References

- David Asperó,
*Coding into $H(\omega _2)$, together (or not) with forcing axioms. A survey*, Computational prospects of infinity. Part II. Presented talks, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 15, World Sci. Publ., Hackensack, NJ, 2008, pp. 23–46. MR**2449458**, DOI 10.1142/9789812796554_{0}002 - David Asperó, Paul Larson, and Justin Tatch Moore,
*Forcing axioms and the continuum hypothesis*, Acta Math.**210**(2013), no. 1, 1–29. MR**3037610**, DOI 10.1007/s11511-013-0089-7 - James E. Baumgartner,
*All $\aleph _{1}$-dense sets of reals can be isomorphic*, Fund. Math.**79**(1973), no. 2, 101–106. MR**317934**, DOI 10.4064/fm-79-2-101-106 - Douglas R. Burke,
*Precipitous towers of normal filters*, J. Symbolic Logic**62**(1997), no. 3, 741–754. MR**1472122**, DOI 10.2307/2275571 - Andrés Eduardo Caicedo and Boban Velic̆ković,
*The bounded proper forcing axiom and well orderings of the reals*, Math. Res. Lett.**13**(2006), no. 2-3, 393–408. MR**2231126**, DOI 10.4310/MRL.2006.v13.n3.a5 - Sean Cox,
*PFA and ideals on $\omega _2$ whose associated forcings are proper*, Notre Dame J. Form. Log.**53**(2012), no. 3, 397–412. MR**2981015**, DOI 10.1215/00294527-1716793 - Ilijas Farah,
*All automorphisms of the Calkin algebra are inner*, Ann. of Math. (2)**173**(2011), no. 2, 619–661. MR**2776359**, DOI 10.4007/annals.2011.173.2.1 - Matthew Foreman,
*Ideals and generic elementary embeddings*, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 885–1147. MR**2768692**, DOI 10.1007/978-1-4020-5764-9_{1}4 - M. Foreman, M. Magidor, and S. Shelah,
*Martin’s maximum, saturated ideals, and nonregular ultrafilters. I*, Ann. of Math. (2)**127**(1988), no. 1, 1–47. MR**924672**, DOI 10.2307/1971415 - U. Fuchs,
*Donder’s version of revised countable support*(1992). - Joel David Hamkins and Benedikt Löwe,
*The modal logic of forcing*, Trans. Amer. Math. Soc.**360**(2008), no. 4, 1793–1817. MR**2366963**, DOI 10.1090/S0002-9947-07-04297-3 - Thomas Jech,
*Set theory*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR**1940513** - Joel David Hamkins and Thomas A. Johnstone,
*Resurrection axioms and uplifting cardinals*, Arch. Math. Logic**53**(2014), no. 3-4, 463–485. MR**3194674**, DOI 10.1007/s00153-014-0374-y - Peter Koellner and W. Hugh Woodin,
*Incompatible $\Omega$-complete theories*, J. Symbolic Logic**74**(2009), no. 4, 1155–1170. MR**2583814**, DOI 10.2178/jsl/1254748685 - Paul B. Larson,
*The stationary tower*, University Lecture Series, vol. 32, American Mathematical Society, Providence, RI, 2004. Notes on a course by W. Hugh Woodin. MR**2069032**, DOI 10.1090/ulect/032 - Paul B. Larson,
*Martin’s maximum and definability in $H(\aleph _2)$*, Ann. Pure Appl. Logic**156**(2008), no. 1, 110–122. MR**2474445**, DOI 10.1016/j.apal.2008.06.012 - Paul B. Larson,
*Forcing over models of determinacy*, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 2121–2177. MR**2768703**, DOI 10.1007/978-1-4020-5764-9_{2}5 - Justin Tatch Moore,
*Set mapping reflection*, J. Math. Log.**5**(2005), no. 1, 87–97. MR**2151584**, DOI 10.1142/S0219061305000407 - Justin Tatch Moore,
*A five element basis for the uncountable linear orders*, Ann. of Math. (2)**163**(2006), no. 2, 669–688. MR**2199228**, DOI 10.4007/annals.2006.163.669 - Saharon Shelah,
*Infinite abelian groups, Whitehead problem and some constructions*, Israel J. Math.**18**(1974), 243–256. MR**357114**, DOI 10.1007/BF02757281 - Saharon Shelah,
*Decomposing uncountable squares to countably many chains*, J. Combinatorial Theory Ser. A**21**(1976), no. 1, 110–114. MR**409196**, DOI 10.1016/0097-3165(76)90053-4 - Stevo Todorcevic,
*Basis problems in combinatorial set theory*, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 43–52. MR**1648055** - Stevo Todorcevic,
*Generic absoluteness and the continuum*, Math. Res. Lett.**9**(2002), no. 4, 465–471. MR**1928866**, DOI 10.4310/MRL.2002.v9.n4.a6 - S. Todorčević,
*The power et of $\omega _1$ and the continuum problem*(2014). Available at http://logic.harvard.edu/Todorcevic_Structure4.pdf. - K. Tsaprounis,
*Large cardinals and resurrection axioms*, 2012. Ph.D. thesis. - Boban Velic̆ković,
*Forcing axioms and stationary sets*, Adv. Math.**94**(1992), no. 2, 256–284. MR**1174395**, DOI 10.1016/0001-8708(92)90038-M - Matteo Viale,
*A family of covering properties*, Math. Res. Lett.**15**(2008), no. 2, 221–238. MR**2385636**, DOI 10.4310/MRL.2008.v15.n2.a2 - Matteo Viale,
*Martin’s maximum revisited*(2013). Available at http://www.personalweb.unito.it/matteo.viale/. - M. Viale and G. Audrito,
*Absoluteness via resurrection*(2014). Available at http://www.personalweb.unito.it/matteo.viale/absoluteness.pdf. - M. Viale, G. Audrito, and S. Steila,
*A boolean algebraic approach to semiproper iterations*(2013). Notes available at http://www.personalweb.unito.it/matteo.viale/. - Matteo Viale and Christoph Weiß,
*On the consistency strength of the proper forcing axiom*, Adv. Math.**228**(2011), no. 5, 2672–2687. MR**2838054**, DOI 10.1016/j.aim.2011.07.016 - W. Hugh Woodin,
*The axiom of determinacy, forcing axioms, and the nonstationary ideal*, De Gruyter Series in Logic and its Applications, vol. 1, Walter de Gruyter & Co., Berlin, 1999. MR**1713438**, DOI 10.1515/9783110804737

## Bibliographic Information

**Matteo Viale**- Affiliation: Department of Mathematics “Giuseppe Peano”, University of Torino, via Carlo Alberto 10, 10125, Torino, Italy
- MR Author ID: 719238
- Email: matteo.viale@unito.it
- 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.
- © Copyright 2015 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**29**(2016), 675-728 - MSC (2010): Primary 03E35; Secondary 03E40, 03E57
- DOI: https://doi.org/10.1090/jams/844
- MathSciNet review: 3486170

Dedicated: To Chiara and to our kids, Pietro and Adele.