In
nonstandard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
, a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s N is not an
internal subset of the internal set *N of
hypernatural
In nonstandard analysis, a hyperinteger ''n'' is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is ...
numbers.
By applying the
induction principle for the standard integers N and the
transfer principle
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the firs ...
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'',
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
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
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 unlimited (infinite) element of *N.
Example
These facts can be used to prove the equivalence of the following two conditions for an ''internal'' hyperreal-valued function ƒ defined on *R.
:
and
:
The proof that the second fact implies the first uses overspill, since given a non-infinitesimal positive ''ε'',
:
Applying overspill, we obtain a positive appreciable δ with the requisite properties.
These equivalent conditions express the property known in nonstandard analysis as S-continuity (or
microcontinuity
In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or ''S''-continuity) of an internal function ''f'' at a point ''a'' is defined as follows:
:for all ''x'' infinitely close to ''a'', the value ''f''(''x'') is in ...
) of ƒ at ''x''. S-continuity is referred to as an external property. The first definition is external because it involves quantification over standard values only. The second definition is external because it involves the external relation of being infinitesimal.
References
*
Robert Goldblatt
__NOTOC__
Robert Ian Goldblatt (born 1949) is a mathematical logician who is emeritus Professor in the School of Mathematics and Statistics at Victoria University, Wellington, New Zealand. His doctoral advisor was Max Cresswell. His most popula ...
(1998). ''Lectures on the hyperreals. An introduction to nonstandard analysis.'' Springer.
{{Infinitesimals
Nonstandard analysis