Overspill
Encyclopedia
In non-standard analysis
, a branch of mathematics
, 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 numbers N is not an internal subset
of the internal set *N of hypernatural
numbers.
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
then
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:
In particular:
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 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.
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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 numbers N is not an internal subset
Internal set
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.Internal set is the key tool in formulating the transfer principle, which concerns the logical relation between the properties of the real numbers R, and the...
of the internal set *N of hypernatural
Hyperinteger
In non-standard analysis, a hyperinteger N is a hyperreal number equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer...
numbers.
By applying the induction principle for the standard integers N and the transfer principle
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...
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 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
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 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.