Prevalence of Generic Laver Diamond

TL;DR

This paper investigates the prevalence and consistency of the Generic Laver Diamond principle, showing it holds under various conditions and weaker hypotheses than previously known, with implications for set-theoretic combinatorics.

Contribution

It weakens the hypothesis needed for the existence of the Generic Laver Diamond and demonstrates its widespread occurrence in different models of set theory.

Findings

PFA implies iamond_{ ext{Lav}}(\u03c4) for c4= 2

iamond_{ ext{Lav}}(c4) holds in models where Chang's Conjecture fails

A specific forcing adds iamond_{ ext{Lav}}(c4) for successor cardinals

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 $\Diamond_{\text{Lav}}(\kappa)$.