- Alternative set theory
Generically, an alternative set theory is an alternative mathematical approach to the concept of set. It is a proposed alternative to the standard set theory.
Some of the alternative set theories are:
*the theory of
semiset s;
*rough set theory;
*fuzzy set theory.
*the set theoryNew Foundations
*Positive set theory
*Internal set theory Specifically, Alternative Set Theory (or AST) refers to a particular set theory developed in the 1970s and 1980s by
Petr Vopěnka and his students. It builds on some ideas of the theory ofsemiset s, but also introduces more radical changes: for example, all sets are "formally" finite, which means that sets in AST satisfy the law ofmathematical induction for set-formulas (more precisely: the part of AST that consists ofaxioms related to sets only is equivalent to the Zermelo-Fraenkel (or ZF) set theory, in which theaxiom of infinity is replaced by its negation). However, some of these sets contain subclasses that are not sets, which makes them different from Cantor (ZF) finite sets and they are called infinite in AST.ee also
*
Non-well-founded set theory References
*Vopěnka, P. Mathematics in the Alternative Set Theory. Teubner, Leipzig, 1979.
*Proceedings of the 1st Symposium "Mathematics in the Alternative Set Theory." JSMF, Bratislava, 1989.
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