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 X whose members are not sets, and form the superstructure S(X) over X. Assume that X includes the natural numbers as elements, let N denote the set of natural numbers as elements of X, and let ^*S(X) be an enlargement of S(X). By the transfer principle, the ordering < on N extends to a strict linear ordering on ^*N, which can be denoted with the symbol "<." Since ^*S(X) is an enlargement of S(X), it satisfies the concurrency principle, so that there is an element nu of ^*N such that if n in N, then n<nu. This follows because the relation < is a concurrent relation on the set of natural numbers.

Any member nu in ^*N that is not also an element of N is called an infinite nonstandard natural number, and for any set A in ^*S(X), if A is in one-to-one correspondence with any element of ^*N, then A is called a hyperfinite set in ^*S(X). Because there are infinite nonstandard natural numbers in any enlargement ^*S(X) of S(X), 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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Hyperfinite Set

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Insall, Matt. "Hyperfinite Set." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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