Low level definability above large cardinals

Abstract

We study connections between definability in generalized descriptive set theory and large cardinals, under ZFC. We show that if $\kappa$ is a limit of measurables then there is no wellorder of a subset of $P(\kappa)$ of length $\geq\kappa^+$ which is $\Sigma_1(V_\kappa\cup\mathrm{OR})$, answering a question of L\"ucke and M\"uller. However, consistently, a Woodin cardinal exists and for every uncountable cardinal $\kappa$ which is not a limit of measurables, there is a $\Sigma_1(H_\kappa\cup\{\kappa\})$-good wellorder of $H_{\kappa^+}$. If $\kappa$ is a limit of measurables and $\kappa$ has uncountable cofinality then there is no $\Sigma_1(V_\kappa\cup\mathrm{OR})$ almost disjoint family $F\subseteq P(\kappa)$ of cardinality $>\kappa$. Consistently, $\Pi_1(\{\kappa\})$ mad families and maximal independent families $F\subseteq P(\kappa)$ exist, $\kappa$ is a limit of measurables, and more. If $\kappa$ is weakly compact and every $\Sigma_1(V_\kappa\cup\{\kappa\})$ subset of $P(\kappa)$ of cardinality $>\kappa$ contains a perfect subset of the right kind, then there is an inner model with a weakly compact limit of measurables. We prove some related facts regarding $\Sigma_1(V_\lambda\cup\{V_\lambda\}\cup\mathrm{OR})$ when $I_2(\lambda)$ holds. These depend on an analysis of fixed points of linear iterations involving $I_2(\lambda)$-extenders.