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

TL;DR

This paper investigates the definability of certain sets related to $ ext{H}( ext{card})$ under large cardinal assumptions, revealing how large cardinals influence the complexity and existence of definable sets and orderings.

Contribution

It demonstrates the impact of large cardinals on the definability of well-orderings, stationary sets, and Bernstein sets within the context of $ ext{H}( ext{card})$, and introduces new consistency results.

Findings

No $oldsymbol{ ext{Sigma}_1(oldsymbol{ ext{omega}_1})}$-definable well-ordering of reals under certain large cardinal assumptions.

The set of stationary subsets of $oldsymbol{ ext{omega}_1}$ is not $oldsymbol{ ext{Sigma}_1(oldsymbol{ ext{omega}_1})}$-definable with large cardinals.

Existence of $oldsymbol{ ext{Sigma}_1(oldsymbol{ ext{omega}_1})}$-definable Bernstein subsets is consistent with certain large cardinal hypotheses.

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.