Axiomatic set theory

Set theory is a branch of mathematics created principally 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.

Axioms for set theory

The axioms for set theory now most often studied and used, although put in their final form by Skolem, are called the Zermelo-Fraenkel set theory (ZF). Actually, this term usually excludes the axiom of choice, which was once more controversial than it is today. When this axiom is included, the resulting system is called ZFC.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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 ten 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. Moreover, the axiom of separation, along with the axiom of replacement, is actually a schema of axioms, one for each proposition). Each axiom has further information in its own article.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Axiom of extensionality: Two sets are the same if and only if they have the same elements.
• Axiom of empty set: There is a set with no elements. We will use {} to denote this empty set.
• Axiom of pairing: If x, y are sets, then so is {x,y}, a set containing x and y as its only elements.
• Axiom of union: Every set has a union. That is, for any set x there is a set y whose elements 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 separation (or subset axiom): Given any set and any proposition P(x), there is a subset of the original set containing precisely those elements x for which P(x) holds.
• 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 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 (or axiom of foundation): Every non-empty set x contains some element y such that x and y are disjoint sets.
• Axiom of choice: (Zermelo's version) Given a set x of mutually disjoint nonempty sets, there is a set y (a choice set for x) containing exactly one element from each member of x.

The axioms of choice and regularity are still controversial today among a minority of mathematicians. Other axiom systems for set theory are Von Neumann-Bernays-Gödel set theory (NBG), the Kripke-Platek set theory (KP), Kripke-Platek set theory with urelements (KPU) and Morse-Kelley set theory.

Related Topics:
Von Neumann-Bernays-Gödel set theory - Kripke-Platek set theory - Kripke-Platek set theory with urelements - Morse-Kelley set theory

~ ~ ~ ~ ~ ~ ~ ~ ~ ~