User:Paul August/Subpage 5
From Wikipedia, the free encyclopedia
Mathematical set theory is many things. It encompasses the everyday notions, introduced in primary school, of collections of objects called sets, and the elements of, and membership in, such collections. It is the language in which mathematical objects are described. It is (along with logic and the predicate calculus) the axiomatic foundations for mathematics, allowing mathematical objects to be constructed formally from the undefined terms of "set", and "set membership". And, it is, in it's own right, a branch of mathematics and an active field of ongoing mathematical research.
Like the concepts of point and line in Euclidian geometry, in mathematics, the terms "set" and "set membership" are fundamental objects used to define other mathematical objects, and so are not themselves formally defined. In Naive set theory, sets are introduced and understood using the intuitive concept of sets as well-defined collections of objects considered as a whole. In modern formal axiomatic set theory, sets and set membership are undefined terms, described by postulating certain axioms, which specify their properties.
"The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics. As a mathematical theory, Set Theory possesses a rich internal structure, and its methods serve as a powerful tool for applications in many other fields of Mathematics.
"The objects of study of Set Theory are sets . As sets are fundamental objects that can be used to define all other concepts in mathematics, they are not defined in terms of more fundamental concepts. Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms.
"The language of set theory is based on a single fundamental relation, called membership " (Stanford Encyclopedia of Philosophy)
