Abstract
In this article ‡ we show how the universe of HST, Hrbaček set theory (a nonstandard set theory of “external” type, which includes, in particular, the ZFC Replacement and Separation schemata for all formulas in the language containing the membership and standardness predicates, and Saturation for “standard size” families of internal sets, but does not include the Power set axiom) admits a system of subuniverses which keep the Replacement, model Power set and Choice (in fact all of ZFC, with the exception of the Regularity axiom, which indeed is replaced by the Regularity over the internal subuniverse), and also keep as much of Saturation as it is necessary.
This gives sufficient tools to develop the most complicated topics in nonstandard analysis, such as Loeb measures.
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References
D. Ballard, Foundational aspects of “non”standard mathematics. Contemporary Math. 1994, 176.
D. Ballard and K. Hrbaček, Standard foundations for nonstandard analysis. J. Symbolic Logic 1992, 57, 741 - 748.
F. Diener and G. Reeb, Analyse non standard (Herrmann Editeurs, Paris, 1989).
P. Fletcher, Nonstandard set theory. J. Symbolic Logic 1989, 54, 1000 - 1008.
C. W. Henson and H. J. Keisler, On the strength of nonstandard analysis. J. Symbolic Logic 1986, 51, 377 - 386.
K. Hrbaček, Axiomatic foundations for nonstandard analysis. Fundamenta Mathematicae 1978, 98, pp 1 - 19.
K. Hrbaček, Nonstandard set theory, Amer. Math. Monthly 1979, 86, 659 - 677.
A. E. Hurd and P. A. Loeb, An introduction to nonstandard real analysis (Academic Press, 1985).
V. Kanovei, Undecidable hypotheses in Edward Nelson's Internal Set Theory. Russian Math. Surveys 1991, 46, no 6, 1 - 54.
V. Kanovei, A course on foundations of nonstandard analysis. (With a preface by M. Reeken.) IPM, Tehran, Iran, 1994.
V. Kanovei, Uniqueness, Collection, and external collapse of cardinals in IST and models of Peano arithmetic. J. Symbolic Logic 1995, 60, no 2, 318 - 324.
V. Kanovei and M. Reeken, Internal approach to external sets and universes, Part 1, Bounded set theory, Studia Logica 1995, 55, no. 2, 229 - 257.
V. Kanovei and M. Reeken, Internal approach to external sets and universes, Part 2, External universes over the universe of bounded set theory, Studia Logica 1995, 55, no. 3, 347 - 376.
T. Kawaï, Nonstandard analysis by axiomatic methods, in: Southeast Asia Conference on Logic, Singapore 1981 (Studies in Logic and Foundations of Mathematics, 111, North Holland, 1983), 55 – 76.
H. J. Keisler, The hyperreal line, in P. Erlich (ed.) Real numbers, generalizations of reals, and theories of continua, Kluwer Academic Publishers, 1994, 207 – 237.
H. J. Keisler, K. Kunen, A. Miller, and S. Leth, Descriptive set theory over hyperfinite sets, J. Symbolic Logic 1989, 54, 1167 - 1180.
T. Lindstrøm, An invitation to nonstandard analysis, in: N. Cutland (ed.) Nonstandard analysis and its applications (London Math. Soc. Student Texts 10, Cambridge Univ. Press, 1988), 1 – 105.
R. Lutz and M. Goze, Nonstandard analysis. A practical guide with applications (Lecture Notes Math. 881, Springer, 1981).
W. A. J. Luxemburg, What is nonstandard analysis? Amer. Math. Monthly 1973, 80 (Supplement), 38 - 67.
W. A. J. Luxemburg, Non-standard analysis, lecture notes (Dept. Math. Cal. Inst. Tech. 1962).
E. Nelson, Internal set theory; a new approach to nonstandard analysis, Bull. Amer. Math. Soc. 1977, 83, 1165- 1198.
M. Reeken, On external constructions in Internal Set Theory. Expositiones Mathematicae 1992, 10, 193 - 247.
A. Robert, Nonstandard analysis (Wiley, 1988).
K. D. Stroyan and J. M. Bayod, Foundations of infinitesimal stochastic analysis (North Holland, 1986).
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Partially supported by AMS grants in 1993 – 1995 and DFG grants in 1994 – 1995.
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Kanovei, V., Reeken, M. Internal approach to external sets and universes. Stud Logica 56, 293–322 (1996). https://doi.org/10.1007/BF00372770
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DOI: https://doi.org/10.1007/BF00372770