arXiv subjects

Jonathan Schilhan

Publications and source records attributed to Jonathan Schilhan.

At least 19 recordsLinked to original sources

arXiv2026

Universal domination and idealized forcing

We introduce the universality property of a definable $\sigma$-ideal on a Polish space, which, on one hand, can serve as a benchmark for the properness of the associated idealized forcing of positive Borel sets ordered by inclusion, and, on the other hand, unifies many of the results that can be found in Zapletal's book. We show that under mild absoluteness assumptions, it implies properness, various dichotomy theorems, and closure under well-ordered unions in the Solovay model and under $\mathsf{AD}^+$, among other things. All major classes of proper idealized forcings studied in the book have this property. Further, we use this viewpoint to answer a question of Khomskii by showing that the naive idealized forcing for adding an eventually different real or a refining real is not proper below some condition. We also answer a question related to the definability of $\sigma$-ideals generated by Borel sets due to Kanovei, Sabok, and Zapletal.

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arXiv2026

Structural Infinite-Exponent Partition Relations and Weak Choice Principles

We investigate infinite-exponent partition relations on arbitrary relational structures, with a focus on linear orders and graphs. Any such relation contradicts the Axiom of Choice. We show that there are some such relations which are consistent with ZF which imply the failure not just of Choice but also of the Kinna-Wagner Selection Principle KWP$_1$ and the Ordering Principle O.

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arXiv2026

Infinite-Exponent Partition Relations on Higher Analogues of the Real Line

We present a number of results concerning infinite-exponent partition relations on linear orders of the form $\langle {}^\alpha 2,<_{\text{lex}}\rangle$ for $\alpha$ an ordinal, generalising the setting of the real line, working throughout in ZF without the Axiom of Choice. As a particular consequence of our results, we obtain a full classification of the relation $\langle {}^\alpha 2,<_{\text{lex}}\rangle \rightarrow (\tau)^\tau$ for $\tau$ countable.

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arXiv2026

A General Theory of Class Symmetric Systems

We develop a general theory for class-sized symmetric systems as a natural extension of symmetric systems with respect to class forcing. In particular, adapting the usual notions of pretameness and tameness for class forcing, we present sufficient conditions for the preservation of the axioms of G\"odel-Bernays set theory (without the axiom of choice), and for the forcing theorem to hold for class-sized symmetric systems.

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arXiv2026

Towards a theory of symmetric extensions

The technique of symmetric extensions is derived from forcing and it is one of the most important tools for studying models without the Axiom of Choice. Despite being incredibly successful since the 1960s, our understanding of the technique remained fairly limited compared to the theory of forcing. Whereas forcing developed products and iterations, no serious attempts at developing any general framework for iterating symmetric extensions were presented before [10], where only finite support iterations are treated. In this paper we develop the theory of symmetric extensions including different types of iterations, quotients, equivalents, and the structural results that can be described in this language. In particular, we give a modern exposition to some of the important theorems of Grigorieff [3], study Kinna--Wagner Principles in symmetric extensions, and show that it is provable from $\mathsf{ZF}$ that every set lies in a symmetric extension of $\operatorname{HOD}$.

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arXiv2025

Transcendence degrees over mutually generic extensions

Let $G_0$,..., $G_{n-1}$ be mutually generic over $V$, each $G_i$ adding at least one new real over $V$. We show that the transcendence degree of the reals of $V[G_0, \dots, G_{n-1}]$ is maximal (of size continuum) over the field generated by reals coming from models $V[ G_i : i \in a]$, for a proper subset $a$ of $n$. This answers a question of Fatalini and Schindler.

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arXiv2025

The Ordering Principle and Higher Dependent Choice

We provide, for any regular uncountable cardinal $\kappa$, a new argument for Pincus' result on the consistency of $\mathrm{ZF}$ with the higher dependent choice principle $\mathrm{DC}_{<\kappa}$ and the ordering principle in the presence of a failure of the axiom of choice. We also generalise his methods and obtain these consistency results in a larger class of models.

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arXiv2025

$\Sigma^1_3$ sets in the Sacks model

We show that in the iterated Sacks model over the constructible universe the Mansfield-Solovay Theorem holds for $\Sigma^1_3$ sets. In particular, every $\mathbf{\Sigma}^1_3$ set is Marczewski measurable and the optimal complexity for a Bernstein set is $\Delta^1_4$. Based on a result by Kanovei, we also briefly show how to separate the Mansfield-Solovay Theorem at non-trivial levels of the projective hierarchy.

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arXiv2025

The Ordering Principle and Dependent Choice

We introduce finite support iterations of symmetric systems, and use them to provide a strongly modernized proof of David Pincus' classical result that the axiom of dependent choice is independent over ZF with the ordering principle together with a failure of the axiom of choice.

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arXiv2024

Intermediate models and Kinna--Wagner Principles

Kinna--Wagner Principles state that every set can be mapped into some fixed iterated power set of an ordinal, and we write $\mathsf{KWP}$ to denote that there is some $\alpha$ for which this holds. The Kinna--Wagner Conjecture, formulated by the first author in [9], states that if $V$ is a model of $\mathsf{ZF}+\mathsf{KWP}$ and $G$ is a $V$-generic filter, then whenever $W$ is an intermediate model of $\mathsf{ZF}$, that is $V\subseteq W\subseteq V[G]$, then $W=V(x)$ for some $x$ if and only if $W$ satisfies $\mathsf{KWP}$. In this work we prove the conjecture and generalise it even further. We include a brief historical overview of Kinna--Wagner Principles and new results about Kinna--Wagner Principles in the multiverse of sets.

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arXiv2024

Upwards homogeneity in iterated symmetric extensions

It is sometimes desirable in choiceless constructions of set theory that one iteratively extends some ground model without adding new sets of ordinals after the first extension. Pushing this further, one may wish to have models $V \subseteq M \subseteq N$ of $\mathsf{ZF}$ such that $N$ contains no subsets of $V$ that do not already appear in $M$. We isolate, in the case that $M$ and $N$ are symmetric extensions (particular inner models of a generic extension of $V$), the exact conditions that cause this behaviour and show how it can broadly be applied to many known constructions. We call this behaviour upwards homogeneity.

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arXiv2024

Wetzel families and the continuum

We provide answers to a question brought up by Erdős about the construction of Wetzel families in the absence of the continuum hypothesis - a Wetzel family is a family $\mathcal{F}$ of entire functions on the complex plane which pointwise assumes fewer than $\vert \mathcal{F} \vert$ values. To be more precise, we show that the existence of a Wetzel family is consistent with all possible values $κ$ of the continuum and, if $κ$ is regular, also with Martin's Axiom. In the particular case of $κ= \aleph_2$ this answers an open question asked by Kumar and Shelah. In the buildup to this result, we are also solving an open question of Zapletal on strongly almost disjoint functions. We also study a strongly related notion of sets exhibiting a universality property via mappings by entire functions and show that these consistently exist while the continuum equals $\aleph_2$.

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arXiv2023

Geometric condition for Dependent Choice

We provide a geometric condition which characterises when the Principle of Dependent Choice holds in a Fraenkel--Mostowski--Specker permutation model. This condition is a slight weakening of requiring the filter of groups to be closed under countable intersections. We show that this condition holds nontrivially in a new permutation model we call "the nowhere dense model" and we study its extensions to uncountable cardinals as well.

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arXiv2022

Maximal sets without Choice

We show that it is consistent relative to ZF, that there is no well-ordering of $\mathbb{R}$ while a wide class of special sets of reals such as Hamel bases, transcendence bases, Vitali sets or Bernstein sets exists. To be more precise, we can assume that every projective hypergraph on $\mathbb{R}$ has a maximal independent set, among a few other things. For example, we get transversals for all projective equivalence relations. Moreover, this is possible while either $\mathsf{DC}_{ω_1}$ holds, or countable choice for reals fails. Assuming the consistency of an inaccessible cardinal, "projective" can even be replaced with "$L(\mathbb{R})$". This vastly strengthens earlier consistency results in the literature.

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arXiv2022

Tree forcing and definable maximal independent sets in hypergraphs

We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over $L$, every analytic hypergraph on a Polish space admits a $\mathbfΔ^1_2$ maximal independent set. As a main application we get the consistency of $\mathfrak{r} = \mathfrak{u} = \mathfrak{i} = ω_2$ together with the existence of a $Δ^1_2$ ultrafilter, a $Π^1_1$ maximal independent family and a $Δ^1_2$ Hamel basis. This solves open problems of Brendle, Fischer and Khomskii and the author. We also show in ZFC that $\mathfrak{d} \leq \mathfrak{i}_{cl}$.

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arXiv2021

Sequential and distributive forcings without choice

In the Zermelo--Fraenkel set theory with the Axiom of Choice a forcing notion is "$\kappa$-distributive" if and only if it is "$\kappa$-sequential". We show that without the Axiom of Choice this equivalence fails, even if we include a weak form of the Axiom of Choice, the Principle of Dependent Choice for $\kappa$. Still, the equivalence may still hold along with very strong failures of the Axiom of Choice, assuming the consistency of large cardinal axioms. We also prove that while a $\kappa$-distributive forcing notion may violate Dependent Choice, it must preserve the Axiom of Choice for families of size $\kappa$. On the other hand, a $\kappa$-sequential can violate the Axiom of Choice for countable families. We also provide a condition of "quasiproperness" which is sufficient for the preservation of Dependent Choice, and is also necessary if the forcing notion is sequential.

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arXiv2021

Some combinatorial properties of splitting trees

We show that splitting forcing does not have the weak Sacks property below any condition, answering a question of Laguzzi, Mildenberger and Stuber-Rousselle. We also show how some partition results for splitting trees hold or fail and we determine the value of cardinal invariants after an $ω_2$-length countable support iteration of splitting forcing.

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arXiv2019

Coanalytic Ultrafilter Bases

We study the definability of ultrafilter bases on $ω$ in the sense of descriptive set theory. As a main result we show that there is no coanalytic base for a Ramsey ultrafilter, while in $L$ we can construct $Π^1_1$ P-point and Q-point bases. We also show that the existence of a $\mathbfΔ^1_{n+1}$ ultrafilter is equivalent to that of a $\mathbfΠ^1_n$ ultrafilter base, for $n \in ω$. Moreover we introduce a Borel version of the classical ultrafilter number and make some observations.

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