Non-standard analysis

Nonstandard Analysis is a branch of mathematics that deals with non- Archimedean parent bodies. The most important difference to the normal analysis is that in the nonstandardanalysis also occur infinitely large and infinitely small numbers.

Theoretical model access

Besides the usual in the standard analysis real numbers called hyper- real numbers are used. The hyper real numbers form an ordered extension field of real numbers, and can not therefore satisfy the Archimedean axiom. A violation of the Archimedean axiom is held here, for example, by the so-called Infinitesimalzahlen; these are numbers that are closer to zero than any real number different from 0.

However, the hyper- real numbers satisfy the Archimedean axiom, if it is instead formulated with the amount of hyper natural numbers. Formulated to the completeness with pictures (instead of strings ), the hyper- real numbers also satisfy this completeness. Thus, the hyper- real numbers form ( if the icon is reinterpreted ) a (* - ) complete, (* - ) Archimedean ordered field, that is a model of the axioms of the real numbers (which is not isomorphic ).

The first model of a non-standard analysis was developed in the 1960s by Abraham Robinson. He used this to show a set of functional analysis, namely that every polynomially compact operator has an invariant subspace in a Hilbert space. However, the construction of model requires of the use of an ultrafilter. From this it is possible to show the existence with the help of the axiom of choice, one can, however, specify no such ultrafilter concrete.

In the nonstandardanalysis the usual in analysis terms such derivative or integral can be defined without limits. In this regard, the nonstandardanalysis is closer to the ideas of the founders of calculus, Newton and Leibniz. The use of " infinitely small quantities " in the nonstandard analysis, however, is in contrast to Newton and Leibniz, logically correctly and with no known contradictions. There are also applications of nonstandard analysis in the stochastic theory and topology.

Axiomatic approaches

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

Note: The existing literature is almost exclusively in English, also the theories are usually referred to by their abbreviations. Therefore, so far partially no German technical terms have prevailed.

Hrbacek'sche set theory

In the HST ( Hrbáček Set Theory ) by Karel Hrbáček the model-theoretic notion is taken almost exactly. On this point, there are three classes of objects, which the well-founded sets, the internal quantities and the standard quantities. The classes, and adhere to a different axioms, eg the axiom of choice is valid only within these sets, but not for amounts that are not included in any of these classes ( external volume ).

The picture that combines the model-theoretic approach, the original with the expanded universe, here is a Strukturisomorphismus, so a figure that combines objects so that logical statements are preserved. For example, a complete, Archimedean secondary body, that is, a complete (with respect to hyper sequences) Archimedean minor (in terms of hyper natural numbers ) body.

In this context, one can build the math as usual from the set theory, while ensuring our entirely automatically the expanded universe.

Internal Set Theories

These theories restrict the considerations to the expanded universe ( the internal quantities ) by default objects are awarded within the "usual mathematics ". How do these standard objects will behave defined by axioms. Widely used is about the transfer axiom: If a statement in the language of classical mathematics applies to all standard objects, then for all objects.

The correspondence in the model- theoretical approach would be: If a statement in the original universe is true, then in ( strukturisomorphen ) expanded universe.

The most common theory of internal sets is the Internal Set Theory by Edward Nelson. But it is not compatible with the theory of Hrbáček, because there IS a lot that contains all the standard objects, but must in HST (see above ) may be a proper class.

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

Example

The continuity of a real function at a point can be defined in the Standard Analysis as:

In the nonstandardanalysis you can define it like this: If a function and a standard point, then is S- continuous at if and only if

Wherein the generated in the nonstandardanalysis extension field of, and means that the ( non-standard ) numbers x and y have an infinitesimal distance.

However, these two definitions describe different concepts: It can be examples of non-standard functions indicate that ( according to the epsilon -delta definition) are discontinuous, eg i have tiny cracks, but ( according to the infinitesimal definition) are S -continuous, or vice versa, eg when a portion of the function has a large pitch i. Only for standard functions are both continuity concepts are equivalent.

Others

The surreal numbers form a non - Archimedean ordered field extension of the real numbers, which is obtained in a completely different way than the body of the hyper- real numbers.