Editing Transfer principle (section)

==Three examples==
The appropriate setting for the hyperreal transfer principle is the world of ''internal'' entities.  Thus, the well-ordering property of the natural numbers by transfer yields the fact that every internal subset of <math>\mathbb{N}</math> has a least element. In this section internal sets are discussed in more detail.
* Every nonempty ''internal'' subset of <sup>*</sup>'''R''' that has an upper bound in <sup>*</sup>'''R''' has a least upper bound in <sup>*</sup>'''R'''.  Consequently the set of all infinitesimals is external.
** The well-ordering principle implies every nonempty ''internal'' subset of <sup>*</sup>'''N''' has a smallest member.  Consequently the set
::: <math> {^*\mathbb{N}} \setminus \mathbb{N}</math>
:: of all infinite integers is external.
* If ''n'' is an infinite integer, then the set {1,&nbsp;...,&nbsp;''n''} (which is not standard) must be internal.  To prove this, first observe that the following is trivially true:
::: <math> \forall n\in\mathbb{N} \ \exists A\subseteq\mathbb{N} \ \forall x\in\mathbb{N} \ [x\in A \text{ iff } x \leq n].</math>
:: Consequently
::: <math> \forall n \in {^*\mathbb{N}} \ \exists \text{ internal } A \subseteq {^*\mathbb{N}} \ \forall x \in {^*\mathbb{N}} \ [x\in A \text{ iff } x\leq n].</math>
* As with internal sets, so with internal functions: Replace
:: <math> \forall f : A \rightarrow \mathbb{R} \dots </math>
: with
:: <math> \forall\text{ internal } f: {^*\!A}\rightarrow {^*\mathbb{R}} \dots</math>
: when applying the transfer principle, and similarly with <math>\exists</math> in place of <math>\forall</math>.
: For example: If ''n'' is an infinite integer, then the complement of the image of any internal [[one-to-one function]] ''&fnof;'' from the infinite set {1,&nbsp;...,&nbsp;''n''} into {1,&nbsp;...,&nbsp;''n'',&nbsp;''n''&nbsp;+&nbsp;1,&nbsp;''n''&nbsp;+&nbsp;2,&nbsp;''n''&nbsp;+&nbsp;3} has exactly three members by the transfer principle.  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 <sup>*</sup>'''R''' is one that can be placed in ''internal'' one-to-one correspondence with {1,&nbsp;...,&nbsp;''n''} for some ''n''&nbsp;&isin;&nbsp;<sup>*</sup>'''N'''.