Prevalence of Generic Laver Diamond
HTML articles powered by AMS MathViewer
- by Sean D. Cox PDF
- Proc. Amer. Math. Soc. 143 (2015), 4045-4058 Request permission
Abstract:
Viale (2012) introduced the notion of Generic Laver Diamond at $\kappa$—which we denote $\Diamond _{\text {Lav}}(\kappa )$—asserting the existence of a single function from $\kappa \to H_\kappa$ that behaves much like a supercompact Laver function, except with generic elementary embeddings rather than internal embeddings. Viale proved that the Proper Forcing Axiom (PFA) implies $\Diamond _{\text {Lav}}(\omega _2)$. We strengthen his theorem by weakening the hypothesis to a statement strictly weaker than PFA. We also show that the principle $\Diamond _{\text {Lav}}(\kappa )$ provides a uniform, simple construction of 2-cardinal diamonds, and prove that $\Diamond _{\text {Lav}}(\kappa )$ is quite prevalent in models of set theory; in particular:
-
$L$ satisfies $\Diamond ^+_{\text {Lav}}(\kappa )$ whenever $\kappa$ is a successor cardinal, or when the appropriate version of Chang’s Conjecture fails.
-
For any successor cardinal $\kappa$, there is a $\kappa$-directed closed class forcing—namely, the forcing from Friedman-Holy (2011)—that forces $\Diamond _{\text {Lav}}(\kappa )$.
References
- Andrés Eduardo Caicedo and Boban Veli ković, 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
- Paul Corazza, Laver sequences for extendible and super-almost-huge cardinals, J. Symbolic Logic 64 (1999), no. 3, 963–983. MR 1779746, DOI 10.2307/2586614
- Sean Cox and Matteo Viale, Martin’s Maximum and tower forcing, Israel J. Math. 197 (2013), no. 1, 347–376. MR 3096619, DOI 10.1007/s11856-013-0004-0
- Hans-Dieter Donder and Jean-Pierre Levinski, Some principles related to Chang’s conjecture, Ann. Pure Appl. Logic 45 (1989), no. 1, 39–101. MR 1024901, DOI 10.1016/0168-0072(89)90030-4
- Hans-Dieter Donder and Pierre Matet, Two cardinal versions of diamond, Israel J. Math. 83 (1993), no. 1-2, 1–43. MR 1239715, DOI 10.1007/BF02764635
- Joel David Hamkins, A class of strong diamond principles, Preprint (2002), available at math/0211419.
- Alex Hellsten, Diamonds on large cardinals, Ann. Acad. Sci. Fenn. Math. Diss. 134 (2003), 48. Dissertation, University of Helsinki, Helsinki, 2003. MR 2026390
- Thomas J. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1972/73), 165–198. MR 325397, DOI 10.1016/0003-4843(73)90014-4
- Matthew Foreman and Menachem Magidor, Large cardinals and definable counterexamples to the continuum hypothesis, Ann. Pure Appl. Logic 76 (1995), no. 1, 47–97. MR 1359154, DOI 10.1016/0168-0072(94)00031-W
- Sy-David Friedman and Peter Holy, Condensation and large cardinals, Fund. Math. 215 (2011), no. 2, 133–166. MR 2860182, DOI 10.4064/fm215-2-3
- Moti Gitik and Saharon Shelah, On certain indestructibility of strong cardinals and a question of Hajnal, Arch. Math. Logic 28 (1989), no. 1, 35–42. MR 987765, DOI 10.1007/BF01624081
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
- Richard Laver, Making the supercompactness of $\kappa$ indestructible under $\kappa$-directed closed forcing, Israel J. Math. 29 (1978), no. 4, 385–388. MR 472529, DOI 10.1007/BF02761175
- David Richard Law, An abstract condensation property, ProQuest LLC, Ann Arbor, MI, 1994. Thesis (Ph.D.)–California Institute of Technology. MR 2691107
- Itay Neeman, Forcing with sequences of models of two types, Notre Dame J. Form. Log. 55 (2014), no. 2, 265–298. MR 3201836, DOI 10.1215/00294527-2420666
- Saharon Shelah, Can you take Solovay’s inaccessible away?, Israel J. Math. 48 (1984), no. 1, 1–47. MR 768264, DOI 10.1007/BF02760522
- Masahiro Shioya, Splitting $\scr P_\kappa \lambda$ into maximally many stationary sets, Israel J. Math. 114 (1999), 347–357. MR 1738689, DOI 10.1007/BF02785587
- Stevo Todorčević, Partition problems in topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, RI, 1989. MR 980949, DOI 10.1090/conm/084
- Matteo Viale, Guessing models and generalized Laver diamond, Ann. Pure Appl. Logic 163 (2012), no. 11, 1660–1678. MR 2959666, DOI 10.1016/j.apal.2011.12.015
- 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
Additional Information
- Sean D. Cox
- Affiliation: Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, 1015 Floyd Avenue, Richmond, Virginia 23284
- MR Author ID: 883409
- Email: scox9@vcu.edu
- Received by editor(s): April 5, 2013
- Received by editor(s) in revised form: April 18, 2014
- Published electronically: February 26, 2015
- Additional Notes: Part of this work was done while the author participated in the Thematic Program on Forcing and its Applications at the Fields Institute, which was partially supported from NSF grant DMS-1162052.
- Communicated by: Mirna Džamonja
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 4045-4058
- MSC (2010): Primary 03E57, 03E55, 03E35, 03E05
- DOI: https://doi.org/10.1090/S0002-9939-2015-12540-3
- MathSciNet review: 3359593
Dedicated: In memory of Richard Laver, 1942-2012