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  1. Incommensurables and Incomparables: On the Conceptual Status and the Philosophical Use of Hyperreal Numbers.Michael White - 1999 - Notre Dame Journal of Formal Logic 40 (3):420-446.
    After briefly considering the ancient Greek and nineteenth-century history of incommensurables (magnitudes that do not have a common aliquot part) and incomparables (magnitudes such that the larger can never be surpassed by any finite number of additions of the smaller to itself), this paper undertakes two tasks. The first task is to consider whether the numerical accommodation of incommensurables by means of the extension of the ordered field of rational numbers to the field of reals is `similar' or analogous to (...)
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  • Internal Approach to External Sets and Universes: Part 3: Partially Saturated Universes.Vladimir Kanovei & Michael Reeken - 1996 - Studia Logica 56 (3):293-322.
    In this article ‡ we show how the universe of HST, Hrbaček set theory admits a system of subuniverses which keep the Replacement, model Power set and Choice, and also keep as much of Saturation as it is necessary. This gives sufficient tools to develop the most complicated topics in nonstandard analysis, such as Loeb measures.
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  • 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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  • A definable nonstandard enlargement.Frederik Herzberg - 2008 - Mathematical Logic Quarterly 54 (2):167-175.
    This article establishes the existence of a definable , countably saturated nonstandard enlargement of the superstructure over the reals. This nonstandard universe is obtained as the union of an inductive chain of bounded ultrapowers . The underlying ultrafilter is the one constructed by Kanovei and Shelah [10].
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  • 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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