We produce, relative to a $\textsf {ZFC}$ model with a supercompact cardinal, a $\textsf {ZFC}$ model of the Proper Forcing Axiom in which the nonstationary ideal on $\omega _1$ is $\Pi _1$-definable in a parameter from $H_{\aleph _2}$.

We answer a question of Woodin [3] by showing that “$\mathrm {NS}_{\omega _1}$ is $\omega _1$-dense” holds in a stationary set preserving extension of any universe with a cardinal $\kappa $ which is a limit of ${<}\kappa $-supercompact cardinals. We introduce a new forcing axiom $\mathrm {Q}$-Maximum, prove it consistent from a supercompact limit of supercompact cardinals, and show that it implies the version of Woodin’s $(*)$-axiom for $\mathbb Q_{\mathrm {max}}$. It follows that $\mathrm {Q}$-Maximum implies “$\mathrm {NS}_{\omega _1}$ is $\omega _1$-dense.” Along the way we produce a number of other new instances of Asperó–Schindler’s $\mathrm {MM}^{++}\Rightarrow (*)$ (see [1]).

To force $\mathrm {Q}$-Maximum, we develop a method which allows for iterating $\omega _1$-preserving forcings which may destroy stationary sets, without collapsing $\omega _1$. We isolate a new regularity property for $\omega _1$-preserving forcings called respectfulness which lies at the heart of the resulting iteration theorem.

In the second part, we show that the $\kappa $-mantle, i.e., the intersection of all grounds which extend to V via forcing of size ${<}\kappa $, may fail to be a model of $\mathrm {AC}$ for various types of $\kappa $. Most importantly, it can be arranged that $\kappa $ is a Mahlo cardinal. This answers a question of Usuba [2].

Abstract prepared by Andreas Lietz

E-mail: andreas.lietz@tuwien.ac.at.

URL: https://andreas-lietz.github.io/resources/PDFs/AJourneyGuidedByThe Stars.pdf.

We show that under $\mathsf {BMM}$ and “there exists a Woodin cardinal,$"$ the nonstationary ideal on $\omega _1$ cannot be defined by a $\Pi _1$ formula with parameter $A \subset \omega _1$. We show that the same conclusion holds under the assumption of Woodin’s $(\ast )$-axiom. We further show that there are universes where $\mathsf {BPFA}$ holds and $\text {NS}_{\omega _1}$ is $\Pi _1(\{\omega _1\})$-definable. Lastly we show that if the canonical inner model with one Woodin cardinal $M_1$ exists, there is a generic extension of $M_1$ in which $\text {NS}_{\omega _1}$ is saturated and $\Pi _1(\{ \omega _1\} )$-definable, and $\mathsf {MA_{\omega _1}}$ holds.

We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal $M_1$, there is a model of $\mathsf {ZFC}$ in which $\mbox {NS}_{\omega _{1}}$ is $\aleph _2$-saturated and ${\Delta }_{1}$-definable with $\omega _1$ as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with $\mbox {NS}_{\omega _{1}}$ saturated and ${\Delta }_{1}$-definable with a ladder system $\vec {C}$ and a full Suslin tree T as parameters. Both results rely on a new coding technique whose presentation is the main goal of this article .

We show that, assuming the existence of the canonical inner model with one Woodin cardinal $M_1 $ , there is a model of $ZFC$ in which the nonstationary ideal on $\omega _1 $ is $\aleph _2 $-saturated and whose reals admit a ${\rm{\Sigma }}_4^1 $-wellorder.

We study Σ1(ω 1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω 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 Σ1(ω 1)-definable, the set of all stationary subsets of ω 1 is not Σ1(ω 1)-definable and the complement of every Σ1(ω 1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω 1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω 1)-definable well-ordering of H(ω 2) and the existence of a Δ1(ω 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 Σ1(ω 1)-definable uniformization of the club filter on ω 1. Moreover, we prove a perfect set theorem for Σ1(ω 1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$ , assuming that there is a measurable cardinal and the nonstationary ideal on ω 1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κ κ, in the case where κ itself has certain large cardinal properties.