Alternative set theory has been created and, together with his colleagues at Charles Univ., developed by P. Vopěnka since the 1970s. In agreement with Husserl’s phenomenology, he based his theory on the natural world and the human view thereof.
The most important for any set theory is the way it treats infinity. A different approach to infinity forms the key difference between AST and classical set theories based on the Cantor set theory (CST). Cantor’s approach led to the creation of a rigid, abstract world with an enormous scale of infinite cardinalities while Vopěnka’s infinity, based on the notion of horizon, is more natural and acceptable.
Another source of inspiration were nonstandard models of Peano arithmetics with infinitely large (nonstandard) numbers. The way to build them in AST is easy and natural.
Classes, Sets and Semisets
AST, as well as CST, builds on notions of’ set’, ‘class’, ‘element of a set’ and, in addition, introduces...
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Mlček, J.: ‘Valuation of structures’, Comment. Math. Univ. Carolinae20 (1979), 681–695.
Mlček, J.: ‘Some automorphisms of natural numbers in AST’, Comment. Math. Univ. Carolinae26 (1985), 467–475.
Sochor, A.: ‘Metamathematics of AST’, From the logical point of view1 (1992), 61–75.
Sochor, A., and Pudlák, P.: ‘Models of AST’, J. Symbolic Logic49 (1984), 570–585.
Sochor, A., and Vopěnka, P.: ‘Endomorfic universes and their standard extensions’, Comm. Math. Univ. Carolinae20 (1979), 605–629.
Sochor, A., and Vopěnka, P.: ‘Ultrafilters of sets’, Comment. Math. Univ. Carolinae22 (1981), 698–699.
Trlifajová, K., and Vopěnka, P.: ‘Utility theory in AST’, Comment. Math. Univ. Carolinae26 (1985), 699–711.
Čuda, K.: ‘The consistency of measurability of projective semisets’, Comment. Math. Univ. Carolinae27 (1986), 103–121.
Čuda, K., Sochor, A., Vopěnka, P., and Zlatoš, P.: ‘Guide to AST’, Proc. First Symp. Mathematics in AST, Assoc. Slovak Mathematicians and Physicists, 1989.
Vopénka, P.: Mathematics in AST, Teubner, 1979.
Vopénka, P.: Introduction to mathematics in AST, Alfa Bratislava, 1989. (In Slovak.)
Vopénka, P.: Calculus infinitesimalis-pars prima, Práh Praha, 1996. (In Czech.)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this entry
Cite this entry
Vopěnka, P., Trlifajová, K. (2001). Alternative Set Theory . In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-306-48332-7_9
Download citation
DOI: https://doi.org/10.1007/0-306-48332-7_9
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-7923-6932-5
Online ISBN: 978-0-306-48332-5
eBook Packages: Springer Book Archive