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In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite. Formally, it states that for every indexed family of nonempty sets there exists an indexed family of elements such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.
In mathematical analysis, a null set is a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set in set theory anticipates the development of Lebesgue measure since a null set necessarily has measure zero. More generally, on a given measure space a null set is a set such that .
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects.
In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin (1920) and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: Solovay & Tennenbaum (1971) showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent.
In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space.
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.
In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in the series On Rings of Operators. One of von Neumann's goals in this paper was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings.
In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.
In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum. Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra A has an essentially unique completion, which is a complete Boolean algebra containing A such that every element is the supremum of some subset of A. As a partially ordered set, this completion of A is the Dedekind–MacNeille completion.
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean algebra.
In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that:
- Each element of the Boolean algebra can be expressed as a finite combination of generators, using the Boolean operations, and
- The generators are as independent as possible, in the sense that there are no relationships among them that do not hold in every Boolean algebra no matter which elements are chosen.
Mikhail Yakovlevich Suslin was a Russian mathematician who made major contributions to the fields of general topology and descriptive set theory.
In mathematics, a measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the Boolean algebra of measurable sets modulo null sets.
A Schröder–Bernstein property is any mathematical property that matches the following pattern
In mathematics, a Jónsson–Tarski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set X onto the product X×X. They were introduced by Bjarni Jónsson and Alfred Tarski. Smirnov (1971), named them after Georg Cantor because of Cantor's pairing function and Cantor's theorem that an infinite set X has the same number of elements as X×X; the term "Cantor algebra" is also occasionally used to mean the Boolean algebra of all clopen subsets of the Cantor set, or the Boolean algebra of Borel subsets of the reals modulo meager sets.
In set theory, the random algebra or random real algebra is the Boolean algebra of Borel sets of the unit interval modulo the ideal of measure zero sets. It is used in random forcing to add random reals to a model of set theory. The random algebra was studied by John von Neumann in 1935 who showed that it is not isomorphic to the Cantor algebra of Borel sets modulo meager sets. Random forcing was introduced by Solovay (1970).
In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure. They were introduced by Dorothy Maharam (1947).
This is a glossary of set theory.
References
- Balcar, Bohuslav; Jech, Thomas (2006), "Weak distributivity, a problem of von Neumann and the mystery of measurability", Bulletin of Symbolic Logic , 12 (2): 241–266, MR 2223923
- von Neumann, John (1998) [1960], Continuous geometry, Princeton Landmarks in Mathematics, Princeton University Press, ISBN 978-0-691-05893-1, MR 0120174