On the Existence of Factor Sets by External Equivalences in IST
We study the possibility of defining the factor sets of the real axis by external equivalences through external formulas on using the IST axiomatics of nonstandard analysis. We consider the case of an additive convex equivalence whose equivalence classes are defined by a formula with external universal quantifiers. We show that in this case there exists an external function selecting one representative from each equivalence class if and only if the relation in question coincides up to translation and dilation with the relation of infinite proximity.
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- 1.Kanove \(\imath\) V. G., “Undecidable hypotheses in Edward Nelson's internal set theory,” Uspekhi Mat. Nauk, 46, No. 6, 3–50 (1991).Google Scholar
- 2.Nelson E., “Internal set theory: a new approach to nonstandard analysis,” Bull. Amer. Math. Soc., 83, No. 6, 1165–1198 (1977).Google Scholar
- 3.Gordon E. I., “Relatively standard elements in E. Nelson's internal set theory,” Sibirsk. Mat. Zh., 30, No. 1, 89–95 (1989).Google Scholar
- 4.Prokhorova M. F., “External cardinality of finite sets in nonstandard analysis,” in: Problems of Theoretical and Applied Mathematics [in Russian], Ural'sk. Otdel. Ross. Akad. Nauk, Ekaterinburg, 1993.Google Scholar
- 5.Prokhorova M. F., “On relative nearstandardness in IST,” Sibirsk. Mat. Zh., 39, No. 3, 600–603 (1998).Google Scholar
- 6.Benninghofen B. and Richter M. M., “A general theory of superinfinitesimals,” Fund. Math., 128, No. 3, 199–215 (1987).Google Scholar
- 7.Prokhorova M. F., “An external analog of the choice axiom in nonstandard analysis,” in: Problems of Theoretical and Applied Mathematics [in Russian], Ural'sk. Otdel. Ross. Akad. Nauk, Ekaterinburg, 1997.Google Scholar
- 8.Prokhorova M. F., “An external analog of the axiom of choice in IST,” submitted to VINITI on October 29, 1999, No. 3246–B99.Google Scholar