Prevalence of Generic Laver Diamond

Abstract

Viale \cite{Viale_GuessingModel} 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 \cite{MR2860182}---that forces $\Diamond_{\text{Lav}}(\kappa)$.