Alternative to the standard Zermelo–Fraenkel set theory

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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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 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.