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Models of the alternative set theory

Published online by Cambridge University Press:  12 March 2014

P. Pudlák  and
A. Sochor
P. Pudlák
Affiliation:
Mathematical Institute, Czechoslovak Academy of Sciences, Prague, Czechoslovakia
A. Sochor
Affiliation:
Mathematical Institute, Czechoslovak Academy of Sciences, Prague, Czechoslovakia
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Extract

Classical set theory is considered as the framework of contemporary mathematics. But there are aspects of our understanding of the real world about what it is not clear that they are described in Cantor's set theory in the best possible way. For example the formalization of some intuitive concepts such as the description of vague properties (which led to the theory of fuzzy sets) and the interpretation of the notion of infinitely small quantities (modelled by nonstandard models) is not quite simple in the classical set theory. Therefore it is natural to look for an alternative to Cantor's set theory which would enable us to formalize these considerations more naturally and which would be a sufficiently strong framework for mathematics at the same time.

The alternative set theory which was created by P. Vopěnka (cf. [V]) is an attempt to construct a theory that could serve as an alternative to Cantor's set theory. The ideas of the alternative set theory make it possible to constitute a new approach to mathematics. An effort to formalize mathematically our intuitive concepts once more is made, and moreover it seems that one can obtain in the alternative set theory mathematical formalizations of concepts which up to now were not adequately formalized.

In the alternative set theory we can build up all essential parts of mathematics. Development of the classical disciplines is described in [V]. Let us mention a few further results obtained in the alternative set theory: a connection between the discrete and the continuous on the basis of which it is possible to define topology (cf. [V]), investigation of motion (cf. [V]), construction of different types of σ-additive ultrafilters (cf. [S-V4]), classification of classes (measurement of vagueness of properties; cf. [Č-V]), possibility of definition of notions of nonstandard methods (cf. [S-V 1]) and valuations of ideals (cf. [M]). A series of articles has been written developing mathematics in the alternative set theory (for the full list of papers cf. [S]). For some remarks concerning the connection between the alternative set theory and nonstandard methods, the reader is refered to [S 4].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 2 , June 1984 , pp. 570 - 585
Copyright
Copyright © Association for Symbolic Logic 1984

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References

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