Overspill

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.