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Nonstandard set theory

Journal of Symbolic Logic 54 (3):1000-1008 (1989)

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  1. 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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  • 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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  • 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.Vladimir Kanovei & Michael Reeken - 1995 - Studia Logica 55 (2):229-257.
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  • Realism, Nonstandard Set Theory, and Large Cardinals.Karel Hrbacek - 2001 - Annals of Pure and Applied Logic 109 (1-2):15-48.
    Mathematicians justify axioms of set theory “intrinsically”, by reference to the universe of sets of their intuition, and “extrinsically”, for example, by considerations of simplicity or usefullness for mathematical practice. Here we apply the same kind of justifications to Nonstandard Analysis and argue for acceptance of BNST+ . BNST+ has nontrivial consequences for standard set theory; for example, it implies existence of inner models with measurable cardinals. We also consider how to practice Nonstandard Analysis in BNST+, and compare it with (...)
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