$Σ_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

Abstract

Given an uncountable cardinal κ, we consider the question whether subsets of the power set of κ that are usually constructed with the help of the Axiom of Choice are definable by Σ 1 -formulas that only use the cardinal κ and sets of hereditary cardinality less than κ 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 κ of length at least κ + implies the existence of a projective wellordering of the reals. In addition, we determine the exact consistency strength of the non-existence of Σ 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 ω 1 .