Michael Reeken
University of Wuppertal
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Studia Logica | 1995
Vladimir Kanovei; Michael Reeken
In this article we show how the universe ofBST,bounded set theory (a modification ofIST which is, briefly, a theory for the family of those sets inIST which are members of standard sets) can be enlarged by definable subclasses of sets (which are not necessarily sest in internal theories likeBST orIST) so that Separation and Replacement are true in the enlargement for all formulas, including those in which the standardness predicate may occur.ThusBST is strong enough to incorporate external sets in the internal universe in a way sufficient to develop topics in nonstandard analysis inaccessible in the framework of a purely internal approach, such as Loeb measures.
Studia Logica | 1995
Vladimir Kanovei; Michael Reeken
AbstractA problem which enthusiasts ofIST, Nelsons internal set theory, usually face is how to treat external sets in the internal universe which does not contain them directly. To solve this problem, we considerBST,bounded set theory, a modification ofIST which is, briefly, a theory for the family of thoseIST sets which are members of standard sets.We show thatBST is strong enough to incorporate external sets in the internal universe in a way sufficient to develop the most advanced applications of nonstandard methods. In particular, we define inBST an enlargement of theBST universe which satisfies the axioms ofHST, an external theory close to a theory introduced by Hrbaček. HST includes Replacement and Saturation for all formulas but contradicts the Power Set and Choice axioms (either of them is incompatible with Replacement plus Saturation), therefore to get an external universe which satisfies all ofZFC minus Regularity one has to pay by a restriction of Saturation. We prove thatHST admits a system of subuniverses which modelZFC (minus Regularity but with Power Set and Choice) and Saturation in a form restricted by a fixed but arbitrary standard cardinal.Thus the proposed system of set theoretic foundations for nonstandard mathematics, based on the simple and natural axioms of the internal theoryBST, provides the treatment of external sets sufficient to carry out elaborate external constructions.
Annals of Pure and Applied Logic | 1997
Vladimir Kanovei; Michael Reeken
Abstract We study models of HST (a nonstandard set theory which includes, in particular, the Replacement and Separation schemata of ZFC 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 , which postulates that any two elementarily equivalent internally presented structures of a well-orderable language are isomorphic. 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 a constructible model of HST by a sufficient number of generic isomorphisms).
Izvestiya: Mathematics | 2003
Vladimir Kanovei; Michael Reeken
We prove that orbit equivalence relations (ERs, for brevity) of generically turbulent Polish actions are not Borel reducible to ERs of a family which includes Polish actions of (the group of all permutations of and is closed under the Fubini product modulo the ideal Fin of all finite sets and under some other operations. We show that (an equivalence relation called the equality of countable sets of reals is not Borel reducible to another family of ERs which includes continuous actions of Polish CLI groups, Borel equivalence relations with classes and some ideals, and is closed under the Fubini product modulo Fin. These results and their corollaries extend some earlier irreducibility theorems of Hjorth and Kechris.
Monatshefte für Mathematik | 2003
Vladimir Kanovei; Michael Reeken
Abstract. We consider, in a nonstandard domain, reducibility of equivalence relations in terms of the Borel reducibility ≤B and the countably determined (CD, for brevity) reducibility ≤CD. This reveals phenomena partially analogous to those discovered in modern “standard” descriptive set theory. The ≤CD-structure of CD sets (partially) and the ≤B-structure of Borel sets (completely) in *ℕ are described. We prove that all “countable” (i.e., those with countable equivalence classes) CD equivalence relations (ERs) are CD-smooth, but not all are B-smooth: the relation x Mℕy iff ∣x−y∣∈ℕ is a counterexample. Similarly to the Silver dichotomy theorem in Polish spaces, any CD equivalence relation on *ℕ either has at most continuum-many classes (and this can be witnessed, in some manner, by a countably determined function) or there is an infinite internal set of pairwise inequivalent elements. Our study of monadic equivalence relations, i.e., those of the form x MU y iff ∣x−y∣∈U, where U is an additive countably determined cut (initial segment) demonstrates that these ERs split in two linearly ≤B-(pre)ordered families, associated with countably cofinal and countably coinitial cuts. The equivalence u FD v iff uΔv is finite, on the set of all hyperfinite subsets of *ℕ, ≤B-reduces all “countably cofinal” ERs but does not ≤CD-reduce any of “countably coinitial” ERs.
Mathematika | 2000
Vladimir Kanovei; Michael Reeken
Farah recently proved that many Borel P-ideals. F on N satisfy the following requirement: any measurable homomorphism F: F(N)/ →F(N)/F has a continuous lifting f:F(N)→F(N) which is a homomorphism itself. Ideals having such a property were called Radon-Nikodym (RN) ideals. Answering some Farahs questions, it is proved that many non-P ideals, including, for instance, Fin ⊗ Fin, are Radon-Nikodym. To prove this result, another property of ideals called the Fubini property, is introduced, which implies RN and is stable under some important transformations of ideals.
Mathematical Logic Quarterly | 2003
Vladimir Kanovei; Michael Reeken
It is known that every Borel hypersmooth but non-smooth equivalence relation is Borel bi-reducible to E1. We prove a ROD version of this result in the Solovay model.
Studia Logica | 2000
Vladimir Kanovei; Michael Reeken
We study those models of ZFCwhich are embeddable, as the class of all standard sets, in a model of internal set theory >ISTor models of some other nonstandard set theories.
Mathematical Logic Quarterly | 2000
Vladimir Kanovei; Michael Reeken
The quotient ℝ/G of the additive group of the reals modulo a countable subgroup G does not admit nontrivial Baire measurable automorphisms.
Topology and its Applications | 2001
J.R.P. Christensen; Vladimir Kanovei; Michael Reeken
Abstract We prove that any Borel Abelian ordered group B , having a countable subgroup G as the largest convex subgroup, and such that the quotient B/G is order isomorphic to R , the reals, is Borel group-order isomorphic to the product R ×G , ordered lexicographically. As a main ingredient of this proof, we show, answering a question of D. Marker, that all Borel cocycles R 2 → Z are Borel coboundaries. A Borel classification theorem for Borel ordered ccc groups is proved.