₁-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

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 nondefinability 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 nonexistence 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).