Abstract
We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. In particular we show that, in combination with class many Woodin cardinals, the forcing axiom MM ++ makes the \({\Pi_2}\)-fragment of the theory of \({H_{\aleph_2}}\) invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a promising generalization to \({H_{\aleph_2}}\) of Woodin’s absoluteness results for \({L(\mathbb{R})}\). In due course of proving this, we shall give a new proof of some of these results of Woodin. Finally we relate our generic absoluteness results with the resurrection axioms introduced by Hamkins and Johnstone and with their unbounded versions introduced by Tsaprounis.
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References
Audrito, G., Viale, M.: Absoluteness via resurrection. arXiv:1404.2111
Bagaria J.: Bounded forcing axioms as principles of generic absoluteness. Arch. Math. Log. 39(6), 393–401 (2000)
Cohen P.J.: The independence of the continuum hypothesis. Proc. Natl. Acad. Sci. USA 50, 1143–1148 (1963)
Cox S.: The diagonal reflection principle. Proc. Am. Math. Soc. 140(8), 2893–2902 (2012)
Farah I.: All automorphisms of the Calkin algebra are inner. Ann. Math. (2) 173(2), 619–661 (2011)
Foreman M., Magidor M., Shelah S.: Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Ann. Math. (2) 127(1), 1–47 (1988)
Hamkins J.D., Johnstone T.A.: Resurrection axioms and uplifting cardinals. Arch. Math. Log. 53(3–4), 463–485 (2014)
Jech, T.: Set theory. Springer Monographs in Mathematics. Springer, Berlin, The third millennium edition, revised and expanded (2003)
Kunen, K.: Set Theory: An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics, vol. 102. North-Holland, Amsterdam (1980)
Larson, P.B.: Forcing over models of determinacy. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, Vols. 1, 2, 3, pp. 2121–2177. Springer, Dordrecht (2010)
Larson, P.B.: The stationary tower. University Lecture Series, vol. 32, American Mathematical Society, Providence, RI. Notes on a course by W. Hugh Woodin (2004)
Moore J.T.: Set mapping reflection. J. Math. Log. 5(1), 87–97 (2005)
Moore J.T.: A five element basis for the uncountable linear orders. Ann. Math. (2) 163(2), 669–688 (2006)
Shelah S.: Infinite abelian groups, Whitehead problem and some constructions. Israel J. Math. 18, 243–256 (1974)
Stavi, J., Väänänen, J.: Reflection principles for the continuum. Logic and algebra, Contemp. Math., vol. 302, Amer. Math. Soc., Providence, RI, pp. 59–84 (2002)
Todorcevic S.: Generic absoluteness and the continuum. Math. Res. Lett. 9(4), 465–471 (2002)
Tsaprounis, K.: Large cardinals and resurrection axioms. (Ph.D. thesis) (2012)
Viale M.: Guessing models and generalized Laver diamond. Ann. Pure Appl. Log. 163(11), 1660–1678 (2012)
Viale, M.: Category forcings, MM +++, and generic absoluteness for the theory of strong forcing axioms. arXiv:1305.2058
Viale M., Weiß C.: On the consistency strength of the proper forcing axiom. Adv. Math. 228(5), 2672–2687 (2011)
Woodin, W.H.: 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)
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Viale, M. Martin’s maximum revisited. Arch. Math. Logic 55, 295–317 (2016). https://doi.org/10.1007/s00153-015-0466-3
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DOI: https://doi.org/10.1007/s00153-015-0466-3