Abstract
In continuation of our study of HST, Hrbaček 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 κ, any set R ⊑ *κ 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 κ-version of HST including the principle of κ+-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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References
Ballard, D., 1994, Foundational aspects of “non” standard mathematics, Contemporary Math. 176.
Ballard, D., and K. HrbaČek, 1992, Standard foundations for nonstandard analysis, J. Symbolic Logic 57, 741–748.
van den Berg, I., 1992, Extended use of IST, Ann. Pure Appl. Logic 58, 73–92.
Chang, C. C., and H. J. Keisler, 1983, Model Theory, North Holland, Amsterdam, (3rd ed. 1990).
Fletcher, P., 1989, Nonstandard set theory, J. Symbolic Logic 54, 1000–1008.
Gordon, E., 1989, Relatively standard elements in the theory of internal sets of E. Nelson. Siberian Math. J. 30, 89–95.
HrbaČek, K., 1978, Axiomatic foundations for nonstandard analysis. Fund. Math. 98, 1–19.
HrbaČek, K. 1979, Nonstandard set theory, Amer. Math. Monthly 86, 659–677.
Kanovei, V., 1991, Undecidable hypotheses in Edward Nelson's Internal Set Theory. Russian Math. Surveys 46,no. 6, 1–54.
Kanovei, V., and M. Reeken, 1995, Internal approach to external sets and universes, Part 1, Studia Logica 55,no. 2, 227–235.
Kanovei, V., and M. Reeken, 1995, Internal approach to external sets and universes, Part 2, Studia Logica 55,no. 3, 347–376.
Kanovei, V., and M. Reeken, 1996, Internal approach to external sets and universes, Part 3, Studia Logica 56,no. 3, 293–322.
Kanovei, V., and M. Reeken, 1997, Mathematics in a nonstandard world, Math. Japonica 45,no. 2, 369–408, and no. 3, 555–571.
KawaÏ, T., 1981, Nonstandard analysis by axiomatic methods, in: Southeast Asia Conference on Logic, Singapore (Studies in Logic and Foundations of Mathematics, 111, North Holland, 1983), 55–76.
Luxemburg, W. A. J., 1962, Non-standard analysis, lecture notes (Dept. Math. Cal. Inst. Tech.).
Nelson, E., 1977, Internal set theory; a new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83, 1165–1198.
Peraire, Y., 1992, Théorie relative des ensembles internes, Osaka J. Math. 29, 267–297.
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Kanovei, V., Reeken, M. Elementary Extensions of External Classes in a Nonstandard Universe. Studia Logica 60, 253–273 (1998). https://doi.org/10.1023/A:1005064032270
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DOI: https://doi.org/10.1023/A:1005064032270