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.
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
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
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.
Other alternative set theories
Other alternative set theories include:
*
Von Neumann–Bernays–Gödel set theory
*
Morse–Kelley set theory
*
Tarski–Grothendieck set theory
*
Ackermann set theory
In mathematics and logic, Ackermann set theory (AST) is an axiomatic set theory proposed by Wilhelm Ackermann in 1956.
The language
AST is formulated in first-order logic. The language L_ of AST contains one binary relation \in denoting set ...
*
Type theory
*
New Foundations
*
Positive set theory
*
Internal set theory
*
Naive set theory
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It ...
*
S (set theory)
*
Kripke–Platek set theory
*
Scott–Potter set theory
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Constructive set theory
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Zermelo set theory
*
General set theory
See also
*
Non-well-founded set theory
*
Notes
References
*{{cite book
, author = Petr Vopěnka
, year = 1979
, title = Mathematics in the Alternative Set Theory
, publisher =
Teubner
, location = Leipzig
, url = https://drive.google.com/file/d/17JRj2orUVDw7lrBEmBS1K6OK06RP32Xa/view?usp=sharing
*Proceedings of the 1st Symposium ''Mathematics in the Alternative Set Theory.'' JSMF, Bratislava, 1989.
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Systems of set theory