User:Michael Hardy/transfer principle

The ordered field ^*R of nonstandard real numbers properly includes the real field R. Like all ordered fields that properly include R, this field is non-Archimedean. It means that some members x ≠ 0 of ^*R are infinitesimal, i.e.,

$\underbrace {\left|x\right|+\cdots +\left|x\right|} _{n{\text{ terms}}}<1{\text{ for every finite cardinal number }}n.\,$

The only infinitesimal in R is 0. Some other members of ^*R, the reciprocals y of the nonzero infinitesimals, are infinite, i.e.,

$\underbrace {1+\cdots +1} _{n{\text{ terms}}}<\left|y\right|{\text{ for every finite cardinal number }}n.\,$

The underlying set of the field ^*R is the image of R under a mapping A $\mapsto$ ^*A from subsets A of R to subsets of ^*R. In every case

$A\subseteq {^{*}\!A},\,$

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

$f:A\rightarrow \mathbb {R} \,$

extends to a function

${^{*}\!f}:{^{*}\!A}\rightarrow {^{*}\mathbb {R} };\,$

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 ^*R can be expressed via functions of finitely many variables (e.g. (x, y) $\mapsto$ x + y), relations among finitely many variables (e.g. x ≤ y), finitary logical connectives such as and, or, not, if...then..., and the quantifiers

$\forall x\in \mathbb {R} {\text{ and }}\exists x\in \mathbb {R} .\,$

For example, one such proposition is

$\forall x\in \mathbb {R} \ \exists y\in \mathbb {R} \ x+y=0.\,$

Such a proposition is true in R if and only if it is true in ^*R when the quantifier

$\forall x\in {^{*}\!\mathbb {R} }\,$

replaces

$\forall x\in \mathbb {R} ,\,$

and similarly for $\exists$.

• Suppose a proposition otherwise expressible as simply as those considered above mentions some particular sets $\scriptstyle A\,\subseteq \,\mathbb {R}$. Such a proposition is true in R if and only if it is true in ^*R with each such "A" replaced by the corresponding ^*A. Here are two examples:
  • The set

$[0,1]^{\ast }=\{\,x\in \mathbb {R} :0\leq x\leq 1\,\}^{\ast }$

must be

$\{\,x\in {^{*}\mathbb {R} }:0\leq x\leq 1\,\},$

including not only members of R between 0 and 1 inclusive, but also members of ^*R between 0 and 1 that differ from those by infinitesimals. To see this, observe that the sentence

$\forall x\in \mathbb {R} \ (x\in [0,1]{\text{ if and only if }}0\leq x\leq 1)$

is true in R, and apply the transfer principle.

• • The set ^*N must have no upper bound in ^*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 + 1 if it contains n, but must not contain anything between n and n + 1. Members of

${^{*}\mathbb {N} }\setminus \mathbb {N} \,$

are "infinite integers".)

• Suppose a proposition otherwise expressible as simply as those considered above contains the quantifier

$\forall A\subseteq \mathbb {R} \dots {\text{ or }}\exists A\subseteq \mathbb {R} \dots \ .$

Such a proposition is true in R if and only if it is true in ^*R after the changes specified above and the replacement of the quantifiers with

$[\forall {\text{ internal }}A\subseteq {^{*}\mathbb {R} }\dots ]\,$

and

$[\exists {\text{ internal }}A\subseteq {^{*}\mathbb {R} }\dots ]\ .$

Here are 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

${^{*}\mathbb {N} }\setminus \mathbb {N} \,$

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:

$\forall n\in \mathbb {N} \ \exists A\subseteq \mathbb {N} \ \forall x\in \mathbb {N} \ [x\in A{\text{ iff }}x\leq n].$

Consequently

$\forall n\in {^{*}\mathbb {N} }\ \exists {\text{ internal }}A\subseteq {^{*}\mathbb {N} }\ \forall x\in {^{*}\mathbb {N} }\ [x\in A{\text{ iff }}x\leq n].$

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

$\forall f:A\rightarrow \mathbb {R} \dots \,$

with

$\forall {\text{ internal }}f:{^{*}\!A}\rightarrow {^{*}\mathbb {R} }\dots ]$

and similarly with $\exists$ in place of $\forall$.

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.