In a general sense, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set theory. More specifically, Alternative Set Theory (or AST) may refer to a particular set theory developed in the 1970s and 1980s by Petr Vopěnka and his students.
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Transcription
Vopěnka's Alternative Set Theory
Vopěnka's Alternative Set Theory builds on some ideas of the theory of semisets, but also introduces more radical changes: for example, all sets are "formally" finite, which means that sets in AST satisfy the law of mathematical induction for setformulas (more precisely: the part of AST that consists of axioms related to sets only is equivalent to the Zermelo–Fraenkel (or ZF) set theory, in which the axiom 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.
Other alternative set theories
Other alternative set theories include:^{[1]}
 Von Neumann–Bernays–Gödel set theory
 Morse–Kelley set theory
 Tarski–Grothendieck set theory
 Ackermann set theory
 Type theory
 New Foundations
 Positive set theory
 Internal set theory
 Naive set theory
 S (set theory)
 Kripke–Platek set theory
 Scott–Potter set theory
 Constructive set theory
See also
Notes
 ^ Holmes, M. Randall. "Alternative Axiomatic Set Theories". Stanford Encyclopedia of Philosophy. Retrieved 17 January 2020.
References
 Petr Vopěnka (1979). Mathematics in the Alternative Set Theory. Leipzig: Teubner.
 Proceedings of the 1st Symposium Mathematics in the Alternative Set Theory. JSMF, Bratislava, 1989.