Overspill
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In mathematics, particularly in non-standard analysis, overspill is a widely used proof technique. It is based on the fact that N is not an internal subset of the nonstandard integers *N. Indeed, by applying the induction principle and transfer principle we get the following general principle
for any internal subset A of *N, ifthen
- 1 is an element of A and
- for every element n of A, n+1 also belongs to A
- A= *N
Instantiating this general principle with N, it would follow N=*N which we know not to be the case.
This principle has a number of extremely 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
We can use these facts to prove equivalence of the following two conditions for an internal hyperreal-valued function f defined on *R.
- <math> \forall \epsilon >\!\!\!> 0, \exists \delta >\!\!\!> 0, |h| \leq \delta \implies |f(x+h) - f(x)| \leq \epsilon<math>
and
- <math> \forall h \cong 0, \ |f(x+h) - f(x)| \cong 0 <math>
The proof that the second fact implies the first uses overspill, since given a non-infinitesimal positive ε
- <math> \forall \mbox{ positive } \delta \cong 0, \ (|h| \leq \delta \implies |f(x+h) - f(x)| < \epsilon) <math>
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 f at x. S-continuity is referred to as an external property, since its extension (e.g. the set of pairs (f, x) such that f is S-continuous at x) is not an internal set.