field of set theory, an ultrafilter on a set X {\displaystyle X} is a maximal filter on the set X . {\displaystyle X.} In other words, it is a collection...
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ultrafilter on a given partially ordered set (or "poset") P {\textstyle P} is a certain subset of P , {\displaystyle P,} namely a maximal filter on P...
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analysis, the notion of a property true almost everywhere is sometimes defined in terms of an ultrafilter. An ultrafilter on a set X is a maximal collection...
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Field (mathematics) (redirect from Additive group of a field)
characteristic. If U is an ultrafilter on a set I, and Fi is a field for every i in I, the ultraproduct of the Fi with respect to U is a field. It is denoted...
86 KB (10,288 words) - 08:55, 28 July 2024
Singleton (mathematics) (redirect from Singleton (set theory))
{\displaystyle X,} which is the set { S ⊆ X : x ∈ S } , {\displaystyle \{S\subseteq X:x\in S\},} is a principal ultrafilter on X . {\displaystyle X.} Moreover...
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In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple...
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a set, set theory — as a branch of mathematics — is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory...
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In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations...
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In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing...
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Shelah cardinal (category Set theory stubs)
{\displaystyle V_{j(f)(\kappa )}\subset N} . A Shelah cardinal has a normal ultrafilter containing the set of weakly hyper-Woodin cardinals below it. Ernest...
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other sets. A set may have a finite number of elements or be an infinite set. There is a unique set with no elements, called the empty set; a set with a single...
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In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence...
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Tychonoff's theorem (category Pages displaying wikidata descriptions as a fallback via Module:Annotated link)
the following criterion: assuming the ultrafilter lemma, a space is compact if and only if each ultrafilter on the space converges. With this in hand...
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equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic...
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century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically...
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In set theory, the complement of a set A, often denoted by A ∁ {\displaystyle A^{\complement }} (or A′), is the set of elements not in A. When all elements...
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set of principal ultrafilters, and the set N* to the set of free ultrafilters. The study of βN, and in particular N*, is a major area of modern set-theoretic...
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ultrafilter lemma/principal/theorem (Tarski) — Every filter on a set X {\displaystyle X} is a subset of some ultrafilter on X . {\displaystyle X.} A consequence...
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Property of Baire (redirect from Baire set (general topology))
is a nonprincipal ultrafilter on the set of natural numbers; each such ultrafilter induces, via binary representations of reals, a set of reals without...
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Cofiniteness (redirect from Cofinite set)
algebra on X . {\displaystyle X.} In the other direction, a Boolean algebra A {\displaystyle A} has a unique non-principal ultrafilter (that is, a maximal...
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topology, all nonempty open sets are dense. In abstract algebra and mathematical logic, if U is an ultrafilter on a set X, "almost all elements of X"...
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set of all cofinite subsets of S, which is called the Fréchet filter, is a free ultrafilter on A. However, the Fréchet filter is not an ultrafilter on...
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Venn diagram (redirect from Set diagram)
A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams...
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mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure...
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A consistent choice of index sets that matter is given by any free ultrafilter U on the natural numbers; these can be characterized as ultrafilters that...
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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...
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mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself...
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Ultraproduct (category Pages displaying wikidata descriptions as a fallback via Module:Annotated link)
every ultrafilter on a finite set is principal, the index set I {\displaystyle I} is consequently also usually infinite. The ultraproduct acts as a filter...
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theory Naive set theory S (set theory) Kripke–Platek set theory Scott–Potter set theory Constructive set theory Zermelo set theory General set theory Non-well-founded...
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Filter (mathematics) (category Pages displaying wikidata descriptions as a fallback via Module:Annotated link)
convention. An ultrafilter is a filter contained in no other proper filter. A subset S of F is a base or basis for F if the upper set generated by S (i...
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