Transfer Principle - Three Examples

Three Examples

• Every nonempty internal subset of *R that has an upper bound in *R has a least upper bound in *R. Consequently the set of all infinitesimals is external.
  • The well-ordering principle implies every nonempty internal subset of *N has a smallest member. Consequently the set

of all infinite integers is external.

• If n is an infinite integer, then the set {1, ..., n} (which is not standard) must be internal. To prove this, first observe that the following is trivially true:

Consequently

• As with internal sets, so with internal functions: Replace

with

and similarly with in place of .

For example: If n is an infinite integer, then the complement of the image of any internal one-to-one function ƒ from the infinite set {1, ..., n} into {1, ..., n, n + 1, n + 2, n + 3} has exactly three members. Because of the infiniteness of the domain, the complements of the images of one-to-one functions from the former set to the latter come in many sizes, but most of these functions are external.

This last example motivates an important definition: A *-finite (pronounced star-finite) subset of *R is one that can be placed in internal one-to-one correspondence with {1, ..., n} for some n ∈ *N.