Zermelo-Fraenkel set theory

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 class 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.
[itex]\forall A, \forall B: A=B \iff (\forall C: C \in A \iff C \in B)[itex]
• Axiom of empty set: There is a set with no elements. We will also use {} to denote this empty set.
[itex]\exist \varnothing, \forall A: \lnot (A \in \varnothing)[itex]
• Axiom of pairing: If x, y are sets, then there exists a set containing x and y as its only elements, which we denote by {x,y} or {x} ∪ {y}.
[itex]\forall A, \forall B, \exist C, \forall D: D \in C \iff (D = A \or D = B)[itex]
• 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.
[itex]\forall A, \exist B, \forall C: C \in B \iff (\exist D: C \in D \and D \in A)[itex]
• Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is y ∪ {y}.
[itex]\exist \mathbf{N}: \varnothing \in \mathbf{N} \and (\forall A: A \in \mathbf{N} \implies A \cup \{A\} \in \mathbf{N})[itex]
• 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.
[itex]\forall A, \exists\; {\mathcal{P}A}, \forall B: B \in {\mathcal{P}A} \iff (\forall C: C \in B \implies C \in A)[itex]
• Axiom of regularity: Every non-empty set x contains some element y such that x and y are disjoint sets.
[itex]\forall A: A \neq \varnothing \implies \exists B: B \in A \land \lnot \exist C: C \in A \land C \in B[itex]
• 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.)
[itex]\forall A, \exist B, \forall C: C \in B \iff C \in A \and P(C)[itex]
• 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.)
[itex](\forall X, \exist!\, Y: P(X, Y)) \rightarrow \forall A, \exist B, \forall C: C \in B \iff \exist D: D \in A \and P(D, C)[itex]
• 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 by 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.

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