The criterion thus obtained can be successfully applied to proofs of ordinary mathematical results. 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 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. 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. 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. 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. Under a suitable construction new, non-standard, elements of the model can be interpreted as limiting "ideal" elements of the original structure. Then one constructs by methods of model theory a non-standard model of the theory of $M$ that is a proper extension of $M$. 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. The basic method of non-standard analysis can roughly be described as follows. 2010 Mathematics Subject Classification: Primary: 26E35 Secondary: 03H05 Ī 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.
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