Non-standard analysis

Nonstandardanalysis is a field of mathematics that deals with non-Archimedean ordered bodies . The most important difference to normal analysis is that in the nonstandard analysis there are also infinitely large and infinitely small numbers, hyperreal numbers .

Model-theoretical approach

In addition to the real numbers customary in standard analysis, so-called hyper - real numbers are used. The hyper-real numbers form an ordered expansion field of the real numbers and thus cannot fulfill the Archimedean axiom . A violation of the Archimedean axiom takes place here, for example, through the so-called infinitesimal numbers; these are numbers that are closer to zero than any real number other than 0.

The first model of non-standard analysis was developed by Abraham Robinson in the 1960s . He used this to show a theorem from functional analysis , namely that every polynomial compact operator in a Hilbert space has an invariant subspace . However, the construction of the model requires the use of a free ultrafilter over . One can prove the existence of this with the help of the axiom of choice , but one cannot specifically specify such an ultrafilter. ${\displaystyle \mathbb {N} }$

In non-standard analysis, the terms common in analysis such as derivative or integral can be defined without limit values . In this regard, nonstandard analysis is closer to the ideas of the founders of calculus , Newton and Leibniz . However, in contrast to Newton and Leibniz, the use of “infinitely small quantities” in non-standard analysis is logically sound and without known contradictions . There are also applications of non-standard analysis in stochastics and topology .

Axiomatic approaches

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In addition to the model-theoretical approach, there are also various axiomatic approaches that differ greatly from one another.

Note: The available literature is almost exclusively in English , and the theories are usually referred to by their abbreviations. Therefore, in some cases, no German technical terms have caught on.

Hrbacek's set theory

In Karel Hrbáček's HST (Hrbacek Set Theory) , the theoretical model is adopted almost exactly. To this end, we introduce three classes of objects , those of well-founded sets , those of internal sets and those of standard sets. The classes , and follow different axioms, e.g. B. the axiom of choice only applies within these sets, but not for sets that are not included in any of these classes (external sets). ${\displaystyle WF}$ ${\displaystyle I}$ ${\displaystyle S}$

The mapping that connects the original with the extended universe in the model-theoretical approach is here a structure isomorphism , i.e. a mapping that connects objects in such a way that logical statements are preserved. For example, is a complete, Archimedean ordered body, so is also a complete (with regard to hypersequences ), Archimedean ordered (with regard to hypernatural numbers ) body. ${\displaystyle \ast }$ ${\displaystyle WF\rightarrow S}$ ${\displaystyle \mathbb {R} \in WF}$ ${\displaystyle {}^{\ast }\mathbb {R} \in S}$ ${\displaystyle {}^{\ast }\mathbb {N} \rightarrow {}^{\ast }\mathbb {R} }$ ${\displaystyle {}^{\ast }\mathbb {N} }$

With this in mind, you can build up mathematics from set theory as usual , but automatically get the expanded universe.

Internal set theories

These theories limit the considerations to the extended universe (of the internal sets) by marking standard objects within "normal mathematics". How these standard objects behave is determined by axioms. The transfer axiom, for example, is widespread: If a statement in the language of classical mathematics applies to all standard objects, then also to all objects.

The equivalent in the theoretical model approach would be: If a statement applies in the original universe, then also in the (structurally isomorphic) extended universe.

The most common theory of internal sets is the Internal set theory by Edward Nelson . However, it is not compatible with Hrbáček's theory, because in IST there is a set that contains all standard objects, but in HST (see above) there must be a real class. ${\displaystyle S}$

Therefore, weaker theories are also considered (Bounded Set Theory, Basic Internal Set Theory and - little noticed in the professional world - the revised version of Nelson's IST), which are also summarized under the collective term "theories of internal sets" ("internal set theories") .

Example: definition of continuity

The continuity of a real function at a point can be defined in standard analysis as follows: ${\displaystyle f}$ ${\displaystyle x_{0}}$

${\displaystyle \forall \varepsilon >0\colon \exists \delta >0\colon \forall x\in \mathbb {R} \colon |x-x_{0}|<\delta \Rightarrow |f(x)-f(x_{0})|<\varepsilon }$

In nonstandard analysis it can be defined as follows: If is a function and a standard point, then in is S-continuous if and only if ${\displaystyle f}$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$

${\displaystyle \forall x\in {}^{*}\mathbb {R} \colon x\approx x_{0}\Rightarrow f(x)\approx f(x_{0})}$,

where the extension field generated in the nonstandard analysis is of and means that the (nonstandard) numbers and have an infinitesimal distance. ${\displaystyle {}^{*}\mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle x\approx y}$ ${\displaystyle x}$ ${\displaystyle y}$

However, these two definitions describe different concepts: Examples of non-standard functions can be given which (according to the epsilon-delta definition) are discontinuous, e.g. B. have i-small jumps, but (according to the infinitesimal definition) are S-continuous, or vice versa, e.g. B. if a portion of the function has an i-large slope. Both concepts of continuity are equivalent only for standard functions.

literature

• Detlef Laugwitz, Curt Schmieden: An extension of the infinitesimal calculus. Mathematische Zeitschrift 69 (1958), pp. 1-39.
• Abraham Robinson : Nonstandard Analysis. Studies in Logic and the Foundations of Mathematics . North Holland, Amsterdam 1966.
• Wilhelmus Anthonius Josephus Luxemburg : A general Theory of Monads, in WAJ Luxemburg (Ed.): Applications of Model Theory of Algebra, Analysis and Probability . Holt, Rinehart & Winston, New York 1969, pp. 18-86.
• Script on set theoretical topology, with a chapter on non-standard topology (PDF, German ) (1.72 MB)
• Dieter Landers, Lothar Rogge: Nonstandard Analysis . Springer, Berlin 1994, ISBN 3-540-57115-9 .
• Vladimir Kanovei, Michael Reeken: Nonstandard Analysis, Axiomatically . Springer, Berlin 2004, ISBN 3-540-22243-X .

Individual evidence

1. ↑ ^{a b} What is nonstandard analysis? What are hyperreal numbers? , Edmund Weitz , HAW Hamburg, 2017-10-27. In particular the section on their axiomatic introduction .
2. ↑ ^{a b} Elementary Calculus, An Infinitesimal Approach , H. Jerome Keisler , University of Wisconsin, 1976, revised 2018.