  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 $\omega_1$.
