Editing Transfer principle (section)

== Statement ==

The [[ordered field]] <sup>*</sup>'''R''' of [[nonstandard real number]]s properly includes the [[real number|real]] field '''R'''.  Like all ordered fields that properly include '''R''', this field is [[Archimedean_property|non-Archimedean]].   It means that some members ''x''&nbsp;≠&nbsp;0 of <sup>*</sup>'''R''' are [[infinitesimal]], i.e.,

: <math> \underbrace{\left|x\right|+\cdots+\left|x\right|}_{n \text{ terms}} < 1 </math> for every finite [[cardinal number]] n.

The only infinitesimal in ''R'' is 0.  Some other members of <sup>*</sup>'''R''', the reciprocals ''y'' of the nonzero infinitesimals, are infinite, i.e.,

: <math>\underbrace{1+\cdots+1}_{n\text{ terms}}<\left|y\right|</math> for every finite [[cardinal number]] n.

The underlying set of the field <sup>*</sup>'''R''' is the image of '''R''' under a mapping ''A''&nbsp;{{mapsto}}&nbsp;<sup>*</sup>''A'' from subsets ''A'' of '''R''' to subsets of <sup>*</sup>'''R'''.  In every case

: <math> A \subseteq {^*\!A}, </math>

with equality if and only if ''A'' is finite.  Sets of the form <sup>*</sup>''A'' for some <math>\scriptstyle A\,\subseteq\,\mathbb{R}</math> are called '''standard''' subsets of <sup>*</sup>'''R'''.  The standard sets belong to a much larger class of subsets of <sup>*</sup>'''R''' called '''internal''' sets.  Similarly each function

: <math>f:A\rightarrow\mathbb{R}</math>

extends to a function

: <math> {^*\! f} : {^*\!A} \rightarrow {^*\mathbb{R}};</math>

these are called '''standard functions''', and belong to the much larger class of '''internal functions'''.  Sets and functions that are not internal are '''external'''.

The importance of these concepts stems from their role in the following proposition and is illustrated by the examples that follow it.

The '''transfer principle:'''

* Suppose a proposition that is true of <sup>*</sup>'''R''' can be expressed via functions of finitely many variables (e.g. (''x'',&nbsp;''y'')&nbsp;{{mapsto}}&nbsp;''x''&nbsp;+&nbsp;''y''), relations among finitely many variables (e.g. ''x''&nbsp;≤&nbsp;''y''), finitary logical connectives such as '''and''', '''or''', '''not''', '''if...then...''', and the quantifiers

:: <math>\forall x\in\mathbb{R}\text{ and }\exists x\in\mathbb{R}.</math>

: For example, one such proposition is

:: <math> \forall x\in\mathbb{R} \ \exists y\in\mathbb{R} \ x+y=0.</math>

: Such a proposition is true in '''R''' if and only if it is true in <sup>*</sup>'''R''' when the quantifier

:: <math> \forall x \in {^*\!\mathbb{R}}</math>

: replaces

:: <math>\forall x\in\mathbb{R},</math>

: and similarly for <math>\exists</math>.

* Suppose a proposition otherwise expressible as simply as those considered above mentions some particular sets <math>\scriptstyle A\,\subseteq\,\mathbb{R}</math>.  Such a proposition is true in '''R''' if and only if it is true in <sup>*</sup>'''R''' with each such "''A''" replaced by the corresponding <sup>*</sup>''A''.  Here are two examples:
:* The set
::: <math> [0,1]^\ast = \{\,x\in\mathbb{R}:0\leq x\leq 1\,\}^\ast</math>
:: must be
::: <math> \{\,x \in {^*\mathbb{R}} : 0 \le x \le 1 \,\},</math>
:: including not only members of '''R''' between 0 and 1 inclusive, but also members of <sup>*</sup>'''R''' between 0 and 1 that differ from those by infinitesimals.  To see this, observe that the sentence
::: <math> \forall x\in\mathbb{R} \ (x\in [0,1] \text{ if and only if } 0\leq x \leq 1)</math>
:: is true in '''R''', and apply the transfer principle.
:* The set <sup>*</sup>'''N''' must have no upper bound in <sup>*</sup>'''R''' (since the sentence expressing the non-existence of an upper bound of '''N''' in '''R''' is simple enough for the transfer principle to apply to it) and must contain ''n''&nbsp;+&nbsp;1 if it contains ''n'', but must not contain anything between ''n'' and ''n''&nbsp;+&nbsp;1.  Members of
::: <math> {^*\mathbb{N}} \setminus \mathbb{N} </math>
:: are "infinite integers".)
* Suppose a proposition otherwise expressible as simply as those considered above contains the quantifier
:: <math> \forall A\subseteq\mathbb{R}\dots\text{ or }\exists A\subseteq\mathbb{R}\dots\ .</math>
: Such a proposition is true in '''R''' if and only if it is true in <sup>*</sup>'''R''' after the changes specified above and the replacement of the quantifiers with
:: <math> [\forall \text{ internal } A\subseteq{^*\mathbb{R}}\dots] </math>
: and
:: <math> [\exists \text{ internal } A\subseteq{^*\mathbb{R}}\dots]\ .</math>