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The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. When the axiom of choice is included, the resulting system is ZFC.

The axioms are the result of work by Thoralf Skolem in 1922, based on earlier work by Abraham Fraenkel in the same year, which was based on the axiom system put forth by Ernst Zermelo in 1908 (Zermelo set theory).

The axiom system is written in first-order logic; it has an infinite number of axioms because an axiom schema is used. An alternative, finite system is given by the von Neumann-Bernays-Gödel axioms (NBG), which add the concept of a classes in addition to that of a set; it is "equivalent" in the sense that any theorem about sets which can be proved in one system can be proven in the other.

The axioms of ZFC are:

  • 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: 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 ∪ {y}.
  • Axiom of replacement: Given any set and any mapping, formally defined as a proposition P(x,y) where P(x,y1) and P(x,y2) implies y1 = y2, there is a set containing precisely the images of the original set's elements. (This is an axiom schema.)
  • 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: Every non-empty set x contains some element y such that x and y are disjoint sets.
  • 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. (This is an axiom schema.)
  • Axiom of choice: Given any set of mutually exclusive non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets.

While most metamathematicians believe that these axioms are consistent (in the sense that no contradiction can be derived from them), this has not been proved. In fact, since they are the basis of ordinary mathematics, their consistency (if true) cannot be proved in ordinary mathematics; this is a consequence of Gödel's second incompleteness theorem. On the other hand, the consistency of ZFC can be proved by assuming the existence of an inaccessible cardinal.

See also

  Results from FactBites:
Zermelo–Fraenkel set theory - Wikipedia, the free encyclopedia (1372 words)
ZFC consists of a single primitive ontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe of discourse (i.e., all mathematical objects) are sets.
ZFC is a first-order theory; hence ZFC includes axioms whose background logic is first-order logic.
Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics.
TA Holding::ZFC Ltd (205 words)
ZFC Ltd is the largest local manufacturer and distributor of fertilizer and the third largest distributor of agrochemicals in Zimbabwe.
ZFC has distribution agreements with several of the world's leading principal manufacturers of agricultural chemicals such as Bayer, FMC, Monsanto and Syngenta.
It is ZFC's objective to continually improve the distribution of its products to promote fertilizer use in order to improve agricultural yields and the general produce output in the country.
  More results at FactBites »



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