Article Outline
Glossary
Definition of the Subject
Introduction
Pre-Robinson Infinitesimals
Hyperreals
General Idea of NSA
Loeb Construction
A Future for Non-standard Analysis?
Bibliography
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsAbbreviations
- Hyperreal:
-
A proper extension of the real numbers, containing infinities and infinitesimal numbers, such that a transfer principle allows first-order theorems in the reals to be extended therein.
- Infinity:
-
An element in an ordered field which is larger than \( { 1+1+1+\cdots+1 } \) for any number of 1's.
- Infinitesimal:
-
An element in an ordered field whose absolute value is less than \( { 1/n } \) for any positive integer n.
- Non-archimedean:
-
Infinitesimal or infinite values exist in an ordered field.
- Transfer principle:
-
A rule which transforms assertions about standard sets, mappings, etc., into one about internal sets, mappings. Intuitively, it is an elementary embedding between structures.
Bibliography
AlbeverioS, Fenstad JE, Hoegh-Krohn R, Lindstrøm T (1986) Nonstandard Methods in StochasticAnalysis and Mathematical Physics. Bull Am Math Soc 17(2):385–389
Bernstein A, Robinson A (eds) (1966) Solution of an invariant subspace problemof KT Smith and PR Halmos. Pacific J Math 16(3):421–431
Halmos P (1966) Invariant subspaces for Polynomially Compact Operators. PacificJ Math 16(3):433–437
Kamae T (1982) A simple proof of the ergodic theorem using nonstandardanalysis. Israel J Math 42(4):284–290
Keisler HJ (1986) An Infinitesimal Approach to Stochastic Analysis. J Symb Logic51(3):822–824 now included In: (1984) Mem Amer Math Soc 297
Keisler HJ (1986) Elementary Calculus: An Approach Using Infinitesimals, 2ndedn. Now available at http://www.math.wisc.edu/~keisler/calc.html
KelleyJL (1975) General Topology. Springer, New York
Loeb PA, Wolff MPH (2000) Nonstandard Analysis for the WorkingMathematician. Math. Appl., vol. 510; Kluwer, Dordrecht
Nelson E (1977) Internal Set Theory A New Approach to Nonstandard Analysis. BullAm Math Soc 83(6):1165–1198; Also see http://www.math.princeton.edu/~nelson/books/1.pdf
NelsonE (1989) Radically Elementary Probability Theory. Bull Am Math Soc 20(2):240–243
Robert A (1985) Nonstandard Analysis. Bull Am Math Soc 16(2):298–306
Robinson A (1966) Non‐standardanalysis, rev edn.North Holland, Amsterdam
Schmieden C, Laugwitz D (1958) Eine Erweiterung derInfinitesimalrechnung. Math Zeit 69:1–39
van den Dries L, Wilkie AJ (1984) Gromov's Theorem on Groups of PolynomialGrowth and Elementary Logic. J Algebra 89:349–374
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag
About this entry
Cite this entry
Yang, WZ. (2012). Non-standard Analysis , an Invitation to. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_133
Download citation
DOI: https://doi.org/10.1007/978-1-4614-1800-9_133
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-1799-6
Online ISBN: 978-1-4614-1800-9
eBook Packages: Computer ScienceReference Module Computer Science and Engineering