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Nonstandard analysis is a branch of mathematical Logic which weakens the axioms of usual Analysis to include only the first-order ones. It also introduces Hyperreal Numbers to allow for the existence of ``genuine Infinitesimals,'' numbers which are less than 1/2, 1/3, 1/4, 1/5, ..., but greater than 0. Abraham Robinson developed nonstandard analysis in the 1960s. The theory has since been investigated for its own sake and has been applied in areas such as Banach Spaces, differential equations, probability theory, microeconomic theory, and mathematical physics (Apps).
See also Ax-Kochen Isomorphism Theorem, Logic, Model Theory
References
Albeverio, S.; Fenstad, J.; Hoegh-Krohn, R.; and Lindstrøom, T. Nonstandard Methods in Stochastic Analysis and Mathematical Physics.
New York: Academic Press, 1986.
Anderson, R. ``Nonstandard Analysis with Applications to Economics.'' In
Handbook of Mathematical Economics, Vol. 4. New York: Elsevier, 1991.
Dauben, J. W. Abraham Robinson: The Creation of Nonstandard Analysis, A Personal and Mathematical Odyssey.
Princeton, NJ: Princeton University Press, 1998.
Davis, P. J. and Hersch, R. The Mathematical Experience. Boston: Birkhäuser, 1981.
Keisler, H. J. Elementary Calculus: An Infinitesimal Approach. Boston: PWS, 1986.
Lindstrøom, T. ``An Invitation to Nonstandard Analysis.'' In Nonstandard Analysis and Its Applications (Ed. N. Cutland).
New York: Cambridge University Press, 1988.
Robinson, A. Non-Standard Analysis. Princeton, NJ: Princeton University Press, 1996.
Stewart, I. ``Non-Standard Analysis.'' In From Here to Infinity: A Guide to Today's Mathematics.
Oxford, England: Oxford University Press, pp. 80-81, 1996.