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A nonstandard set theory in the $\displaystyle\in$ -language

A nonstandard set theory in the $\displaystyle\in$ -language Arch. Math. Logic (2000) 39: 403–416 c Springer-Verlag 2000 A nonstandard set theory in the ∈-language 1, 2 Vladimir Kanovei , Michael Reeken Department of Mathematics (Vychislitelnaya Matematika), Moscow Transport Engineer- ing Institute, Obraztsova 15, Moscow 101475, Russia DepartmentofMathematics,UniversityofWuppertal,GaussStrasse20,Wuppertal42097, Germany Received: 17 March 1998 / Revised version: 30 October 1998 Abstract. Wedemonstratethatacomprehensivenonstandardsettheorycan be developed in the standard ∈-language. As an illustration, a nonstandard Law of Large Numbers is obtained. Introduction NonstandardanalysiswasintroducedbyA.Robinsoninthebeginningofthe 60s as a concept in foundations of mathematics which allowed to develop such notions as an infinitesimal real or infinitely large natural number ade- quately and with full mathematical rigor. Those new mathematical objects, called “nonstandard”, brought some benefits to several branches of mathe- matics. However it was soon discovered that this idea naturally led to more and more complicated nonstandard mathematical objects, which could not be effectively tackled in the framework of Robinson’s original approach. Nonstandard set theories represent one of the two known ways of how to develop “nonstandard” mathematics in unified way. (The other setup, called the model theoretic version of nonstandard analysis, employs nonstandard extensions of mathematical structures in the “standard” Zermelo – Fraenkel universe,seeLindstrøm[13],whichisclosertoRobinson’sapproach.)Any such theory arranges the set universe in such a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

A nonstandard set theory in the $\displaystyle\in$ -language

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Publisher
Springer Journals
Copyright
Copyright © 2000 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s001530050155
Publisher site
See Article on Publisher Site

Abstract

Arch. Math. Logic (2000) 39: 403–416 c Springer-Verlag 2000 A nonstandard set theory in the ∈-language 1, 2 Vladimir Kanovei , Michael Reeken Department of Mathematics (Vychislitelnaya Matematika), Moscow Transport Engineer- ing Institute, Obraztsova 15, Moscow 101475, Russia DepartmentofMathematics,UniversityofWuppertal,GaussStrasse20,Wuppertal42097, Germany Received: 17 March 1998 / Revised version: 30 October 1998 Abstract. Wedemonstratethatacomprehensivenonstandardsettheorycan be developed in the standard ∈-language. As an illustration, a nonstandard Law of Large Numbers is obtained. Introduction NonstandardanalysiswasintroducedbyA.Robinsoninthebeginningofthe 60s as a concept in foundations of mathematics which allowed to develop such notions as an infinitesimal real or infinitely large natural number ade- quately and with full mathematical rigor. Those new mathematical objects, called “nonstandard”, brought some benefits to several branches of mathe- matics. However it was soon discovered that this idea naturally led to more and more complicated nonstandard mathematical objects, which could not be effectively tackled in the framework of Robinson’s original approach. Nonstandard set theories represent one of the two known ways of how to develop “nonstandard” mathematics in unified way. (The other setup, called the model theoretic version of nonstandard analysis, employs nonstandard extensions of mathematical structures in the “standard” Zermelo – Fraenkel universe,seeLindstrøm[13],whichisclosertoRobinson’sapproach.)Any such theory arranges the set universe in such a

Journal

Archive for Mathematical Logic – Springer Journals

Published: Aug 1, 2000

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