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Vladimir Kanovei

Publications and source records attributed to Vladimir Kanovei.

At least 37 records · Page 2Linked to original sources
arXiv2019

On mathematical realism and the applicability of hyperreals

We argue that Robinson's hyperreals have just as much claim to applicability as the garden variety reals. In a recent text, Easwaran and Towsner (ET) analyze the applicability of mathematical techniques in the sciences, and introduce a distinction between techniques that are applicable and those that are merely instrumental. Unfortunately the authors have not shown that their distinction is a clear and fruitful one, as the examples they provide are superficial and unconvincing. Moreover, their analysis is vitiated by a reliance on a naive version of object realism which has long been abandoned by most philosophical realists in favor of truth-value realism. ET's argument against the applicability of hyperreals based on automorphisms of hyperreal models involves massaging the evidence and is similarly unconvincing. The purpose of the ET text is to argue that Robinson's infinitesimal analysis is merely instrumental rather than applicable. Yet in spite of Robinson's techniques being applied in physics, probability, and economics, ET don't bother to provide a meaningful analysis of even a single case in which these techniques are used. Instead, ET produce page after page of speculations mainly imitating Connesian chimera-type arguments `from first principles' against Robinson. In an earlier paper Easwaran endorsed real applicability of the sigma-additivity of measures, whereas the ET text rejects real applicability of the axiom of choice, voicing a preference for ZF. Since it is consistent with ZF that the Lebesgue measure is not sigma-additive, Easwaran is thereby walking back his earlier endorsement. We note a related inaccuracy in the textbook Measure Theory by Paul Halmos.

math.HO↗
arXiv2019

19th century real analysis, forward and backward

19th century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy historians work in a conceptual scheme dominated by an assumption of a teleological nature of the evolution of real analysis toward a preordained outcome. Thus, Gilain and Siegmund-Schultze assume that references to limite in Cauchy's work necessarily imply that Cauchy was working with an Archi-medean continuum, whereas infinitesimals were merely a convenient figure of speech, for which Cauchy had in mind a complete justification in terms of Archimedean limits. However, there is another formalisation of Cauchy's procedures exploiting his limite, more consistent with Cauchy's ubiquitous use of infinitesimals, in terms of the standard part principle of modern infinitesimal analysis. We challenge a misconception according to which Cauchy was allegedly forced to teach infinitesimals at the Ecole Polytechnique. We show that the debate there concerned mainly the issue of rigor, a separate one from infinitesimals. A critique of Cauchy's approach by his contemporary de Prony sheds light on the meaning of rigor to Cauchy and his contemporaries. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis, and indicates that he was a pioneer of infinitesimal techniques as much as a harbinger of the Epsilontik.

math.HO↗
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 $n\ge3$.

math.LO↗
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.

math.LO↗
arXiv2018

A model of second-order arithmetic satisfying AC but not DC

We show that there is a $β$-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a $Π^1_2$-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of ${\rm ZFC}^-$. This work is a rediscovery by the first two authors of a result obtained by the third author.

math.LO↗
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 $x\,\mathsf F \,y$.

math.LO↗
arXiv2018

Monotone subsequence via ultrapower

An ultraproduct can be a helpful organizing principle in presenting solutions of problems at many levels, as argued by Terence Tao. We apply it here to the solution of a calculus problem: every infinite sequence has a monotone infinite subsequence, and give other applications. Keywords: ordered structures; monotone subsequence; ultrapower; saturation; compactness

math.CA↗
arXiv2018

What makes a theory of infinitesimals useful? A view by Klein and Fraenkel

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.

math.HO↗
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.

math.LO↗
arXiv2017

Cauchy, infinitesimals and ghosts of departed quantifiers

Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz's distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson's framework, while Leibniz's law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz's infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson's framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler's own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson's framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy's procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson's framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. Keywords: historiography; infinitesimal; Latin model; butterfly model

math.HO↗
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.

math.LO↗
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.

math.LO↗
arXiv2017

Cauchy's infinitesimals, his sum theorem, and foundational paradigms

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy's proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy's proof closely and show that it finds closer proxies in a different modern framework. Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation; uniform convergence; foundational paradigms.

math.HO↗
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.

math.LO↗