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.

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.