$\Sigma_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

Abstract

Given an uncountable cardinal $\kappa$, we consider the question of whether subsets of the power set of $\kappa$ that are usually constructed with the help of the Axiom of Choice are definable by $\Sigma_1$-formulas that only use the cardinal $\kappa$ and sets of hereditary cardinality less than $\kappa$ as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical non-definability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of $\kappa$ of length at least $\kappa^+$ implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the non-existence of $\Sigma_1$-definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at the first uncountable cardinal $\omega_1$.