Search results
Results from the WOW.Com Content Network
In set theory, the complement of a set A, often denoted by (or A′), [1] is the set of elements not in A. [2]When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set U, the absolute complement of A is the set of elements in U that are not in A.
In mathematical logic, 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.
Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes. There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets.
The motivation for the hierarchy is to follow the way in which a Borel set could be constructed from open sets using complementation and countable unions. A Borel set is said to have finite rank if it is in Σ α 0 {\displaystyle \mathbf {\Sigma } _{\alpha }^{0}} for some finite ordinal α {\displaystyle \alpha } ; otherwise it has infinite rank .
1. Naive set theory can mean set theory developed non-rigorously without axioms 2. Naive set theory can mean the inconsistent theory with the axioms of extensionality and comprehension 3. Naive set theory is an introductory book on set theory by Halmos natural The natural sum and natural product of ordinals are the Hessenberg sum and product NCF
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.