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  1. Forcing in nonstandard analysis.Masanao Ozawa - 1994 - Annals of Pure and Applied Logic 68 (3):263-297.
    A nonstandard universe is constructed from a superstructure in a Boolean-valued model of set theory. This provides a new framework of nonstandard analysis with which methods of forcing are incorporated naturally. Various new principles in this framework are provided together with the following applications: An example of an 1-saturated Boolean ultrapower of the real number field which is not Scott complete is constructed. Infinitesimal analysis based on the generic extension of the hyperreal numbers is provided, and the hull completeness theorem (...)
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  • Possible size of an ultrapower of $\omega$.Renling Jin & Saharon Shelah - 1999 - Archive for Mathematical Logic 38 (1):61-77.
    Let $\omega$ be the first infinite ordinal (or the set of all natural numbers) with the usual order $<$ . In § 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of $\omega$ , whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [1], modulo the assumption of supercompactness. In § 2 we construct several $\lambda$ -Archimedean ultrapowers of $\omega$ under some large cardinal (...)
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  • Game sentences and ultrapowers.Renling Jin & H. Jerome Keisler - 1993 - Annals of Pure and Applied Logic 60 (3):261-274.
    We prove that if is a model of size at most [kappa], λ[kappa] = λ, and a game sentence of length 2λ is true in a 2λ-saturated model ≡ , then player has a winning strategy for a related game in some ultrapower ΠD of . The moves in the new game are taken in the cartesian power λA, and the ultrafilter D over λ must be chosen after the game is played. By taking advantage of the expressive power of (...)
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  • Maharam spectra of Loeb spaces.Renling Jin & H. Jerome Keisler - 2000 - Journal of Symbolic Logic 65 (2):550-566.
    We characterize Maharam spectra of Loeb probability spaces and give some applications of the results.
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  • The generic filter property in nonstandard analysis.Mauro Di Nasso - 2001 - Annals of Pure and Applied Logic 111 (1-2):23-37.
    In this paper two new combinatorial principles in nonstandard analysis are isolated and applications are given. The second principle provides an equivalent formulation of Henson's isomorphism property.
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  • Combinatorial principles in nonstandard analysis.Mauro Di Nasso & Karel Hrbacek - 2003 - Annals of Pure and Applied Logic 119 (1-3):265-293.
    We study combinatorial principles related to the isomorphism property and the special model axiom in nonstandard analysis.
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