Alternative set theory
Some of the alternative set theories are: 
- Von Neumann–Bernays–Gödel set theory
- Ackermann set theory
- Type theory
- New Foundations
- Positive set theory
- Internal set theory
- Naive set theory
- the theory of semisets (see below)
Vopěnka’s Alternative Set Theory
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. It 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 set-formulas (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.
- Holmes, M. Randall. "Alternative Axiomatic Set Theories". Stanford Encyclopedia of Philosophy. Retrieved 17 January 2020.
- 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.