Forum of Mathematics, Sigma. Volume 11, 2023. e103.

DOI: 10.1017/fms.2023.102. PDF. arXiv.
Bibtex.

Given an uncountable cardinal $\kappa$, we consider the question 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 $\omega_1$.