Non-standard analysis

2020 Mathematics Subject Classification: Primary: 26E35 Secondary: 03H05 [MSN][ZBL]

A branch of mathematical logic concerned with the application of the theory of non-standard models to investigations in traditional domains of mathematics: mathematical analysis, function theory, the theory of differential equations, probability theory, and others. The basic method of non-standard analysis can roughly be described as follows. One considers a certain mathematical structure $M$ and constructs a first-order logico-mathematical language that reflects those aspects of this structure that are of interest to the investigator. Then one constructs by methods of model theory a non-standard model of the theory of $M$ that is a proper extension of $M$. Under a suitable construction new, non-standard, elements of the model can be interpreted as limiting "ideal" elements of the original structure. For example, if as the original structure one takes the field of real numbers, then it is natural to treat the non-standard elements of the model as "infinitesimals" , that is, as infinitely large or infinitely small, but non-zero, real numbers. Then all the usual relations between real numbers carry over to the non-standard elements, with the preservation of all their properties that can be expressed in the logico-mathematical language. Similarly, in the theory of filters on a given set the intersection of all non-empty elements of the filter determines a non-standard element; in topology this gives rise to a family of non-standard points situated "infinitely close" to a given point. The interpretation of the non-standard elements of a model often makes it possible to give convenient criteria for ordinary concepts in terms of non-standard elements. For example, it can be proved that a standard real-valued function $f$ is continuous at a standard point $x_0$ if and only if $f(x)$ is infinitely close to $f(x_0)$ for all (non-standard) points $x$ infinitely close to $x_0$. The criterion thus obtained can be successfully applied to proofs of ordinary mathematical results.

Naturally, results obtained by methods of non-standard analysis can be, in principle, proved in the standard theory, but the consideration of a non-standard model has the distinct advantage that it allows one to actually introduce into the argument "ideal" elements, making it possible to give lucid statements for many concepts connected with a limit transition from the finite to the infinite. Non-standard analysis places the ideas of G. Leibniz and his followers, about the existence of infinitely small non-zero quantities, on a strict mathematical basis, a circle of ideas (the infinitesimal calculus) which in the subsequent development of mathematical analysis was rejected in favour of the precise concept of the limit of a variable quantity.

A number of new facts have been discovered by means of non-standard analysis. Many classical proofs gain substantially in clarity when presented by means of non-standard analysis. Non-standard analysis has been used successfully in constructing a rigorous theory of certain semi-empirical methods of mechanics and physics.

References

[Da]	M. Davis, "Applied nonstandard analysis", Wiley (1977)
[Ro]	A. Robinson, "Non-standard analysis", North-Holland (1966)

Comments

In recent years numerous developments involving non-standard analysis, especially in stochastic analysis, the theory of dynamical systems and mathematical physics, have taken place. The references below cover such aspects.

References

[AlFeHøLi]	S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrøm, "Nonstandard methods in stochastic analysis and mathematical physics", Acad. Press (1986)
[Cu]	N. Cutland (ed.), Nonstandard analysis and its applications, Cambridge Univ. Press (1988)
[HuLo]	A.E. Hurd, P.A. Loeb, "An introduction to nonstandard real analysis", Acad. Press (1985)
[LuRo]	W. Luxemburg (ed.) A. Robinson (ed.), Contributions to non-standard analysis, North-Holland (1972)
[LuSt]	W. Luxemburg, K. Stroyan, "Introduction to the theory of infinitesimals", Acad. Press (1976)
[Ne]	E. Nelson, "Radically elementary probability theory", Princeton Univ. Press (1987)
[StBa]	K.D. Stroyan, J.M. Bayod, "Foundations of infinitesimal stochastic analysis", North-Holland (1986)