Low level definability above large cardinals
Farmer Schlutzenberg
Abstract
We study some connections between definability in generalized descriptive set theory and large cardinals, particularly measurable cardinals and limits thereof, working in ZFC. We show that if $κ$ is a limit of measurable cardinals then there is no $Σ_1(H_κ\cup\mathrm{OR})$ wellorder of a subset of $P(κ)$ of length $\geqκ^+$; this answers a question of Lücke and Müller. However, in $M_1$, the minimal proper class mouse with a Woodin cardinal, 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 then there is no $Σ_1(H_κ\cup\mathrm{OR})$ mad family $F\subseteq P(κ)$ of cardinality $>κ$, and if also $\mathrm{cof}(κ)>ω$ then there is no $Σ_1(H_κ\cup\mathrm{OR})$ almost disjoint family $F\subseteq P(κ)$ of cardinality $>κ$. However, relative to the consistency of large cardinals, $Π_1(\{κ\})$ mad families and maximal independent families $F\subseteq P(κ)$ can exist, when $κ$ is a limit of measurables, and even more. We also examine some of the features of $L[U]$, and answer another question of Lücke and Müller, showing that if $κ$ is a weakly compact cardinal such that every $Σ_1(H_κ\cup\{κ\})$ subset of $P(κ)$ of cardinality $>κ$ has a subset which is the range of a perfect function, then there is an inner model satisfying "there is a weakly compact limit of measurable cardinals".