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 $\alpha < \ ... [...More Info...] [...Related Items...]$ 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 nonempty. Therefore, (S \cap C) must be station ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Normal Function In axiomatic set theory, a function is called normal (or a normal function) 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 (i.e. is neither zero nor a successor), it is the case that . # For all ordinals , it is the case that . Examples A simple normal function is given by (see ordinal arithmetic). But is ''not'' normal because it is not continuous at any limit ordinal (for example, f(\omega) = \omega+1 \ne \omega = \sup \). If is a fixed ordinal, then the functions , (for ), and (for ) are all normal. More important examples of normal functions are given by the aleph numbers f(\alpha) = \aleph_\alpha, which connect ordinal and cardinal numbers, and by the beth numbers f(\alpha) = \beth_\alpha. Properties If is normal, then for any ordinal , :. Proof: If not, choose minimal such that . Since is strictly monotonically increasing, , contr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Azriel Levy Azriel or Ezriel may refer to: People Azriel the father of Seraiah in the Bible, see Jeremiah 36#Verse 26 Azriel of Gerona (c. 1160–c. 1238), Catalan kabbalist * Azriel Graeber (born 1948), Talmudic Scholar and founder of the Jewish Scholarship Society * Azriel Hildesheimer (1820–1899), German rabbi * Azriel Lévy (born 1934), Logician, Hebrew University, Jerusalem * Azriel Rabinowitz (1905–1941), Lithuanian rabbi and Holocaust victim * Azriel Rosenfeld (1931–2004), American professor and expert on computer image analysis * Asriel Günzig (also known as Azriel Günzig, Ezriel Günzig, or other spellings), a rabbi, scholar, bookseller, editor and writer * Ezriel Carlebach (1909–1956), Israeli journalist Fictional characters the title character's name in the Anne Rice novel '' Servant of the Bones'' Azriel, a character in '' A Court of Thorns and Roses'' by Sarah J. Maas Azriel, the supernatural antagonist in the Netflix series ''Warrior Nun'' (TV series) Other use ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Thomas Jech Thomas J. Jech (, ; born 29 January 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 topology a ... [...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 : $\displaystyle\bigcap_ (, \alpha \cup X_\alpha ), [...More Info...] [...Related Items...]$ OR: [Wikipedia] [Google] [Baidu]
picture info	Poset In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is Reflexive relation, reflexive, antisymmetric relation, antisymmetric, and Transitive relation, transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relatio ... [...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\subset B\subset X and A\in \mathcal, then B\in \mathcal A filter on a set may be thought of as representing a "collection of large subsets", one intuitive example being the neighborhood filter. Filters appear in order theory, model theory, and 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 s ... [...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 so the following are equivalent: # \kappa is a regular cardinal. # If \kappa = \textstyle\sum_ \lambda_i and \lambda_i < \kappa for all $i$ , then $, I, \ge \kappa$ . # If $S = \textstyle\bigcup_ S_i$ , and if $, I, < \kappa$ and $, S_i, < \kappa$ for all $i$ , then $, S, < \kappa$ . That is, every union of fewer than $\kappa$ sets smaller than $\kappa$ is smaller than $\kappa$ . # The [...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''. Formally, :\operatorname(A) = \inf \ 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Cardinal Number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter \aleph (aleph) marked with subscript indicating their rank among the infinite cardinals. 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 number of elements. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for two infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Initial Ordinal The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set ''U'', we define its cardinal number to be the smallest ordinal number equinumerous to ''U'', using the von Neumann definition of an ordinal number. More precisely: :, U, = \mathrm(U) = \inf \, where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal. That such an ordinal exists and is unique is guaranteed by the fact that ''U'' is well-orderable and that the class of ordinals is well-ordered, using the axiom of replacement. With the full axiom of choice, every set is well-orderable, so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily found to coincide with the ordering via ≤''c''. This is a well-ordering of cardinal numbers. Initial ordinal of a cardinal Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]