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 $κ$ is a limit of measurables then there is no wellorder of a subset of $P(κ)$ of length $\geqκ^+$ which is $Σ_1(V_κ\cup\mathrm{OR})$, answering a question of Lücke and Müller. However, consistently, a Woodin cardinal exists and for every uncountable cardinal $κ$ which is not a limit of measurables, there is a $Σ_1(H_κ\cup\{κ\})$-good wellorder of $H_{κ^+}$. If $κ$ is a limit of measurables and $κ$ has uncountable cofinality then there is no $Σ_1(V_κ\cup\mathrm{OR})$ almost disjoint family $F\subseteq P(κ)$ of cardinality $>κ$. Consistently, $Π_1(\{κ\})$ mad families and maximal independent families $F\subseteq P(κ)$ exist, $κ$ is a limit of measurables, and more. If $κ$ is weakly compact and every $Σ_1(V_κ\cup\{κ\})$ subset of $P(κ)$ of cardinality $>κ$ 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 $Σ_1(V_λ\cup\{V_λ\}\cup\mathrm{OR})$ when $I_2(λ)$ holds. These depend on an analysis of fixed points of linear iterations involving $I_2(λ)$-extenders.