Overspill

This article is about mathematics. For housing estates, see overspill estate.

In non-standard analysis, a branch of mathematics, overspill (referred to as overflow by Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N of hypernatural numbers.

By applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction:

For any internal subset A of *N, if

1. 1 is an element of A, and
2. for every element n of A, n + 1 also belongs to A,

then

A = *N

If N were an internal set, then instantiating the internal induction principle with N, it would follow N = *N which is known not to be the case.

The overspill principle has a number of useful consequences:

• The set of standard hyperreals is not internal.
• The set of bounded hyperreals is not internal.
• The set of infinitesimal hyperreals is not internal.

In particular:

• If an internal set contains all infinitesimal non-negative hyperreals, it contains a positive non-infinitesimal (or appreciable) hyperreal.
• If an internal set contains N it contains an unbounded element of *N.

Example

These facts can be used to prove the equivalence of the following two conditions for an internal hyperreal-valued function ƒ defined on *R.

$\forall \epsilon >\!\!\!> 0, \exists \delta >\!\!\!> 0, |h| \leq \delta \implies |f(x+h) - f(x)| \leq \varepsilon$

and

$\forall h \cong 0, \ |f(x+h) - f(x)| \cong 0$

The proof that the second fact implies the first uses overspill, since given a non-infinitesimal positive ε,

$\forall \mbox{ positive } \delta \cong 0, \ (|h| \leq \delta \implies |f(x+h) - f(x)| < \varepsilon).\,$

Applying overspill, we obtain a positive appreciable δ with the requisite properties.

These equivalent conditions express the property known in non-standard analysis as S-continuity of ƒ at x. S-continuity is referred to as an external property, since its extension (e.g. the set of pairs (ƒ, x) such that ƒ is S-continuous at x) is not an internal set.

References

• Robert Goldblatt (1998). Lectures on the hyperreals. An introduction to nonstandard analysis. Springer.