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# Hyperfinite Set

One of the most useful tools in nonstandard analysis is the concept of a hyperfinite set. To understand a hyperfinite set, begin with an arbitrary infinite set whose members are not sets, and form the superstructure over . Assume that includes the natural numbers as elements, let denote the set of natural numbers as elements of , and let be an enlargement of . By the transfer principle, the ordering on extends to a strict linear ordering on , which can be denoted with the symbol "." Since is an enlargement of , it satisfies the concurrency principle, so that there is an element of such that if , then . This follows because the relation is a concurrent relation on the set of natural numbers.

Any member that is not also an element of is called an infinite nonstandard natural number, and for any set , if is in one-to-one correspondence with any element of , then is called a hyperfinite set in . Because there are infinite nonstandard natural numbers in any enlargement of , there are hyperfinite sets that are not finite, in any such enlargement. Such hyperfinite sets can be used to study infinite structures satisfying various finiteness conditions.

This entry contributed by Matt Insall (author's link)

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## 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. M. "Nonstandard Analysis with Applications to Economics." Ch. 39 in Handbook of Mathematical Economics, Vol. 4 (Ed. W. Hildenbrand and H. Sonnenschein). New York: Elsevier, pp. 2145-2208, 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, MA: Birkhäuser, 1981.Insall, M. "Nonstandard Methods and Finiteness Conditions in Algebra." Zeitschr. f. Math., Logik, und Grundlagen d. Math. 37, 525-532, 1991.Keisler, H. J. Elementary Calculus: An Infinitesimal Approach. Boston, MA: PWS, 1986. http://www.math.wisc.edu/~keisler/calc.html.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.

Hyperfinite Set

## Cite this as:

Insall, Matt. "Hyperfinite Set." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HyperfiniteSet.html