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In non-standard analysis, a branch of mathematics, overspill 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 hyperintegers. ## Example

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

- # 1 is an element of A and

- # for every element n of A, n + 1 also belongs to A

- A = *N

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:

- 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.

We can use these facts to prove equivalence of the following two conditions for an internal hyperreal-valued function ƒ defined on *R.

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

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

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

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

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.

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Last updated on Monday September 15, 2008 at 07:10:00 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday September 15, 2008 at 07:10:00 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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