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In a general sense, an alternative set theory is any of the alternative mathematical approaches to the concept of set and an alternative to standard set theory.

Some of the alternative set theories are: [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
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.

Non-well-founded set theory
List_of_first-order_theories#Set_theories

References

Holmes, M. Randall. "Alternative Axiomatic Set Theories". Stanford Encyclopedia of Philosophy. Retrieved 17 January 2020.

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.