arXiv subjects
Publications and source records attributed to Vladimir Kanovei.
We prove that in some cases definable chains of Borel partial orderings are necessarily countably cofinal. This includes the following cases: analytic chains, ROD chains in the Solovay model, and $Σ^1_2$ chains in the assumption that $ω_1^{L[x]}<ω_1$ for all reals $x$.
We examine prevailing philosophical and historical views about the origin of infinitesimal mathematics in light of modern infinitesimal theories, and show the works of Fermat, Leibniz, Euler, Cauchy and other giants of infinitesimal mathematics in a new light. We also detail several procedures of the historical infinitesimal calculus that were only clarified and formalized with the advent of modern infinitesimals. These procedures include Fermat's adequality; Leibniz's law of continuity and the transcendental law of homogeneity; Euler's principle of cancellation and infinite integers with the associated infinite products; Cauchy's infinitesimal-based definition of continuity and "Dirac" delta function. Such procedures were interpreted and formalized in Robinson's framework in terms of concepts like microcontinuity (S-continuity), the standard part principle, the transfer principle, and hyperfinite products. We evaluate the critiques of historical and modern infinitesimals by their foes from Berkeley and Cantor to Bishop and Connes. We analyze the issue of the consistency, as distinct from the issue of the rigor, of historical infinitesimals, and contrast the methodologies of Leibniz and Nieuwentijt in this connection.
We examine some of Connes' criticisms of Robinson's infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes' own earlier work in functional analysis. Connes described the hyperreals as both a "virtual theory" and a "chimera", yet acknowledged that his argument relies on the transfer principle. We analyze Connes' "dart-throwing" thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being "virtual" if it is not definable in a suitable model of ZFC. If so, Connes' claim that a theory of the hyperreals is "virtual" is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren't definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in his Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes' criticism of virtuality. We analyze the philosophical underpinnings of Connes' argument based on Goedel's incompleteness theorem, and detect an apparent circularity in Connes' logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace (featured on the front cover of Connes' magnum opus) and the Hahn-Banach theorem, in Connes' own framework. We also note an inaccuracy in Machover's critique of infinitesimal-based pedagogy.
We prove several theorems on sigma-bounded and sigma-compact pointsets. We start with a known theorem by Kechris, saying that any lightface Σ^1_1 set of the Baire space either is effectively sigma-bounded (that is, covered by a countable union of compact lightface Δ^1_1 sets), or contains a superperfect subset (and then the set is not sigma-bounded, of course). We add different generalizations of this result, in particular, 1) such that the boundedness property involved includes covering by compact sets and equivalence classes of a given finite collection of lightface Δ^1_1 equivalence relations, 2) generalizations to lightface Σ^1_2 sets, 3) generalizations true in the Solovay model. As for effective sigma-compactness, we start with a theorem by Louveau, saying that any lightface Δ^1_1 set of the Baire space either is effectively sigma-compact (that is, is equal to a countable union of compact lightface Δ^1_1 sets), or it contains a relatively closed superperfect subset. Then we prove a generalization of this result to lightface Σ^1_1 sets.
Using the Gandy -- Harrington topology and other methods of effective descriptive set theory, we prove several theorems on compact and sigma-compact pointsets. In particular we show that any $Σ^1_1$ set $A$ of the Baire space $N^N$ either is covered by a countable union of compact $Δ^1_1$ sets, or $A$ contains a subset closed in $N^N$ and homeomorphic to $N^N$ (and then $A$ is not covered by a sigma-compact set, of course).
It is known that the assumption that ``GCH first fails at \aleph_ω'' leads to large cardinals in ZFC. Gitik and Koepke have demonstrated that this is not so in ZF: namely there is a generic cardinal-preserving extension of L (or any universe of ZFC + GCH in which all ZF axioms hold, the axiom of choice fails, GCH holds for all cardinals \aleph_n, but there is a surjection from PowerSet(\aleph_ω) onto λ, where λ is any previously chosen cardinal in L greater than \aleph_ω, for instance, \aleph_{ω+17}. In other words, in such an extension GCH holds in proper sense for all cardinals \aleph_n but fails at \aleph_ω in Hartogs' sense. The goal of this note is to analyse the system of automorphisms involved in the Gitik -- Koepke proof.
The following is true in the Solovay model. 1. If $\le$ is a Borel partial order on a set $D$ of the reals, and $X$ is a ROD subset of $D$ linearly ordered by $\le$, then the restriction of $\le$ onto $X$ is countably cofinal. 2. If in addition every countable set $Y$ of $D$ has a strict upper bound in the sense of $\le$ then the ordering $< D ; \le >$ has no maximal chains that are ROD sets.
For any abelian Polish sigma-compact group H there exist a sigma-ideal Z over N and a Borel Z-approximate homomorphism f : H --> H^N which is not Z-approximable by a continuous true homomorphism g : H --> H^N.
If E is an equivalence relation Borel reducible to E_1 \times E_3 then either E is Borel reducible to the equality of countable sets of reals or E_1 is Borel reducible to E. The "either" case admits further strengthening.
We present a selection of basic results on Borel reducibility of Borel ideals and equivalence relations, especially those with comparably short proofs. The focal point are reducibility/irreducibility results related to some special equivalences like $E_0, E_1, E_2, E_3, E_\infty, Z_0,$ and Banach-induced equivalences $l_p,$ in particular several dichotomy theorems. The bulk of results included in the book were obtained in the 1990s, but some rather recent theorems are presented as well, like Rosendal's proof that Borel ideals are cofinal within Borel equivalences of general form.
A selection of basic results on Borel reducibility of ideals and equivalence relations, especially those with comparably short proofs. This is an unfinished text as yet. Some proofs have missing parts and loose ends. kanovei@mccme.ru and vkanovei@math.uni-wuppertal.de are my contact addresses.
We study the structure of effective cardinals in the nonstandard set universe of Hrbacek set theory HST. Some results resemble those known in descriptive set theory in the domain of Borel reducibility of equivalence relations.
We prove in ZFC the existence of a definable, countably saturated elementary extension of the reals. It seems that it has been taken for granted that there is no distinguished, definable nonstandard model of the reals. (This means a countably saturated elementary extension of the reals.) Of course if V=L then there is such an extension (just take the first one in the sense of the canonical well-ordering of L), but we mean the existence provably in ZFC. There were good reasons for this: without Choice we cannot prove the existence of any elementary extension of the reals containing an infinitely large integer. Still there is one. Theorem (ZFC). There exists a definable, countably saturated extension R* of the reals R, elementary in the sense of the language containing a symbol for every finitary relation on R.
We prove that orbit equivalence relations (ERs, for brevity) of generically turbulent Polish actions are not Borel reducible to ER s of a family which includes Polish actions of S_\infty, the group of all permutations of N, and is closed under the Fubini product modulo the ideal Fin of all finite sets, and some other operations. Our second main result shows that T_2, the equivalence relation called ``the equality of countable sets of the reals'', is not Borel reducible to another family of ERs which includes continuous actions of Polish CLI groups, Borel equivalence relations with G_δclasses, some ideals, and is closed under the Fubini product over $Fin.$ Both results and their corollaries extend some earlier irreducibility theorems by Hjorth and Kechris.
We consider reducibility of equivalence relations (ERs, for brevity), in a nonstandard domain, in terms of the Borel reducibility and the countably determined (CD, for brevity) reducibility. This reveals phenomena partially analogous to those discovered in descriptive set theory. The Borel reducibility structure of Borel sets and (partially) CD reducibility structure of CD sets in *N is described. We prove that all CD ERs with countable equivalence classes are CD-smooth, but not all are B-smooth, for instance, the ER of having finite difference on *N. Similarly to the Silver dichotomy theorem in Polish spaces, any CD ER on *N either has at most continuum-many classes or there is an infinite internal set of pairwise inequivalent elements. Our study of monadic ERs on *N, i.e., those of the form x E y iff |x-y| belongs to a given additive Borel cut in *N, shows that these ERs split in two linearly families, associated with countably cofinal and countably coinitial cuts, each of which is linearly ordered by Borel reducibility. The relationship between monadic ERs and the ER of finite symmetric difference on hyperfinite subsets of *N is studied.
Theorem: Let $n\ge 2.$ There is a CCC in $L$ forcing notion $P=P_n\in L$ such that $P$-generic extensions of $L$ are of the form $L[a],$ where $a\subseteqω$ and 1) $a$ is $Δ^1_{n+1}$ in $L[a]$; and 2) if $b\in L[a],$ $b\subseteqω$ is $Σ^1_n$ in $L[a]$ then $b\in L$ and $b$ is $Σ^1_n$ in $L$. In addition, if a model $M$ extends $L$ and contains two different $P$-generic sets $a,\,a'\subseteqω,$ then $ω^M_1 > ω^L_1$. Comment: For $n=2,$ this is a result of Jensen and Johnsbråten, 1974. In this case, 2) is a corollary of the Shoenfield absoluteness theorem.
We characterize those standard models M of ZFC which are embeddable, as the class of all standard sets, in a model of internal set theory IST. The necessary and sufficient condition is that 1) there is a wellordering < of M which does not destroy the ZFC schemata, and 2) the truth relation for (M,<) does not destroy Separation. The result is interpreted as the answer for the question in the title.
We prove that if $\leq$ is an analytic partial order then either $\leq$ can be extended to a (boldface) $Δ^1_2$ linear order similar to an antichain in $2^{<ω_1}$ ordered lexicographically or a certain Borel partial order $\leq_0$ embeds in $\leq.$ Some corollaries for analytic equivalence relations are given, for instance, if $E$ is a $Σ^1_1[z]$ equivalence relation such that $E_0$ does not embed in $E$ then $E$ is determined by intersections with E-invariand Borel sets coded in $L[z]$.