Abstract
To accommodate readers with little or no background in mathematical logic, this chapter gives a simple construction of a nonstandard number system associated with a very simple system of logic. The simplicity of the system of logic helps to illuminate the way logic is used to obtain mathematical results. It also simplifies the proof of the “transfer principle”, which is the basic Property of nonstandard number systems. The skills the reader needs for more advanced work are developed through applications of this simple system to calculus and elementary real analysis.To accommodate readers with little or no background in mathematical logic, this chapter gives a simple construction of a nonstandard number system associated with a very simple system of logic. The simplicity of the system of logic helps to illuminate the way logic is used to obtain mathematical results. It also simplifies the proof of the “transfer principle”, which is the basic Property of nonstandard number systems. The skills the reader needs for more advanced work are developed through applications of this simple system to calculus and elementary real analysis.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R.M. Anderson, A non-standard representation for Brownian motion and Itô integration. Isr. J. Math. 25, 15–46 (1976)
A.E. Hurd, P.A. Loeb, An Introduction to Nonstandard Real Analysis (Academic Press, Orlando, 1985)
H.J. Keisler, Elementary Calculus, An Infinitesimal Approach, 1st edn. 1976, 2nd edn. 1986 (Prindle, Weber & Smith, Boston, 1976)
P.A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans. Am. Math. Soc. 211, 113–122 (1975)
P.A. Loeb, A lost theorem of calculus. Math. Intell. 24, 15–18 (2002)
A. Robinson, Non-Standard Analysis (North-Holland, Amsterdam, 1966)
A.E. Taylor, Advanced Calculus (Ginn & Company, Boston, 1955)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Loeb, P.A. (2015). Simple Nonstandard Analysis and Applications. In: Loeb, P., Wolff, M. (eds) Nonstandard Analysis for the Working Mathematician. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7327-0_1
Download citation
DOI: https://doi.org/10.1007/978-94-017-7327-0_1
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-017-7326-3
Online ISBN: 978-94-017-7327-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)