arXiv subjects

Yurii Khomskii

Publications and source records attributed to Yurii Khomskii.

9 recordsLinked to original sources

arXiv2026

Combinatorial Properties of the Raisonnier Filter

The Raisonnier Filter is a combinatorial object isolated by Jean Raisonnier in order to simplify Shelah's proof that if all $\boldsymbolΣ^1_3$ sets are Lebesgue-measurable then there is an inner model with an inaccessible cardinal. In this paper, we study the combinatorics of a general version of the Raisonnier filter, with an eye to potential applications in descriptive set theory. Among the most interesting of our results is a partial converse to Raisonnier's theorem, which can be used to provide a new characterisation of the statement "all $\boldsymbolΣ^1_2$ sets are measurable". We also introduce an ideal on the Cantor Space induced by the Raisonnier filter and study its cardinal characteristics, connecting them to the well-known characteristics in Cichoń's Diagram.

math.LO↗

arXiv2023

Paraconsistent and Paracomplete Zermelo-Fraenkel Set Theory

We present a novel treatment of set theory in a four-valued paraconsistent and paracomplete logic, i.e., a logic in which propositions can be both true and false, and neither true nor false. Our approach is a significant departure from previous research in paraconsistent set theory, which has almost exclusively been motivated by a desire to avoid Russell's paradox and fulfil naive comprehension. Instead, we prioritise setting up a system with a clear ontology of non-classical sets, which can be used to reason informally about incomplete and inconsistent phenomena, and is sufficiently similar to ZFC to enable the development of interesting mathematics. We propose an axiomatic system BZFC, obtained by analysing the ZFC-axioms and translating them to a four-valued setting in a careful manner, avoiding many of the obstacles encountered by other attempted formalizations. We introduce the anti-classicality axiom postulating the existence of non-classical sets, and prove a surprising results stating that the existence of a single non-classical set is sufficient to produce any other type of non-classical set. Our theory is naturally bi-interpretable with ZFC, and provides a philosophically satisfying view in which non-classical sets can be seen as a natural extension of classical ones, in a similar way to the non-well-founded sets of Peter Aczel. Finally, we provide an interesting application concerning Tarski semantics, showing that the classical definition of the satisfaction relation yields a logic precisely reflecting the non-classicality in the meta-theory.

math.LO↗

arXiv2020

Laver Trees in the Generalized Baire Space

We prove that any suitable generalization of Laver forcing to the space $ κ^κ$, for uncountable regular $κ$, necessarily adds a Cohen $κ$-real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. Using this dichotomy, we prove the following stronger result: if $ κ^{<κ}=κ$, then every $<κ$-distributive tree forcing on $κ^κ$ adding a dominating $κ$-real which is the image of the generic under a continuous function in the ground model, adds a Cohen $κ$-real. This is a contribution to the study of generalized Baire spaces and answers a question from arXiv:1611.08140

math.LO↗

arXiv2020

Bounded Symbiosis and Upwards Reflection

Bagaria and Väänänen developed a framework for studying the large cardinal strength of downwards Löwenheim-Skolem theorems and related set theoretic reflection properties. The main tool was the notion of symbiosis, originally introduced by the third author. Symbiosis provides a way of relating model theoretic properties of strong logics to definability in set theory. In this paper we continue the systematic investigation of symbiosis and apply it to upwards Löwenheim-Skolem theorems and reflection principles. To achieve this, we need to adapt the notion of symbiosis to a new form, called bounded symbiosis. As one easy application, we obtain upper and lower bounds for the large cardinal strength of upwards Löwenheim-Skolem-type principles for second order logic.

math.LO↗

arXiv2019

Definable Maximal Independent Families

We study maximal independent families (m.i.f.) in the projective hierarchy. We show that (a) the existence of a $\boldsymbolΣ^1_2$ m.i.f. is equivalent to the existence of a $\boldsymbolΠ^1_1$ m.i.f., (b) in the Cohen model, there are no projective maximal independent families, and (c) in the Sacks model, there is a $\boldsymbolΠ^1_1$ m.i.f. We also consider a new cardinal invariant related to the question of destroying or preserving maximal independent families.

math.LO↗

arXiv2016

Filter-Laver Measurability

We study sigma-ideals and regularity properties related to the "filter-Laver" and "dual-filter-Laver" forcing partial orders. An important innovation which enables this study is a dichotomy theorem proved recently by Miller [1]. [1] Arnold Miller, "Hechler and Laver Trees", Preprint 2012 (arXiv:1204.5198 [math.LO]).

math.LO↗

arXiv2016

Cofinalities of Marczewski-like ideals

We show that the cofinalities of both the Miller ideal m^0 (the sigma-ideal naturally related to Miller forcing) and the Laver ideal ell^0 (related to Laver forcing) are larger than the size of the continuum in ZFC.

math.LO↗

arXiv2015

Almost disjoint refinements and mixing reals

We investigate families of subsets of $ω$ with almost disjoint refinements in the classical case as well as with respect to given ideals on $ω$. More precisely, we study the following topics and questions: 1) Examples of projective ideals. 2) We prove the following generalization of a result due to J. Brendle: If $V\subseteq W$ are transitive models, $ω_1^W\subseteq V$, $\mathcal{P}(ω)\cap V\not = \mathcal{P}(ω)\cap W$, and $\mathcal{I}$ is an analytic or coanalytic ideal coded in $V$, then there is an $\mathcal{I}$-almost disjoint refinement ($\mathcal{I}$-ADR) of $\mathcal{I}^+\cap V$ in $W$, that is, a family $\{A_X:X\in\mathcal{I}^+\cap V\}\in W$ such that (i) $A_X\subseteq X$, $A_X\in \mathcal{I}^+$ for every $X$ and (ii) $A_X\cap A_Y\in\mathcal{I}$ for every distinct $X$ and $Y$. 3) The existence of perfect $\mathcal{I}$-almost disjoint ($\mathcal{I}$-AD) families, and the existence of a "nice" ideal $\mathcal{I}$ on $ω$ with the property: Every $\mathcal{I}$-AD family is countable but $\mathcal{I}$ is nowhere maximal. 4) The existence of $(\mathcal{I},\text{Fin})$-almost disjoint refinements of families of $\mathcal{I}$-positive sets in the case of everywhere meager (e.g. analytic or coanalytic) ideals. We prove a positive result under Martin's Axiom. 5) Connections between classical properties of forcing notions and adding mixing reals (and mixing injections), that is, a (one-to-one) function $f:ω\toω$ such that $|f[X]\cap Y|=ω$ for every $X,Y\in [ω]^ω\cap V$.

math.LO↗

arXiv2014

Regularity Properties on the Generalized Reals

We investigate regularity properties derived from tree-like forcing notions in the setting of "generalized descriptive set theory", i.e., descriptive set theory on $κ^κ$ and $2^κ$, for regular uncountable cardinals $κ$.

math.LO↗