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Prevalence of Generic Laver Diamond

Author: Sean D. Cox
Journal: Proc. Amer. Math. Soc. 143 (2015), 4045-4058
MSC (2010): Primary 03E57, 03E55, 03E35, 03E05
Published electronically: February 26, 2015
MathSciNet review: 3359593
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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:

  1. $ L$ satisfies $ \Diamond ^+_{\text {Lav}}(\kappa )$ whenever $ \kappa $ is a successor cardinal, or when the appropriate version of Chang's Conjecture fails.
  2. 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 )$.

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Additional Information

Sean D. Cox
Affiliation: Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, 1015 Floyd Avenue, Richmond, Virginia 23284

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.
Dedicated: In memory of Richard Laver, 1942-2012
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2015 American Mathematical Society

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