Realism, nonstandard set theory, and large cardinals

Annals of Pure and Applied Logic 109 (1-2):15-48 (2001)
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Abstract

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 other existing nonstandard set theories

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Citations of this work

Standard Sets in Nonstandard Set Theory.Petr Andreev & Karel Hrbacek - 2004 - Journal of Symbolic Logic 69 (1):165-182.
More on Regular and Decomposable Ultrafilters in ZFC.Paolo Lipparini - 2010 - Mathematical Logic Quarterly 56 (4):340-374.

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References found in this work

Believing the Axioms. I.Penelope Maddy - 1988 - Journal of Symbolic Logic 53 (2):481-511.
Standard Foundations for Nonstandard Analysis.David Ballard & Karel Hrbacek - 1992 - Journal of Symbolic Logic 57 (2):741-748.
The Syntax of Nonstandard Analysis.Edward Nelson - 1988 - Annals of Pure and Applied Logic 38 (2):123-134.
Nonstandard Set Theory.Peter Fletcher - 1989 - Journal of Symbolic Logic 54 (3):1000-1008.
Internal Set Theory: A New Approach to Nonstandard Analysis.Edward Nelson - 1977 - Journal of Symbolic Logic 48 (4):1203-1204.

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