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...

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...

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...

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...

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...

{\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...

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...

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...

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...

{\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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

is a nonprincipal ultrafilter on the set of natural numbers; each such ultrafilter induces, via binary representations of reals, a set of reals without...

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...

mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable...

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...

(as corollaries of Lindemann–Weierstrass) Tychonoff's theorem (to do) Ultrafilter lemma Ultraparallel theorem Urysohn's lemma Van der Waerden's theorem...

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...

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"...

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...

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...

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...

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...