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On relative nearstandardness in IST. (English. Russian original) Zbl 0924.03116

Sib. Math. J. 39, No. 3, 518-521 (1998); translation from Sib. Mat. Zh. 39, No. 3, 600-603 (1998).
E. I. Gordon [Sib. Math. J. 30, No. 1, 68-73 (1989); translated from Sib. Mat. Zh. 30, No. 1, 89-95 (1989; Zbl 0697.03037)] has shown that, for some nonstandard natural number \(N\), not all points of the interval \([0,1]\) are \(N\)-nearstandard (i.e., there exists an \(x\in [0,1]\) such that there is no \(N\)-standard number \(N\)-infinitely close to \(x\)). The following question arises: Is it possible to choose a nonstandard natural number \(N\) so that each point of the interval \([0,1]\) has an \(N\)-standard part. In the article under review, it is shown that the answer to this question is negative. Moreover, it remains negative in a more general case when we replace the set of naturals by an arbitrary set of nonmeasurable cardinality and the interval by an arbitrary Hausdorff space other than a rare compact set.

MSC:

03H05 Nonstandard models in mathematics

Citations:

Zbl 0697.03037
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References:

[1] E. I. Gordon, ”Relatively standard elements in E. Nelson’s internal set theory,” Sibirsk. Mat. Zh.,30, No. 1, 89–95 (1989). · Zbl 0697.03037
[2] B. Benninghofen and M. M. Richter, ”A general theory of superinfinitesimals,” Fund. Math.,128, No. 3, 199–215 (1987). · Zbl 0633.03067
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[4] V. G. Kanoveî, ”Undecidable hypotheses in Edward Nelson’s internal set theory,” Uspekhi Mat. Nauk,46, No. 6, 3–50 (1991).
[5] K. Kuratowski and A. Mostowski, Theory of Sets [Russian translation], Mir, Moscow (1970). · Zbl 0204.31301
[6] E. Nelson, ”Internal set theory: a new approach to nonstandard analysis,” Bull. Amer. Math. Soc.,83, No. 6, 1165–1198 (1977). · Zbl 0373.02040
[7] A. V. Arkhangel’skiî, Topological Function Spaces [in Russian], Moscow Univ., Moscow (1989).
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