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Vassily Lyubetsky

Publications and source records attributed to Vassily Lyubetsky.

At least 19 recordsLinked to original sources
arXiv2026

Locally countable graphs of second projective class not generated by countably many projective functions

To answer a question by Rettich and Serafin, we define a model of set theory in which there exists a locally countable $\varPi^1_2$ graph on a subset of the real line, which is not generated by a countable family of projective (or even real-ordinal definable, ROD) functions. We also prove that the $\varSigma^1_2$ equi-constructibility graph on the reals is not generated by a countable family of ROD functions in the Solovay model.

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arXiv2026

On Petr Novikov's problem of ordered systems of uniform sets

We prove that every ordinal $α<ω_2$ is the order type of a certain system of uniform Borel sets in the sense of a well-ordering relation defined by Petr Novikov. This result gives a positive answer to a problem posed by Nicolas Luzin in 1935.

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arXiv2025

On the significance of parameters and the projective level in the Choice and Comprehension axioms

We make use of generalized iterations of Jensen forcing to define a cardinal-preserving generic model of ZF for any $n\ge 1$ and each of the following four Choice hypotheses: (1) $\text{DC}(\mathbfΠ^1_n)\land\neg\text{AC}_ω(\varPi^1_{n+1})\,;$ (2) $\text{AC}_ω(\text{OD})\land\text{DC}(\varPi^1_{n+1})\land \neg\text{AC}_ω(\mathbfΠ^1_{n+1});$ (3) $\text{AC}_ω\land\text{DC}(\mathbfΠ^1_n)\land\neg\text{DC}(\varPi^1_{n+1});$ (4) $\text{AC}_ω\land\text{DC}(\varPi^1_{n+1})\land\neg\text{DC}(\mathbfΠ^1_{n+1}).$ Thus if ZF is consistent and $n\ge1$ then each of these four conjunctions (1)--(4) is consistent with ZF. As for the second main result, let PA$^0_2$ be the 2nd-order Peano arithmetic without the Comprehension schema $\text{CA}$. For any $n\ge1$, we define a cardinal-preserving generic model of ZF, and a set $M\subseteq\mathcal P(ω)$ in this model, such that $\langleω, M\rangle$ satisfies (5) PA$^0_2$ + $\text{AC}_ω(\varSigma^1_{\infty})$ + $\text{CA}(\mathbfΣ^1_{n+1})$ + $\neg\text{CA}(\mathbfΣ^1_{n+1})$. Thus $\text{CA}(\mathbfΣ^1_{n+1})$ does not imply $\text{CA}(\mathbfΣ^1_{n+2})$ in PA$^0_2$ even in the presence of the full parameter-free (countable) Choice $\text{AC}_ω(\varSigma^1_{\infty}).$

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arXiv2025

Notes on the equiconsistency of ZFC without the Power Set axiom and second order PA

We demonstrate that theories $\text{Z}^-$, $\text{ZF}^-$, $\text{ZFC}^-$ (minus means the absence of the Power Set axiom) and $\text{PA}_2$, $\text{PA}_2^-$ (minus means the absence of the Countable Choice schema) are equiconsistent to each other. The methods used include the interpretation of a power-less set theory in $\text{PA}_2^-$ via well-founded trees, as well as the Gödel constructibility in the said power-less set theory.

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arXiv2022

The parameterfree Comprehension does not imply the full Comprehension in the 2nd order Peano arithmetic

The parameter-free part $\text{PA}_2^\ast$ of $\text{PA}_2$, the 2nd order Peano arithmetic, is considered. We make use of a product/iterated Sacks forcing to define an $ω$-model of $\text{PA}_2^\ast + \text{CA}(Σ^1_2)$, in which an example of the full Comprehension schema $\text{CA}$ fails. Using Cohen's forcing, we also define an $ω$-model of $\text{PA}_2^\ast$, in which not every set has its complement, and hence the full $\text{CA}$ fails in a rather elementary way.

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arXiv2022

A model in which the Separation principle holds for a given effective projective Sigma-class

In this paper, we prove the following: If $n\ge3$, there is a generic extension of $L$ -- the constructible universe -- in which it is true that the Separation principle holds for both effective (lightface) classes $\varSigma^1_n$ and $\varPi^1_n$ for sets of integers. The result was announced long ago by Leo Harrington with a sketch of the proof for $n=3$; its full proof has never been presented. Our methods are based on a countable product of almost-disjoint forcing notions independent in the sense of Jensen--Solovay.

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arXiv2021

A product forcing model in which the Russell-nontypical sets satisfy ZFC strictly between HOD and the universe

A set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD $\subseteq$ HNT $\subseteq$ V holds. Solving a problem recently proposed by Tzouvaras, a generic extension L$[a,x]$ of L, by two reals $a,x$, is presented in which L=HOD $\subsetneqq$ L$[a]$=HNT $\subsetneqq$ V=L$[a,x]$, so that HNT is a model of ZFC strictly between HOD and the universe.

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arXiv2021

On Russell typicality in Set Theory

By Tzouvaras, a set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD$\subseteq$HNT$\subseteq$V holds. Several questions about the nature of such sets, recently proposed by Tzouvaras, are solved in this paper. In particular, a model of ZFC is presented in which HOD$\subsetneqq$HNT$\subsetneqq$V, and another model of ZFC, in which HNT does not satisfy the axiom of choice.

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arXiv2019

Indiscernible pairs of countable sets of reals at a given projective level

Using an invariant modification of Jensen's "minimal $\varPi^1_2$ singleton" forcing, we define a model of ZFC, in which, for a given $n\ge2$, there exists a lightface $\varPi^1_n$ unordered pair of non-OD (hence, OD-indiscernible) countable sets of reals, but there is no $\varSigma^1_n$ unordered pairs of this kind.

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arXiv2019

Models of set theory in which separation theorem fails

We make use of a finite support product of the Jensen minimal forcing to define a model of set theory in which the separation theorem fails for projective classes $\mathbfΣ^1_n$ and $\mathbfΠ^1_n$, for a given $n\ge3$.

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arXiv2018

On intermediate extensions of generic extensions by a random real

The paper is the second of our series of notes aimed to bring back in circulation some bright ideas of early modern set theory, mainly due to Harrington and Sami, which have never been adequately presented in set theoretic publications. We prove that if a real $a$ is random over a model $M$ and $x\in M[a]$ is another real then either (1) $x\in M$, or (2) $M[x]=M[a]$, or (3) $M[x]$ is a random extension of $M$ and $M[a]$ is a random extension of $M[x]$. This is a less-known result of old set theoretic folklore, and, as far as we know, has never been published. As a corollary, we prove that $Σ^1_n$-Reduction holds for all $n\ge3$, in a model extending the constructible universe $L$ by $\aleph_1$-many random reals.

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arXiv2018

Canonization of smooth equivalence relations on infinite-dimensional perfect cubes

A canonization scheme for smooth equivalence relations on $\mathbb R^ω$ modulo restriction to infinite perfect products is proposed. It shows that given a pair of Borel smooth equivalence relations $\mathsf E,\mathsf F$ on $\mathbb R^ω$, there is an infinite perfect product $P\subseteq\mathbb R^ω$ such that either ${\mathsf F}\subseteq{\mathsf E}$ on $P$, or, for some $j<ω$, the following is true for all $x,y\in P$: $x\,\mathsf E \,y$ implies $x(j)=y(j)$, and $x\restriction{(ω\smallsetminus\{j\})}=y\restriction{(ω\smallsetminus\{j\})}$ implies $x\,\mathsf F \,y$.

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arXiv2018

Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy

We present a model of set theory, in which, for a given $n\ge2$, there exists a non-ROD-uniformizable planar lightface $\varPi^1_n$ set in $\mathbb R\times\mathbb R$, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar boldface $\bfΣ^1_n$ sets with countable cross-sections are $\bfΔ^1_{n+1}$-uniformizable. Thus it is true in this model, that the ROD-uniformization principle for sets with countable cross-sections first fails precisely at a given projective level.

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arXiv2017

Definable minimal collapse functions at arbitrary projective levels

Using a non-Laver modification of Uri Abraham's minimal $\varDelta^1_3$ collapse function, we define a generic extension $L[a]$ by a real $a$, in which, for a given $n\ge3$, $\{a\}$ is a lightface $\varPi^1_n$ singleton, $a$ effectively codes a cofinal map $ω\toω_1^L$ minimal over $L$, while every $\varSigma^1_n$ set $X\subseteqω$ is still constructible.

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arXiv2017

Minimal axiomatic frameworks for definable hyperreals with transfer

We modify the definable ultrapower construction of Kanovei and Shelah (2004) to develop a ZF-definable extension of the continuum with transfer provable using countable choice only, with an additional mild hypothesis on well-ordering implying properness. Under the same assumptions, we also prove the existence of a definable, proper elementary extension of the standard superstructure over the reals. Keywords: definability; hyperreal; superstructure; elementary embedding.

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arXiv2017

Definable ${\mathsf E}_0$ classes at arbitrary projective levels

Using a modification of the invariant Jensen forcing, we define a model of ZFC, in which, for a given $n\ge3$, there exists a lightface $\varPi^1_n$ set of reals, which is a ${\mathsf E}_0$ equivalence class, hence a countable set, and which does not contain any OD element, while every non-empty countable $\varSigma^1_n$ set of reals is necessarily constructible, hence contains only OD reals.

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