In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P {\textstyle P} is a certain subset of P , {\displaystyle...
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In the mathematical 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...
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property true almost everywhere is sometimes defined in terms of an ultrafilter. An ultrafilter on a set X is a maximal collection F of subsets of X such that:...
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ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical...
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be extended to an ultrafilter, but the proof uses the axiom of choice. The existence of a nontrivial ultrafilter (the ultrafilter lemma) can be added...
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Filter quantifier (redirect from Ultrafilter quantifier)
filter F {\displaystyle {\mathcal {F}}} on X {\displaystyle X} is an ultrafilter if, for every A ⊆ X , {\displaystyle A\subseteq X,} either A ∈ F {\displaystyle...
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principal ultrafilter on X . {\displaystyle X.} Moreover, every principal ultrafilter on X {\displaystyle X} is necessarily of this form. The ultrafilter lemma...
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element i ∈ I {\displaystyle i\in I} (all of the same signature), and an ultrafilter U {\displaystyle {\mathcal {U}}} on I . {\displaystyle I.} For any two...
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then the ultrafilter U witnessing that κ is measurable will be in Vκ+2 and thus in M. So for any α < κ, we have that there exist an ultrafilter U in j(Vκ)...
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ultrafilter is called the ultrafilter lemma and cannot be proven in Zermelo–Fraenkel set theory (ZF), if ZF is consistent. Within ZF, the ultrafilter...
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to 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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million 2002 - Acquired industrial parts maker Ultrafilter International AG for $68 million. Ultrafilter had sales of $100 million and operated in 30 countries...
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Ultralimit (section Ultrafilters)
configurations in the X n {\displaystyle X_{n}} spaces employing an ultrafilter to bypass the need for repeated consideration of subsequences to ensure...
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the set of all ultrafilters on X , {\displaystyle X,} with the elements of X {\displaystyle X} corresponding to the principal ultrafilters. The topology...
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Generic filter (redirect from Generic ultrafilter)
Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann...
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former is equivalent in ZF to Tarski's 1930 ultrafilter lemma: every filter is a subset of some ultrafilter. One of the most interesting aspects of the...
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Normal measure (redirect from Normal ultrafilter)
where the property holds is a member of the ultrafilter, i.e. has measure 1.) Also equivalent, the ultrafilter (set of sets of measure 1) is closed under...
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} However, an ultrafilter (and any other non-degenerate filter) is free if and only if it includes the Fréchet filter. The ultrafilter lemma states that...
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dual to a filter is an order ideal. Special cases of filters include ultrafilters, which are filters that cannot be enlarged, and describe nonconstructive...
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Finite intersection property (section Ultrafilters)
proving that certain perfect sets are uncountable, and the construction of ultrafilters. Let X {\textstyle X} be a set and A {\textstyle {\mathcal {A}}} a nonempty...
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Filters in topology (section Ultrafilters)
to a point or a set, and many others. Special types of filters called ultrafilters have many useful technical properties and they may often be used in place...
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Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann...
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{\displaystyle X} is open. For every x ∈ X , {\displaystyle x\in X,} the fixed ultrafilter at x {\displaystyle x} converges only to x . {\displaystyle x.} For every...
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size κ with A ∈ M, and a nonprincipal ultrafilter U on the Boolean algebra P(κ) ∩ M such that: U is an M-ultrafilter: for any sequence ⟨Xβ : β < κ⟩ ∈ M of...
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provably equivalent to a weak form of the axiom of choice known as the ultrafilter lemma. In particular, no theory extending ZF can prove either the completeness...
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Y} has a neighborhood on which f {\displaystyle f} is constant. Every ultrafilter U {\displaystyle {\mathcal {U}}} on a non-empty set X {\displaystyle...
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choice, the uniqueness of the cardinality of the basis requires only the ultrafilter lemma, which is strictly weaker (the proof given below, however, assumes...
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direction, a Boolean algebra A {\displaystyle A} has a unique non-principal ultrafilter (that is, a maximal filter not generated by a single element of the algebra)...
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generally require the axiom of choice or a weaker form of it, such as the ultrafilter lemma. If V is a vector space over a field F, then: If L is a linearly...
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{\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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