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Problems of set-theoretic non-standard analysis. (English. Russian original) Zbl 1226.03067

Russ. Math. Surv. 62, No. 1, 45-111 (2007); translation from Usp. Mat. Nauk 62, No. 1, 51-122 (2007).
Summary: The main results in the area of set-theoretic nonstandard analysis (nonstandard set and class theories) obtained over the last few years are presented. It is demonstrated how a universe of a comparably simple theory (beginning with the usual Zermelo-Fraenkel set theory ZFC) can be extended to a universe of a more complicated nonstandard set or class theory. The last section develops the foundations of Boolean-valued analysis as a part of set-theoretic nonstandard analysis.

MSC:

03H05 Nonstandard models in mathematics
03E40 Other aspects of forcing and Boolean-valued models
03E70 Nonclassical and second-order set theories
46S20 Nonstandard functional analysis
54J05 Nonstandard topology
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