Elementary extensions of external classes in a nonstandard universe

Studia Logica 60 (2):253-273 (1998)
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Abstract

In continuation of our study of HST, Hrbaek set theory (a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the st--language, and Saturation for well-orderable families of internal sets), we consider the problem of existence of elementary extensions of inner "external" subclasses of the HST universe.We show that, given a standard cardinal , any set R * generates an "internal" class S(R) of all sets standard relatively to elements of R, and an "external" class L[S(R)] of all sets constructible (in a sense close to the Gödel constructibility) from sets in S(R). We prove that under some mild saturation-like requirements for R the class L[S(R)] models a certain -version of HST including the principle of +-saturation; moreover, in this case L[S(R)] is an elementary extension of L[S(R)] in the st--language whenever sets R R satisfy the requirements.

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Citations of this work

Realism, nonstandard set theory, and large cardinals.Karel Hrbacek - 2001 - Annals of Pure and Applied Logic 109 (1-2):15-48.

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References found in this work

Standard foundations for nonstandard analysis.David Ballard & Karel Hrbacek - 1992 - Journal of Symbolic Logic 57 (2):741-748.
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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