Club Set

	Club Set In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name ''club'' is a contraction of "closed and unbounded". Formal definition Formally, if \kappa is a limit ordinal, then a set C\subseteq\kappa is ''closed'' in \kappa if and only if for every \alpha < \kappa, if $\sup(C \cap \alpha) = \alpha \neq 0,$ then $\alpha \in C.$ Thus, if the limit of some sequence from $C$ is less than $\kappa,$ then the limit is also in $C.$ If $\kappa$ is a limit ordinal and $C \subseteq \kappa$ then $C$ is unbounded in $\kappa$ if for any $\alpha < \kappa,$ there is some $\beta \in C$ such that $\alpha < \be ... [...More Info...] [...Related Items...]$ OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Normal Function In axiomatic set theory, a function ''f'' : Ord → Ord is called normal (or a normal function) if and only if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions: # For every limit ordinal ''γ'' (i.e. ''γ'' is neither zero nor a successor), it is the case that ''f''(''γ'') = sup . # For all ordinals ''α'' < ''β'', it is the case that ''f''(''α'') < ''f''(''β''). Examples A simple normal function is given by (see ordinal arithmetic ). But is ''not'' normal because it is not continuous at any limit ordinal; that is, the inverse image of the one-point open set is the set , which is not open when ''λ'' is a limit ordinal. If ''β'' is a fixed ordinal, then the functions , (for ), and (for ) are all normal. More important exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Azriel Levy Azriel, Asriel or Ezriel may refer to: People * Azriel of Gerona (c. 1160–c. 1238), Catalan kabbalist * Azriel Hildesheimer (1820–1899), German rabbi * Azriel Rabinowitz (1905–1941), Lithuanian rabbi and Holocaust victim * Azriel Rosenfeld (1931–2004), American professor and expert on computer image analysis * Azriel Graeber (born 1948), Talmudic Scholar and founder of the Jewish Scholarship Society * Azriel Lévy (born 1934), Logician, Hebrew University, Jerusalem * Ezriel Carlebach (1909–1956), Israeli journalist Fictional characters * Lord Asriel, a character in ''His Dark Materials'' by Philip Pullman * the title character's name in the Anne Rice novel ''Servant of the Bones'' * Asriel, a character in the 2015 indie game ''Undertale'' * Azrael, a character in the novel series ''No Game No Life'' * Azriel, a character in ''A Court of Thorns and Roses'' by Sarah J Maas * Azriel the father of Seraiah in the Bible, see Jeremiah 36#Verse 26 Other uses * Azrael, the tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Thomas Jech Thomas J. Jech ( cs, Tomáš Jech, ; born January 29, 1944 in Prague) is a mathematician specializing in set theory who was at Penn State for more than 25 years. Life He was educated at Charles University (his advisor was Petr Vopěnka) and from 2000 is at thInstitute of Mathematicsof the Academy of Sciences of the Czech Republic. Work Jech's research also includes mathematical logic, algebra, analysis, topology, and measure theory. Jech gave the first published proof of the consistency of the existence of a Suslin line. With Karel Prikry, he introduced the notion of precipitous ideal. He gave several models where the axiom of choice failed, for example one with ω1 measurable. The concept of a Jech–Kunen tree is named after him and Kenneth Kunen Herbert Kenneth Kunen (August 2, 1943August 14, 2020) was a professor of mathematics at the University of Wisconsin–Madison who worked in set theory and its applications to various areas of mathematics, such as set-theoretic to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Diagonal Intersection Diagonal intersection is a term used in mathematics, especially in set theory. If \displaystyle\delta is an ordinal number and \displaystyle\langle X_\alpha \mid \alpha<\delta\rangle is a sequence of subsets of $\displaystyle\delta$ , then the ''diagonal intersection'', denoted by : $\displaystyle\Delta_ X_\alpha,$ is defined to be : $\displaystyle\.$ That is, an ordinal $\displaystyle\beta$ is in the diagonal intersection $\displaystyle\Delta_ X_\alpha$ if and only if it is contained in the first $\displaystyle\beta$ members of the sequence. This is the same as : $\cup X_\alpha ),$ wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of Comparability, comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Filter (set Theory) In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A,B\subset X,A\in \mathcal, and A\subset B, then B\in \mathcal A filter on a set may be thought of as representing a "collection of large subsets". Filters appear in order, model theory, set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal. Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book ''Topologie Générale'' as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specifically, a filter on a set is just a proper order filter in the special case where the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Regular Cardinal In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that \kappa is a regular cardinal if and only if every unbounded subset C \subseteq \kappa has cardinality \kappa. Infinite well-ordered cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular. In the presence of the axiom of choice, any cardinal number can be well-ordered, and then the following are equivalent for a cardinal \kappa: # \kappa is a regular cardinal. # If \kappa = \sum_ \lambda_i and \lambda_i < \kappa for all $i$ , then $, I, \ge \kappa$ . # If $S = \bigcup_ S_i$ , and if $, I, < \kappa$ and $, S_i, < \kappa$ for all $i$ , then $, S, < \kappa$ . # The category [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Cofinality In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''. This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set ''A'' can alternatively be defined as the least ordinal ''x'' such that there is a function from ''x'' to ''A'' with cofinal image. This second definition makes sense without the axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent. Cofinality can be similarly defined for a directed set and is used to generalize the notion of a subsequence in a net. Examples * The cofinality of a partially ordered set with greatest element is 1 as the set consisting only of the greatest element is cofinal (and must be contained in every other cofinal subse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Stationary Set In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in measure theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset. Classical notion If \kappa is a cardinal of uncountable cofinality, S \subseteq \kappa, and S intersects every club set in \kappa, then S is called a stationary set.Jech (2003) p.91 If a set is not stationary, then it is called a thin set. This notion should not be confused with the notion of a thin set in number theory. If S is a stationary set and C is a club set, then their intersection S \cap C is also stationary. This is because if D is any club set, then C \cap D is a club set, thus (S \cap C) \cap D = S \cap (C \cap D) is non empty. Therefore, (S \cap C) must be stati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Cardinal Number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ''transfinite'' cardinal numbers, often denoted using the Hebrew symbol \aleph ( aleph) followed by a subscript, describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Initial Ordinal In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph that is larger than the rest of the text. The word is derived from the Latin ''initialis'', which means ''standing at the beginning''. An initial is often several lines in height and in older books or manuscripts are known as "inhabited" initials. Certain important initials, such as the Beatus initial or "B" of ''Beatus vir...'' at the opening of Psalm 1 at the start of a vulgate Latin. These specific initials in an illuminated manuscript were also called initiums. In the present, the word "initial" commonly refers to the first letter of any word or name, the latter normally capitalized in English usage and is generally that of a first given name or a middle one or ones. History The classical tradition was slow to use capital letters fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

Examples