Axiom of union

In axiomatic set theory, the axiom of union is one of the axioms of Zermelo–Fraenkel set theory. This axiom was introduced by Ernst Zermelo. The axiom states that for each set x there is a set y whose elements are precisely the elements of the elements of x. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: or in words: Given any set A, there is a set B such that, for any element c, c is a member of B if and only if there is a set D such that c is a member of D and D is a member of A. or, more simply: For any set , there is a set which consists of just the elements of the elements of that set . The axiom of union allows one to unpack a set of sets and thus create a flatter set. Together with the axiom of pairing, this implies that for any two sets, there is a set (called their union) that contains exactly the elements of the two sets. The axiom of replacement allows one to form many unions, such as the union of two sets. However, in its full generality, the axiom of union is independent from the rest of the ZFC-axioms: Replacement does not prove the existence of the union of a set of sets if the result contains an unbounded number of cardinalities. Together with the axiom schema of replacement, the axiom of union implies that one can form the union of a family of sets indexed by a set. In the context of set theories which include the axiom of separation, the axiom of union is sometimes stated in a weaker form which only produces a superset of the union of a set. For example, Kunen states the axiom as which is equivalent to Compared to the axiom stated at the top of this section, this variation asserts only one direction of the implication, rather than both directions. There is no corresponding axiom of intersection. If is a nonempty set containing , it is possible to form the intersection using the axiom schema of specification as so no separate axiom of intersection is necessary. (If A is the empty set, then trying to form the intersection of A as {c: for all D in A, c is in D} is not permitted by the axioms.

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Axiom of infinity

In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: In words, there is a set I (the set that is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I.

Zermelo set theory

Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering.

Ernst Zermelo

Ernst Friedrich Ferdinand Zermelo (zɜrˈmɛloʊ, tsɛɐ̯ˈmeːlo; 27 July 1871 21 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem. Furthermore, his 1929 work on ranking chess players is the first description of a model for pairwise comparison that continues to have a profound impact on various applied fields utilizing this method.

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