arXiv subjects
Publications and source records attributed to Vladimir Kanovei.
We prove the following classification theorem of the ``Glimm -- Effros'' type for Borel order relations: a Borel partial order on the reals either is Borel linearizable or includes a copy of a certain Borel partial order $\meo$ which is not Borel linearizable.
A new elementary nonstandard proof of the Jordan curve theorem is given. The proof (the technical part consists of 4 pages) is self-contained, except for the Jordan theorem for polygons taken for granted.
We prove that in the Solovay model every OD graph G on reals satisfies one and only one of the following two conditions: (I) G admits an OD colouring by ordinals; (II) there exists a continuous homomorphism of G_0 into G, where G_0 is a certain F_sigma locally countable graph which is not R-OD colourable by ordinals in the Solovay model. If the graph G is locally countable or acyclic then (II) can be strengthened by the requirement that the homomorphism is a 1-1 map, i.e. an embedding. As the second main result we prove that Sigma^1_2 graphs admit the dichotomy (I) vs. (II) in set--generic extensions of the constructible universe L (although now (I) and (II) may be in general compatible). In this case (I) can be strengthened to the existence of a Delta^1_3 colouring by countable ordinals provided the graph is locally countable. The proofs are based on a topology generated by $\od$ sets.
We study models of HST, a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the language containing the membership and standardness predicates, and Saturation for well-orderable families of internal sets. This theory admits an adequate formulation of the isomorphism property IP: "any two elementarily equivalent internally presented structures of a wellorderable language are isomorphic." IP implies, for instance, that all infinite internal sets are equinumerous, and there exists a unique (modulo isomorphisms) internal elementary extension of the standard reals. We prove that IP is independent of HST (using the class of all sets constructible from internal sets) and consistent with HST (using generic extensions of $\HST$ models by a sufficient number of generic isomorphisms).
We prove that if $\bI$ is a p.\ o. set in a countable transitive model $\gM$ of $\ZFC$ then $\gM$ can be extended by a generic sequence of reals $\a_ı,$ $ı\in\bI,$ such that $\aleph_1^\gM$ is preserved and every $\a_ı$ is Sacks generic over $\gM[\ang{\a_\j:\j<ı}]$. The structure of the degrees of \dd\gM constructibility of reals in the extension is investigated. As an application, we obtain a model in which the $\is12$ equivalence relation $x \E y$ iff $\rL[x]=\rL[y]$ ($x,\,y$ are reals) does not admit a reasonable form of the Glimm -- Effros theorem.
We prove that if every real belongs to a set generic extension of the constructible universe then every Σ_1^1 equivalence E on reals either admits a Delta_1^HC reduction to the equality on the set 2^{<\om_1} of all countable binary sequences, or continuously embeds E_0, the Vitali equivalence. The proofs are based on a topology generated by OD sets.
We prove that in Solovay model every OD equivalence E on reals either admits an OD reduction to the equality on the set of all countable (of length < omega_1) binary sequences, or continuously embeds E_0, the Vitali equivalence. If E is a Sigma_1^1 (resp. Sigma_1^2) relation then the reduction in the ``either'' part can be chosen in the class of all Delta_1 (resp. Delta_2) functions. The proofs are based on a topology generated by OD sets.
We prove that a Spector--like ultrapower extension $\gN$ of a countable Solovay model $\gM$ (where all sets of reals are Lebesgue measurable) is equal to the set of all sets constructible from reals in a generic extension $\gM[\al]$ where $\al$ is a random real over $\gM.$ The proof involves an almost everywhere uniformization theorem in the Solovay model.
There is a sentence in the language of IST, Nelson's internal set theory, which is not equivalent in IST to a sentence in the ZFC language. Thus the Reduction algorithm of Nelson, that converts bounded IST formulas with standard parameters to provably (in IST) equivalent formulas in the ZFC language, cannot be extended to all formulas of the IST language. Therefore, certain IST sentences are "meaningless" from the point of view of the standard universe.