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Encyclopedia > Overspill

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 Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... In mathematical logic, in particular in model theory and non-standard analysis, an internal set is a set that is a member of a model. ... In mathematics particularly in non-standard analysis, the transfer principle is a rule which transforms assertions about standard sets, mappings etc. ...

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
then
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.

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

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)| < epsilon)

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. In metaphysics, extension is the property of taking up space; see Extension (metaphysics). ...


  Results from FactBites:
 
Overspill - Wikipedia, the free encyclopedia (268 words)
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.
By overspill a positive appreciable δ with the requisite properties exists.
Hampshire County Council (1551 words)
The third application is for the formation of an overspill/ replacement student car park area which is intended to be provided on the northern campus near a cul-de-sac called Lower Brook Street.
Regarding the application for the overspill car park to be constructed on the northern campus I note the comments expressed by the objector.
On balance, therefore, I consider the applications for the two-storey extension, construction of an overspill car park area and the relocation of the four temporary classroom units to be acceptable and planning permission should be granted.
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