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

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of

Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...

. More specifically, Alternative Set Theory (or AST) may refer to a particular set theory developed in the 1970s and 1980s by

Petr Vopěnka Petr Vopěnka (16 May 1935 – 20 March 2015) was a Czech mathematician. In the early seventies, he developed alternative set theory (i.e. alternative to the classical Cantor theory), which he subsequently developed in a series of articles and m ...

and his students.

Vopěnka's Alternative Set Theory

Vopěnka's Alternative Set Theory builds on some ideas of the theory of

semiset {{distinguish, Semialgebraic set In set theory, a semiset is a proper class that is a subclass of a set. The theory of semisets was proposed and developed by Czech mathematicians Petr Vopěnka and Petr Hájek (1972). It is based on a modificat ...

s, but also introduces more radical changes: for example, all sets are "formally"

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...

, 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 In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

(more precisely: the part of AST that consists of

axioms An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

related to sets only is equivalent to the Zermelo–Fraenkel (or ZF) set theory, in which the

axiom of infinity In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing th ...

is replaced by its negation). However, some of these sets contain subclasses that are not sets, which makes them different from

Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer. In formal Jewish worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds. In Judaism, a cantor sings and lead ...

(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 In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a colle ...

* Morse–Kelley set theory *

Tarski–Grothendieck set theory Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiom ...

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 formal language, language L_ of AST contains one binary relation \in ...

Type theory In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a fou ...

* New Foundations *

Positive set theory In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension holds for at least the positive formulas \phi (the smallest class of formulas containing atomic membership and equal ...

Internal set theory Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, N ...

* Naive set theory *

S (set theory) S is an axiomatic set theory set out by George Boolos in his 1989 article, "Iteration Again". S, a first-order theory, is two-sorted because its ontology includes “stages” as well as sets. Boolos designed S to embody his understanding of the ...

Kripke–Platek set theory The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can be thought of as roughly the predicative part of ZFC and is considerably weaker than it. Axioms In its fo ...

Scott–Potter set theory An approach to the foundations of mathematics that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician D ...

Constructive set theory Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "=" and "\in" of classical set theory is usually used, so this is not to be confused with a con ...

Zermelo set theory Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It be ...

General set theory General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms. ...

Notes

References

*{{cite book , author = Petr Vopěnka , year = 1979 , title = Mathematics in the Alternative Set Theory , publisher =

Teubner The Bibliotheca Teubneriana, or ''Bibliotheca Scriptorum Graecorum et Romanorum Teubneriana'', also known as Teubner editions of Greek and Latin texts, comprise one of the most thorough modern collection published of ancient (and some medieval) ...

, 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. __NOTOC__ Systems of set theory

Vopěnka's Alternative Set Theory

Other alternative set theories

See also

Notes

References