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Set Theory

  Sets are among the most fundamental and general objects in the whole of mathematics. At the most basic level (in what is known as naive set theory), sets are collections of objects (numbers, points, other sets, apples), though they are subject to certain restrictions (see paradoxes). The objects contained in a set are known as members or elements.

The originator of the subject, Georg Cantor (1845 - 1918), did not really have a clear, rigorous idea of what was meant by the term. Sets are general enough to be used as the basic mathematical object; in modern views of mathematics, everything is a set (or possibly a class, which is a group of sets too large to be a set itself). A number is a set; a group is a set; and anything that mathematics can effectively talk about should be a set or a collection of sets.

Today, the concept of a set has been given a more rigorous definition, mainly by the axiomatization of set theory by , Ernest Zermelo (1871 - 1953) and , Adolf Abraham Fraenkel (1891 - 1965). There are twelve axioms, which assert the existence of various sets (the empty set, containing no members; an infinite set), which allow the construction of new sets from old ones (the union of a set, which is the set of all things in members of the set; a subset for each sentence of symbolic logic, consisting of all the things in the set for which the sentence is true; the set which is a pair of given sets and so on) and which define properties of sets (the axiom of foundation, banning infinite chains of the form A1 containing A2 as an member, which contains A3, which contains A4 and so on; and the axiom of choice). Sets, in this view, are any classes which can be proved to be sets using the axioms. Some of the axioms can be dispensed with; alternative set theories have been developed in which the axiom of choice is false.

Set theory in its modern context, with the inclusion of infinite sets in its domain of discourse is set against the views of the infinite held by the intuitionists, who claimed that Cantor\'s results about infinite sets were against common sense. Intuitionistic versions of set theory have been developed, but they have not been very successful, because they are necessarily limited in scope by their avoidance of direct use of the infinite. Today\'s set theory is very much the product of logicism, and, to a lesser extent, of formalism. However, it is now independent of such philosophical backgrounds, and takes its place in today\'s mathematics, which is so confused about the philosophical meaning of what is actually being done in the subject. SMcL



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