Indeed, 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
If N were an internal set, then instantiating the internal induction principle with N, it would follow N = *N which we know not to be the case.
The overspill principle has a number of extremely useful consequences:
We can use these facts to prove equivalence of the following two conditions for an internal hyperreal-valued function ƒ defined on *R.
The proof that the second fact implies the first uses overspill, since given a non-infinitesimal positive ε,
By overspill a positive appreciable δ with the requisite properties exists.
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.