Prevalence of Generic Laver Diamond

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 $\Diamond _{\text {Lav}}(\kappa )$.