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.