Diagonal Intersection

	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	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]
picture info	Set Theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into 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 was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Ordinal Number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega that is greater than every natural number, along with ordinal numbers \omega + 1, \omega + 2, etc., which are even greater than \omega. A linear order such that every subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Club Filter In mathematics, particularly in set theory, if \kappa is a regular uncountable cardinal then \operatorname(\kappa), the filter of all sets containing a club subset of \kappa, is a \kappa-complete filter closed under diagonal intersection called the club filter. To see that this is a filter, note that \kappa \in \operatorname(\kappa) since it is thus both closed and unbounded (see club set). If x\in\operatorname(\kappa) then any subset of \kappa containing x is also in \operatorname(\kappa), since x, and therefore anything containing it, contains a club set. It is a \kappa-complete filter because the intersection of fewer than \kappa club sets is a club set. To see this, suppose \langle C_i\rangle_ is a sequence of club sets where \alpha < \kappa. Obviously $C = \bigcap C_i$ is closed, since any sequence which appears in $C$ appears in every $C_i,$ and therefore its [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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]
	Fodor's Lemma In mathematics, particularly in set theory, Fodor's lemma states the following: If \kappa is a regular, uncountable cardinal, S is a stationary subset of \kappa, and f:S\rightarrow\kappa is regressive (that is, f(\alpha)<\alpha for any $\alpha\in S$ , $\alpha\neq 0$ ) then there is some $\gamma$ and some stationary $S_0\subseteq S$ such that $f(\alpha)=\gamma$ for any $\alpha\in S_0$ . In modern parlance, the nonstationary ideal is ''normal''. The lemma was first proved by the Hungarian set theorist, Géza Fodor in 1956. It is sometimes also called "The Pressing Down Lemma". Proof We can assume that $0\notin S$ (by removing 0, if necessary). If Fodor's lemma is false, for every $\alpha<\kappa$ there is some [...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]
picture info	Akihiro Kanamori is a Japanese-born American mathematician. He specializes in set theory and is the author of the monograph on large cardinal property, large cardinals, ''The Higher Infinite''. He has written several essays on the history of mathematics, especially set theory. Kanamori graduated from California Institute of Technology and earned a Ph.D. from the University of Cambridge (King's College, Cambridge, King's College). He is a professor of mathematics at Boston University. With Matthew Foreman he is the editor of the ''Handbook of Set Theory'' (2010). Selected publications * A. Kanamori, Menachem Magidor, M. MagidorThe evolution of large cardinal axioms in set theory in: ''Higher set theory'' (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977), Lecture Notes in Mathematics, 669, Springer, 99–275. * Robert Solovay, R. M. Solovay, W. N. Reinhardt, A. KanamoriStrong axioms of infinity and elementary embeddings ''Annals of Mathematical Logic'', 13(1978), 73–116. * A. Kan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	The Higher Infinite ''The Higher Infinite: Large Cardinals in Set Theory from their Beginnings'' is a monograph in set theory by Akihiro Kanamori, concerning the history and theory of large cardinals, infinite sets characterized by such strong properties that their existence cannot be proven in Zermelo–Fraenkel set theory (ZFC). This book was published in 1994 by Springer-Verlag in their series Perspectives in Mathematical Logic, with a second edition in 2003 in their Springer Monographs in Mathematics series, and a paperback reprint of the second edition in 2009 (). Topics Not counting introductory material and appendices, there are six chapters in ''The Higher Infinite'', arranged roughly in chronological order by the history of the development of the subject. The author writes that he chose this ordering "both because it provides the most coherent exposition of the mathematics and because it holds the key to any epistemological concerns". In the first chapter, "Beginnings", the material includes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Ordinal Numbers In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega that is greater than every natural number, along with ordinal numbers \omega + 1, \omega + 2, etc., which are even greater than \omega. A linear order such that every subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

Proof