Low level definability above large cardinals

Farmer Schlutzenberg

Published 2024-01-03Version 1

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 $\kappa$ is a limit of measurable cardinals then there is no $\Sigma_1(H_\kappa\cup\mathrm{OR})$ wellorder of a subset of $P(\kappa)$ of length $\geq\kappa^+$; this answers a question of L\"ucke and M\"uller. However, in $M_1$, the minimal proper class mouse with a Woodin cardinal, 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 then there is no $\Sigma_1(H_\kappa\cup\mathrm{OR})$ mad family $F\subseteq P(\kappa)$ of cardinality $>\kappa$, and if also $\mathrm{cof}(\kappa)>\omega$ then there is no $\Sigma_1(H_\kappa\cup\mathrm{OR})$ almost disjoint family $F\subseteq P(\kappa)$ of cardinality $>\kappa$. However, relative to the consistency of large cardinals, $\Pi_1(\{\kappa\})$ mad families and maximal independent families $F\subseteq P(\kappa)$ can exist, when $\kappa$ is a limit of measurables, and even more. We also examine some of the features of $L[U]$, and answer another question of L\"ucke and M\"uller, showing that if $\kappa$ is a weakly compact cardinal such that every $\Sigma_1(H_\kappa\cup\{\kappa\})$ subset of $P(\kappa)$ of cardinality $>\kappa$ 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".