Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

Request Permissions   Purchase Content 
 
 

 

Category forcings, $ {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
DOI: https://doi.org/10.1090/jams/844
Published electronically: August 19, 2015
MathSciNet review: 3486170
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 {{\sf 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 {{\sf MM}}^{+++}$ is consistent relative to large cardinal axioms and that $ \text {{\sf MM}}^{+++}$ makes the theory of the Chang model $ L([$$ \text {{\rm Ord}}]^{\leq \aleph _1})$ with parameters in $ P(\omega _1)$ generically invariant for stationary set preserving forcings that preserve this axiom. We also show that our results give a close to optimal extension to the Chang model $ L([$$ \text {{\rm Ord}}]^{\leq \aleph _1})$ of Woodin's generic absoluteness results for the Chang model $ L([$$ \text {{\rm Ord}}]^{\aleph _0})$ and give an a posteriori explanation of the success forcing axioms have met in set theory.


References [Enhancements On Off] (What's this?)

  • [1] 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 (2010a:03058), https://doi.org/10.1142/9789812796554_0002
  • [2] David Asperó, Paul Larson, and Justin Tatch Moore, Forcing axioms and the continuum hypothesis, Acta Math. 210 (2013), no. 1, 1-29. MR 3037610, https://doi.org/10.1007/s11511-013-0089-7
  • [3] James E. Baumgartner, All $ \aleph _{1}$-dense sets of reals can be isomorphic, Fund. Math. 79 (1973), no. 2, 101-106. MR 0317934 (47 #6483)
  • [4] Douglas R. Burke, Precipitous towers of normal filters, J. Symbolic Logic 62 (1997), no. 3, 741-754. MR 1472122 (2000d:03114), https://doi.org/10.2307/2275571
  • [5] Andrés Eduardo Caicedo and Boban Veličković, The bounded proper forcing axiom and well orderings of the reals, Math. Res. Lett. 13 (2006), no. 2-3, 393-408. MR 2231126 (2007d:03076), https://doi.org/10.4310/MRL.2006.v13.n3.a5
  • [6] 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, https://doi.org/10.1215/00294527-1716793
  • [7] Ilijas Farah, All automorphisms of the Calkin algebra are inner, Ann. of Math. (2) 173 (2011), no. 2, 619-661. MR 2776359 (2012e:03103), https://doi.org/10.4007/annals.2011.173.2.1
  • [8] Matthew Foreman, Ideals and generic elementary embeddings, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 885-1147. MR 2768692, https://doi.org/10.1007/978-1-4020-5764-9_14
  • [9] 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 (89f:03043), https://doi.org/10.2307/1971415
  • [10] U. Fuchs, Donder's version of revised countable support (1992).
  • [11] Joel David Hamkins and Benedikt Löwe, The modal logic of forcing, Trans. Amer. Math. Soc. 360 (2008), no. 4, 1793-1817. MR 2366963 (2009h:03068), https://doi.org/10.1090/S0002-9947-07-04297-3
  • [12] Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513 (2004g:03071)
  • [13] Joel David Hamkins and Thomas A. Johnstone, Resurrection axioms and uplifting cardinals, Arch. Math. Logic 53 (2014), no. 3-4, 463-485. MR 3194674, https://doi.org/10.1007/s00153-014-0374-y
  • [14] Peter Koellner and W. Hugh Woodin, Incompatible $ \Omega $-complete theories, J. Symbolic Logic 74 (2009), no. 4, 1155-1170. MR 2583814 (2011a:03054), https://doi.org/10.2178/jsl/1254748685
  • [15] 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 (2005e:03001)
  • [16] Paul B. Larson, Martin's maximum and definability in $ H(\aleph _2)$, Ann. Pure Appl. Logic 156 (2008), no. 1, 110-122. MR 2474445 (2009k:03085), https://doi.org/10.1016/j.apal.2008.06.012
  • [17] Paul B. Larson, Forcing over models of determinacy, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 2121-2177. MR 2768703, https://doi.org/10.1007/978-1-4020-5764-9_25
  • [18] Justin Tatch Moore, Set mapping reflection, J. Math. Log. 5 (2005), no. 1, 87-97. MR 2151584 (2006c:03076), https://doi.org/10.1142/S0219061305000407
  • [19] Justin Tatch Moore, A five element basis for the uncountable linear orders, Ann. of Math. (2) 163 (2006), no. 2, 669-688. MR 2199228 (2007d:03085), https://doi.org/10.4007/annals.2006.163.669
  • [20] Saharon Shelah, Infinite abelian groups, Whitehead problem and some constructions, Israel J. Math. 18 (1974), 243-256. MR 0357114 (50 #9582)
  • [21] Saharon Shelah, Decomposing uncountable squares to countably many chains, J. Combin. Theory Ser. A 21 (1976), no. 1, 110-114. MR 0409196 (53 #12958)
  • [22] 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 (2000c:03039)
  • [23] Stevo Todorcevic, Generic absoluteness and the continuum, Math. Res. Lett. 9 (2002), no. 4, 465-471. MR 1928866 (2003f:03067), https://doi.org/10.4310/MRL.2002.v9.n4.a6
  • [24] S. Todorčević, The power et of $ \omega _1$ and the continuum problem (2014). Available at http://logic.harvard.edu/TodorcevicStructure4.pdf.
  • [25] K. Tsaprounis, Large cardinals and resurrection axioms, 2012. Ph.D. thesis.
  • [26] Boban Veličković, Forcing axioms and stationary sets, Adv. Math. 94 (1992), no. 2, 256-284. MR 1174395 (93k:03045), https://doi.org/10.1016/0001-8708(92)90038-M
  • [27] Matteo Viale, A family of covering properties, Math. Res. Lett. 15 (2008), no. 2, 221-238. MR 2385636 (2009b:03130), https://doi.org/10.4310/MRL.2008.v15.n2.a2
  • [28] Matteo Viale, Martin's maximum revisited (2013). Available at http://www.personalweb.unito.it/matteo.viale/.
  • [29] M. Viale and G. Audrito, Absoluteness via resurrection (2014). Available at http://www.personalweb.unito.it/matteo.viale/absoluteness.pdf.
  • [30] 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/.
  • [31] Matteo Viale and Christoph Weiß, On the consistency strength of the proper forcing axiom, Adv. Math. 228 (2011), no. 5, 2672-2687. MR 2838054 (2012m:03131), https://doi.org/10.1016/j.aim.2011.07.016
  • [32] 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 (2001e:03001)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 03E35, 03E40, 03E57

Retrieve articles in all journals with MSC (2010): 03E35, 03E40, 03E57


Additional Information

Matteo Viale
Affiliation: Department of Mathematics “Giuseppe Peano”, University of Torino, via Carlo Alberto 10, 10125, Torino, Italy
Email: matteo.viale@unito.it

DOI: https://doi.org/10.1090/jams/844
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

American Mathematical Society