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Vladimir Kanovei

Publications and source records attributed to Vladimir Kanovei.

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

A philosophical history of infinitesimals

We explore the issue of providing a foundational framework for Leibnizian infinitesimals in the light of modern standard and nonstandard approaches. We outline a trichotomy of ordinals, cardinals and ringinals as a historiographic tool. A ringinal is a concept of infinite number, arithmetic in nature, different from Cantor's transfinite ordinals and cardinals. The continuum is not necessarily identifiable with R; even if one seeks such an identification, infinitesimals are not ruled out. Analysis with unlimited numbers (via the predicate standard) is possible in a conservative extension of Zermelo-Fraenkel set theory and in this sense is epistemologically 'safe'. We sketch a recent theory of infinitesimal analysis that formalizes Leibnizian definitions and heuristic principles while eschewing both the axiom of choice and ultrafilters, thus challenging received philosophical views on the nature of infinitesimals.

math.HO↗
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 $\text{AC}_ω(\varSigma^1_{\infty}).$

math.LO↗
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.

math.LO↗
arXiv2023

Jensen $\varDelta^1_3$ reals by means of ZFC$^-$ or second order Peano arithmetic

It was established by Jensen in 1970 that there is a generic extension $L[a]$ of the constructible universe $L$ by a real $a\not\in L$ such that $a$ is $\varDelta^1_3$ in $L[a]$. Jensen's forcing construction has found a number of applications in modern set theory. A problem has been recently discussed whether Jensen's construction can be reproduced entirely within second order Peano arithmetic or equivalently FC$^-$ (minus the Power Set axiom). The obstacle is that the proof of the key CCC property (whether by Jensen's original argument or a later proof using $\Diamond$) essentially involves countable elementary submodels of $L_{ω_2}$, which is way beyond ZFC$^-$. We show how to circumwent this difficulty by means of killing only definable antichains in the course of a Jensen-like transfinite construction of the forcing, and then define a model with a minimal $\varPi^1_2$ singleton as a class-forcing extension of a model of ZFC$^-$ plus $V=L$.

math.LO↗
arXiv2023

Historical infinitesimalists and modern historiography of infinitesimals

In the history of infinitesimal calculus, we trace innovation from Leibniz to Cauchy and reaction from Berkeley to Mansion and beyond. We explore 19th century infinitesimal lores, including the approaches of Simeon-Denis Poisson, Gaspard-Gustave de Coriolis, and Jean-Nicolas Noel. We examine contrasting historiographic approaches to such lores, in the work of Laugwitz, Schubring, Spalt, and others, and address a recent critique by Archibald et al. We argue that the element of contingency in this history is more prominent than many modern historians seem willing to acknowledge.

math.HO↗
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.

math.LO↗
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.

math.LO↗
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.

math.LO↗
arXiv2020

Continuity between Cauchy and Bolzano: Issues of antecedents and priority

In a paper published in 1970, Grattan-Guinness argued that Cauchy, in his 1821 book Cours d'Analyse, may have plagiarized Bolzano's book Rein analytischer Beweis (RB), first published in 1817. That paper was subsequently discredited in several works, but some of its assumptions still prevail today. In particular, it is usually considered that Cauchy did not develop his notion of the continuity of a function before Bolzano developed his in RB, and that both notions are essentially the same. We argue that both assumptions are incorrect, and that it is implausible that Cauchy's initial insight into that notion, which eventually evolved to an approach using infinitesimals, could have been borrowed from Bolzano's work. Furthermore, we account for Bolzano's interest in that notion and focus on his discussion of a definition by Kästner (in Section 183 of his 1766 book), which the former seems to have misrepresented at least partially. Cauchy's treatment of continuity goes back at least to his 1817 course summaries, refuting a key component of Grattan-Guinness' plagiarism hypothesis (that Cauchy may have lifted continuity from RB after reading it in a Paris library in 1818). We explore antecedents of Cauchy and Bolzano continuity in the writings of Kästner and earlier authors.

math.HO↗
arXiv2020

Metric completions, the Heine-Borel property, and approachability

We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space. As mentioned by do Carmo, a nonextendible Riemannian manifold can be noncomplete, but in the broader category of metric spaces it becomes extendible. We give a short proof of a characterisation of the Heine-Borel property of the metric completion of a metric space M in terms of the absence of inapproachable finite points in *M.

math.DG↗
arXiv2020

Cauchy's work on integral geometry, centers of curvature, and other applications of infinitesimals

Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry where infinitesimals are used neither as variable quantities nor as sequences but rather as numbers. He also applied infinitesimals in an 1832 article on integral geometry, similarly as numbers. We explore these and other applications of Cauchy's infinitesimals as used in his textbooks and research articles. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis and geometry. We demonstrate the viability of Cauchy's infinitesimal techniques in fields as diverse as geometric probability, differential geometry, elasticity, Dirac delta functions, continuity and convergence. Keywords: Cauchy--Crofton formula; center of curvature; continuity; infinitesimals; integral geometry; limite; standard part; de Prony; Poisson

math.HO↗
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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