Set theory

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Set theory is a branch of mathematics created by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rigor in proofs. At the same time the basic concepts of set theory are used throughout mathematics, while the subject is pursued in its own right as a speciality by a comparatively small group of mathematicians and logicians. It should be mentioned that there are also mathematicians using and promoting different approaches to the foundations of mathematics.

The basic concepts of set theory are set and membership. A set is thought of as any collection of objects, called the members (or elements) of the set. In mathematics, the members of sets are any mathematical objects, and in particular can themselves be sets. Thus one speaks of the set of natural numbers (0,1,2,3,4,...), the set of real numbers, the set of functions from the natural numbers to the natural numbers, but also, for example, of the set {0,2,N} which has as members the numbers 0 and 2 and the set N of natural numbers.

For an informal treatment of 'intuitive set theory', click here.

Cantor's basic discovery, which got set theory going as a new field of study, was that if we define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B, then the set N of natural numbers has the same cardinality as the set Q of rational numbers (they are both said to be countably infinite), but the set R of real numbers does not have the same cardinality as N or Q. Cantor gave two proofs that R is not countable, and the second of these, using what is known as the diagonal construction, has been extraordinarily influential and has had manifold applications in logic and mathematics. Cantor went right ahead and constructed infinite hierarchies of infinite sets, the ordinal and cardinal numbers.

An important subfield in set theory is the study of cofinality.

Axioms for set theory

The appearance around the turn of the century of the so-called set-theoretical paradoxes, such as Russell's Paradox, prompted the formulation in 1908 by Ernst Zermelo of an axiomatic theory of sets. The axioms for set theory now most often studied and used are those called the Zermelo-Fraenkel axioms, usually together with the axiom of choice. The Zermelo-Fraenkel axioms are commonly abbreviated to ZF, or ZFC if the axiom of choice is included.

An important feature of ZFC is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, such as numbers, must be subsequently defined in terms of sets.

The axioms of ZFC are listed below. (Strictly speaking, the axioms of ZFC are just strings of logical symbols. What follows should therefore be viewed only as an attempt to express the intended meaning of these axioms in English.)

  • Axiom of extensionality. Two sets are the same if and only if they have the same elements.
  • Axiom of the null set. There is a set with no elements. We will use {} to denote this empty set.
  • Axiom of unordered pairs. If x, y are sets, then so is {x,y}, a set containing x and y as its only elements.
  • Axiom of union. For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x.
  • Axiom of infinity. There exists a set x such that {} is in x and whenever y is in x, so is the union y U {y}.
  • Axiom of replacement. Given any set and any mapping, formally defined as a proposition P(x,y) where P(x,y) and P(x,z) implies y = z, there is a set containing precisely the images of the original set's elements.
  • Axiom of the power set. Every set has a power set. That is, for any set x there exists a set y, such that the elements of y are precisely the subsets of x.
  • Axiom of regularity. Every non-empty set x contains some element y such that x and y are disjoint.
  • Axiom of choice. Any product of nonempty sets is nonempty.

See also the individual articles on the axiom of choice and the axiom of regularity.