arXiv subjects
Publications and source records attributed to Stefan Hoffelner.
We show that there are models of MA where the boldface $Σ^1_3$-uniformization property holds. Further we show that BPFA and the assertion $\aleph_1$ is accessible to reals outright implies that the boldface $Σ^1_3$-uniformization property is true.
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 $\ZFC$ in which $\NS$ is $\aleph_2$-saturated and $Δ_1$-definable with $ω_1$ as a parameter which answers a question of Sy-David Friedman and Liuzhen Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with $\NS$ saturated and $Δ_1$-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.
It is shown that the existence of a measurable cardinal is equiconsistent to a model of ZFC in which there is no ordinal-definable, stationary, costationary subset of $ω_1$
We show that, under the assumption of the existence of $M_1^{\#}$, there exists a model on which the restricted nonstationary ideal $\hbox{NS} \upharpoonright A$ is $\aleph_2$-saturated, for $A$ a stationary co-stationary subset of $ω_1$, while the full nonstationary ideal $\hbox{NS}$ can be made $Δ_1$ definable with $K_{ω_1}$ as a parameter. Further we show, again under the assumption of the existence of $M_1^{\#}$ that there is a model of set theory such that $\hbox{NS}$ is $\aleph_2$-saturated and such that there is lightface $Σ^1_4$-definable well-order on the reals. This result is optimal in the presence of a measurable cardinal.