$\Sigma_1(\kappa)$-definable subsets of $\mathrm{H}(\kappa^+)$

Abstract

We study $\Sigma_1(\omega_1)$-definable sets (i.e. sets that are equal to the collection of all sets satisfying a certain $\Sigma_1$-formula with parameter $\omega_1$) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is $\Sigma_1(\omega_1)$-definable, the set of all stationary subsets of $\omega_1$ is not $\Sigma_1(\omega_1)$-definable and the complement of every $\Sigma_1(\omega_1)$-definable Bernstein subset of ${}^{\omega_1}\omega_1$ is not $\Sigma_1(\omega_1)$-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a $\Sigma_1(\omega_1)$-definable well-ordering of $\mathrm{H}({\omega_2})$ and the existence of a $\Delta_1(\omega_1)$-definable Bernstein subset of ${}^{\omega_1}\omega_1$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no $\Sigma_1(\omega_1)$-definable uniformization of the club filter on $\omega_1$. Moreover, we prove a perfect set theorem for $\Sigma_1(\omega_1)$-definable subsets of ${}^{\omega_1}\omega_1$, assuming that there is a measurable cardinal and the non-stationary ideal on $\omega_1$ is saturated. The proofs of these results use iterated generic ultrapowers and Woodin's $\mathbb{P}_{\mathrm{max}}$-forcing. Finally, we also prove variants of some of these results for $\Sigma_1(\kappa)$-definable subsets of ${}^{\kappa}\kappa$, in the case where $\kappa$ itself has certain large cardinal properties.