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Stefan Hoffelner

Publications and source records attributed to Stefan Hoffelner.

At least 19 recordsLinked to original sources
arXiv2026

Forcing the $Π^1_n$-Uniformization Property

We generically construct a model in which the ${Π^1_3}$-uniformization property is true, thus lowering the best known consistency strength from the existence of $M_1^{\#}$ to just $\mathsf{ZFC}$. The forcing construction can be adapted to work over canonical inner models with Woodin cardinals, which yields, for the first time, universes where the $Π^1_{2n}$-uniformization property holds for $n >1$, thus producing models which contradict the natural $\mathsf{PD}$-induced pattern.

math.LO↗
arXiv2026

Forcing upper $Σ$-uniformization in the presence of lower $Π$-reduction or uniformization

We present a method which allows the combination of forcing uniformization on the $Π$- and the $Σ$-side of the projective hierarchy to a certain extent. Using this method we construct a universe where $Π^1_3$-reduction holds, $Π^1_3$-uniformization fails, yet $Σ^1_n$ uniformization is true for $n \ge 4$. We also construct a universe where $Π^1_3$-uniformization holds and for every $n \ge 4, $ $ Σ^1_4$-uniformization holds, lowering best known upper bound for this statement from the existence of two Woodin cardinals to $Con(\ZFC)$.

math.LO↗
arXiv2026

On a local variant of the 12th Delfino problem -- the $Σ$-side

Assuming that $M_n$, the canonical inner model with $n$ Woodin cardinals, exists, we force a model in which every $\boldsymbolΣ^1_{n+2}$ set is Lebesgue measurable and has the Baire property, and in which $Σ^1_{n+2+m}$-uniformization holds for every $m\inω$. Additionally, this universe has a $Δ^1_{n+3}$-definable wellorder of the reals. This answers a question of S. D. Friedman and R. Schindler from 1999. In the case $n=1$, the construction also gives a model with one Woodin cardinal in which all $Σ^1_3$ sets are measurable with respect to the random, Cohen, Sacks and Miller notions of measurability, while a $Δ^1_4$-definable wellorder of the reals exists answering an instance of a question of S. D. Friedman and D. Schrittesser.

math.LO↗
arXiv2026

On a local variant of the 12th Delfino problem -- the $Π$-side

Assume that \(M_n\), the canonical inner model with \(n\) Woodin cardinals, exists. We force a model with continuum \(\aleph_2\) in which every \(\boldsymbolΣ^1_{n+2}\) set of reals is Lebesgue measurable and has the Baire property, the \(Σ^1_{n+2}\)- and \(Π^1_{n+3}\)-uniformization properties hold, and the reals admit a \(Δ^1_{n+3}\)-definable well-order. Thus regularity up to a fixed finite projective level, together with a definable well-order of the reals at the adjacent level, does not force the determinacy strength which would normally explain that regularity, even when this package is strengthened by adjacent \(Σ\)- and \(Π\)-uniformization. In particular, this gives a negative answer to a local form of Woodin's twelfth Delfino problem asked by Friedman-Schindler.

math.LO↗
arXiv2026

Forcing $\mathbfΣ^1_1$-Separation on $ω_1^{ω_1}$

We prove that it is consistent that every two disjoint boldface $\mathbfΣ^1_1$ subsets of $ω_1^{ω_1}$ can be separated by a boldface $\mathbfΔ^1_1$ set. The forcing starts from $L$ and preserves CH and therefore also $ω_1^{<ω_1}=ω_1$.

math.LO↗
arXiv2026

On graphs of total projective functions

It is well known that the graph of a total $\mathbfΣ^1_n$-function is $\mathbfΠ^1_n$. We prove the consistency of the dual assertion at the third projective level: there is a model of $\ZFC$ in which the graph of every total $\mathbfΠ^1_3$-function is $\mathbfΣ^1_3$. This principle is incompatible with $\mathbfΠ^1_3$-uniformization and hence with the usual projective-determinacy picture. The construction also repairs the final step of the failure-of-uniformization argument from~\cite{HOFFELNER2023103292}.

math.LO↗
arXiv2026

On $\boldsymbolΣ^1_3$- and $Σ^1_4$-uniformization

Assuming the consistency of $\mathsf{ZFC}$, we construct a model of set theory in which the boldface $\mathbfΣ^1_3$-uniformization property holds, yet the lightface $Σ^1_4$-uniformization property fails, separating these two principles for the first time. We also indicate how to create a universe where $Σ^1_3$-uniformization holds, but $Σ^1_4$-uniformization fails using inner models with large cardinals.

math.LO↗
arXiv2026

A Failure of $Π^1_{n+3}$-Reduction in the Presence of $Σ^1_{n+3}$-Separation

We show that one can force over $L$ that $Σ^1_3$-separation holds, while $Π^1_3$-reduction fails, thus separating these two principles for the first time. The construction can be lifted to canonical inner models $M_n$ with $n$-many Woodin cardinals, yielding that assuming the existence of $M_n$, $Σ^1_{n+3}$-separation can hold, yet $Π^1_{n+3}$-reduction fails.

math.LO↗
arXiv2025

$\text{NS}_{ω_1}$ saturated, $Δ_1 ( \{ ω_1 \} )$-definable and a $Δ^1_4$-definable well-order of the reals

Assuming $M_1$, the canonical inner model with one Woodin cardinal exists, we construct a model in which the nonstationary ideal on $ω_1$ is $\aleph_2$-saturated, $Δ_1$-definable with $ω_1$ as the only parameter and there is a $Σ^1_{4}$-definable well-order of the reals. This implies that contrary to the assumption that $NS_{ω_1}$ is $\aleph_1$-dense, the assumption of $NS_{ω_1}$ being saturated and $Δ_1$-definable does not imply any nice structural properties for the projective subsets of the reals.

math.LO↗
arXiv2025

$\mathsf{MA} (\mathcal{I}$) and a Failure of Separation on the third Level

We present a method which forces the failure of $Π^1_3$ and $Σ^1_3$-separation, while $\mathsf{MA} (\mathcal{I}$) holds, for $\mathcal{I}$ the family of indestructible ccc forcings. This shows that, in contrast to the assumption $\mathsf{BPFA}$ and $\aleph_1=\aleph_1^L$ which implies $Π^1_3$-separation, that weaker forcing axioms do not decide separation on the third projective level.

math.LO↗
arXiv2025

A Universe with a $Δ^1_n$-definable well-order of the reals, $\mathsf{CH}$ and $Π^1_n$-Uniformization

This paper details the construction of a universe where $Π^1_3$-uniformization is true, the Continuum Hypothesis holds yet it possesses a $Δ^1_3$-definable well-order of its reals. The method can be lifted to canonical inner models with finitely many Woodin cardinals to produce universes of $\mathsf{CH}$, $Π^1_n$-uniformization and where additionally a $Δ^1_n$-definable well-order of the reals exist.

math.LO↗
arXiv2023

PFA and the definability of the nonstationary ideal

We produce, relative to a ${\sf ZFC}$ model with a supercompact cardinal, a ${\sf ZFC}$ model of the Proper Forcing Axiom in which the nonstationary ideal on $ω_1$ is $Π_1$-definable in a parameter from $H_{\aleph_2}$.

math.LO↗
arXiv2022

The Consistency of the $\bf{Σ^1_3}$-Separation Property

We generically construct a model in which the $\bf{Σ^1_3}$-separation property is true, i.e. every pair of disjoint $\bf{Σ^1_3}$-sets can be separated by a $\bf{Δ^1_3}$-definable set. This answers an old question from the problem list $"$Surrealist landscape with figures$"$ by A. Mathias from 1968. We also construct a model in which the (lightface) $Σ^1_3$-separation property is true.

math.LO↗
arXiv2022

Forcing Axioms and the Definabilty of the Nonstationary Ideal on $ω_1$

We show that under $\BMM$ and "there exists a Woodin cardinal$"$, the nonstationary ideal on $ω_1$ can not be defined by a $Σ_1$ formula with parameter $A \subset ω_1$. We show that the same conclusion holds under the assumption of Woodin's $(\ast)$-axiom. We further show that there are universes where $\BPFA$ holds and $\NS$ is $Σ_1(ω_1)$-definable. Last we show that if the canonical inner model with one Woodin cardinal $M_1$ exists, there is a universe where $\NS$ is saturated, $Σ_1(ω_1)$-definable and $\MA$ holds.

math.LO↗