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  1.  47
    Internal Approach to External Sets and Universes.Vladimir Kanovei & Michael Reeken - 1995 - Studia Logica 55 (2):347 - 376.
    In this article we show how the universe of BST, bounded set theory can be enlarged by definable subclasses of sets so that Separation and Replacement are true in the enlargement for all formulas, including those in which the standardness predicate may occur. Thus BST 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.
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  2.  20
    A Nonstandard Set Theory in the Displaystylein-Language.Vladimir Kanovei & Michael Reeken - 2000 - Archive for Mathematical Logic 39 (6):403-416.
    . We demonstrate that a comprehensive nonstandard set theory can be developed in the standard $\displaystyle{\in}$ -language. As an illustration, a nonstandard ${\sf Law of Large Numbers}$ is obtained.
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  3.  28
    Internal Approach to External Sets and Universes.Vladimir Kanovei & Michael Reeken - 1996 - Studia Logica 56 (3):293 - 322.
    In this article we show how the universe of HST, Hrbaek set theory (a nonstandard set theory of external type, which includes, in particular, the ZFC Replacement and Separation schemata for all formulas in the language containing the membership and standardness predicates, and Saturation for standard size families of internal sets, but does not include the Power set axiom) admits a system of subuniverses which keep the Replacement, model Power set and Choice (in fact all of ZFC, with the exception (...)
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  4.  15
    Elementary Extensions of External Classes in a Nonstandard Universe.Vladimir Kanovei & Michael Reeken - 1998 - Studia Logica 60 (2):253-273.
    In continuation of our study of HST, Hrbaek set theory (a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the st--language, and Saturation for well-orderable families of internal sets), we consider the problem of existence of elementary extensions of inner "external" subclasses of the HST universe.We show that, given a standard cardinal , any set R * generates an "internal" class S(R) of all sets standard relatively to elements of R, and an "external" class (...)
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  5.  5
    A Nonstandard Set Theory in the [Mathematical Formula]-Language.Vladimir Kanovei & Michael Reeken - 2000 - Archive for Mathematical Logic 6.
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  6.  4
    Isomorphism Property in Nonstandard Extensions of theZFC Universe.Vladimir Kanovei & Michael Reeken - 1997 - Annals of Pure and Applied Logic 88 (1):1-25.
    We study models of HST . This theory admits an adequate formulation of the isomorphism propertyIP, 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 and consistent with HST.
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  7.  13
    Extending Standard Models of ZFC to Models of Nonstandard Set Theories.Vladimir Kanovei & Michael Reeken - 2000 - Studia Logica 64 (1):37-59.
    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.
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  8.  3
    Loeb Measure From the Point of View of a Coin Flipping Game.Vladimir Kanovei & Michael Reeken - 1996 - Mathematical Logic Quarterly 42 (1):19-26.
    A hyperfinitely long coin flipping game between the Gambler and the Casino, associated with a given set A, is considered. It turns out that the Gambler has a winning strategy if and only if A has Loeb measure 0. The Casino has a winning strategy if and only if A contains an internal subset of positive Loeb measure.
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  9.  2
    Special Model Axiom in Nonstandard Set Theory.Vladimir Kanovei & Michael Reeken - 1999 - Mathematical Logic Quarterly 45 (3):371-384.
    We demonstrate that the special model axiom SMA of Ross admits a natural formalization in Kawai's nonstandard set theory KST but is independent of KST. As an application of our methods to classical model theory, we present a short proof of the consistency of the existence of a k+ like k-saturated model of PA for a given cardinal k.
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  10.  1
    On Baire Measurable Homomorphisms of Quotients of the Additive Group of the Reals.Vladimir Kanovei & Michael Reeken - 2000 - Mathematical Logic Quarterly 46 (3):377-384.
    The quotient ℝ/G of the additive group of the reals modulo a countable subgroup G does not admit nontrivial Baire measurable automorphisms.
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