Alternative set theory

Alternative set theory

Generically, an alternative set theory is an alternative mathematical approach to the concept of set. It is a proposed alternative to the standard set theory.

Some of the alternative set theories are:

*the theory of semisets;
*rough set theory;
*fuzzy set theory.
*the set theory New Foundations
*Positive set theory
*Internal set theory

Specifically, Alternative Set Theory (or AST) refers 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.

ee also

* Non-well-founded set theory

References

*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.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • set theory — the branch of mathematics that deals with relations between sets. [1940 45] * * * Branch of mathematics that deals with the properties of sets. It is most valuable as applied to other areas of mathematics, which borrow from and adapt its… …   Universalium

  • Set theory — This article is about the branch of mathematics. For musical set theory, see Set theory (music). A Venn diagram illustrating the intersection of two sets. Set theory is the branch of mathematics that studies sets, which are collections of objects …   Wikipedia

  • List of set theory topics — Logic portal Set theory portal …   Wikipedia

  • 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* {x mid phi} exists holds for at least the positive formulas phi (the smallest class of formulas containing atomic… …   Wikipedia

  • Implementation of mathematics in set theory — This article examines the implementation of mathematical concepts in set theory. The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC (the dominant set theory) and in NFU, the version of Quine s New… …   Wikipedia

  • Zermelo–Fraenkel set theory — Zermelo–Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.ZFC consists of a single primitive ontological notion, that of… …   Wikipedia

  • Causal set theory bibliography — Main article: Causal Sets This Causal Set Theory Bibliography is intended to aid causal set research. It gathers together academic papers, books, talks and PhD theses related to causal set theory and is intended to help readers find references… …   Wikipedia

  • Set (mathematics) — This article gives an introduction to what mathematicians call intuitive or naive set theory; for a more detailed account see Naive set theory. For a rigorous modern axiomatic treatment of sets, see Set theory. The intersection of two sets is… …   Wikipedia

  • Set-theoretic definition of natural numbers — Several ways have been proposed to define the natural numbers using set theory.The contemporary standardIn standard (ZF) set theory the natural numbersare defined recursively by 0 = {} (the empty set) and n +1 = n ∪ { n }. Then n = {0,1,..., n… …   Wikipedia

  • Alternative theories of the bombing of Pan Am Flight 103 — suggest that Abdelbaset Ali Mohmed Al Megrahi, a Libyan agent who was convicted on 270 counts of murder for the bombing of Pan Am Flight 103, may actually be innocent. At the end of the Pan Am Flight 103 bombing trial an international observer… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”