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.
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.
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.
