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Cardinal Characteristic Of The Continuum
In the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal number that may consistently lie strictly between Aleph null, \aleph_0 (the cardinality of the set of natural numbers), and the cardinality of the continuum, that is, the cardinality of the set \mathbb R of all real numbers. The latter cardinal is denoted 2^ or \mathfrak c. A variety of such cardinal characteristics arise naturally, and much work has been done in determining what relations between them are provable, and constructing models of set theory for various consistency, consistent configurations of them. Background Cantor's diagonal argument shows that \mathfrak c is strictly greater than \aleph_0, but it does not specify whether it is the ''least'' cardinal greater than \aleph_0 (that is, \aleph_1). Indeed the assumption that \mathfrak c=\aleph_1 is the well-known Continuum Hypothesis, which was shown to be independent of the standard Zermelo–Fraenkel set t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Non-measurable Set
In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Zermelo–Fraenkel set theory, the axiom of choice entails that non-measurable subsets of \mathbb exist. The notion of a non-measurable set has been a source of great controversy since its introduction. Historically, this led Borel and Kolmogorov to formulate probability theory on sets which are constrained to be measurable. The measurable sets on the line are iterated countable unions and intersections of intervals (called Borel sets) plus-minus null sets. These sets are rich enough to include every conceivable definition of a set that arises in standard mathematics, but they require a lot of formalism to prove that sets are measurable. In 1970, Robert M. Solovay constructed the Solovay model, which shows that it is consistent with standard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cichoń's Diagram
In set theory, Cichoń's diagram or Cichon's diagram is a table of 10 infinite cardinal numbers related to the set theory of the reals displaying the provable relations between these Cardinal characteristic of the continuum, cardinal characteristics of the continuum. All these cardinals are greater than or equal to \aleph_1, the smallest uncountable cardinal, and they are bounded above by 2^, the cardinality of the continuum. Four cardinals describe properties of the ideal (order theory), ideal of sets of Lebesgue measure, measure zero; four more describe the corresponding properties of the ideal of meagre set, meager sets (first category sets). Definitions Let ''I'' be an ideal (set theory), ideal of a fixed infinite set ''X'', containing all finite subsets of ''X''. We define the following "Cardinal function, cardinal coefficients" of ''I'': *\operatorname(I)=\min\. ::The "additivity" of ''I'' is the smallest number of sets from ''I'' whose union is not in ''I'' any more. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saharon Shelah
Saharon Shelah ( he, שהרן שלח; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey. Biography Shelah was born in Jerusalem on July 3, 1945. He is the son of the Israeli poet and political activist Yonatan Ratosh. He received his PhD for his work on stable theories in 1969 from the Hebrew University. Shelah is married to Yael, and has three children. His brother, magistrate judge Hamman Shelah was murdered along with his wife and daughter by an Egyptian soldier in the Ras Burqa massacre in 1985. Shelah planned to be a scientist while at primary school, but initially was attracted to physics and biology, not mathematics. Later he found mathematical beauty in studying geometry: He said, "But when I reached the ninth grade I began studying geometry and my eyes opened to that beauty—a system of demonstration and theorems based on a very small number of axioms which impr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eric Van Douwen
Eric Karel van Douwen (April 25, 1946 in Voorburg, South Holland, Netherlands – July 28, 1987 in Athens, Ohio, United States) was a Dutch mathematician specializing in set-theoretic topology. He received his Ph.D. in 1975 from Vrije Universiteit under the supervision of Maarten Maurice and Johannes Aarts, both of whom were in turn students of Johannes de Groot. He began his academic career studying physics, but became dissatisfied partway through the program. His wife helped inspire his choice to switch to mathematics by asking, "Why not mathematics? It's what you work on all the time anyway". He produced the content of his dissertation unsupervised, and seeking better credentials, he transferred to Vrije to defend, a maneuver permitted by the Dutch university rules. References External links Eric van Douwen's papersIncludes a short bio. From Scott Williams's pages at SUNY Buffalo The State University of New York at Buffalo, commonly called the University at Buffalo (UB ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Disjoint
In mathematics, two sets are almost disjoint Kunen, K. (1980), "Set Theory; an introduction to independence proofs", North Holland, p. 47Jech, R. (2006) "Set Theory (the third millennium edition, revised and expanded)", Springer, p. 118 if their intersection is small in some sense; different definitions of "small" will result in different definitions of "almost disjoint". Definition The most common choice is to take "small" to mean finite. In this case, two sets are almost disjoint if their intersection is finite, i.e. if :\left, A\cap B\ < \infty. (Here, ', ''X'', ' denotes the of ''X'', and '< ∞' means 'finite'.) For example, the closed intervals and |
Richard Laver
Richard Joseph Laver (October 20, 1942 – September 19, 2012) was an American mathematician, working in set theory. Biography Laver received his PhD at the University of California, Berkeley in 1969, under the supervision of Ralph McKenzie, with a thesis on ''Order Types and Well-Quasi-Orderings''. The largest part of his career he spent as Professor and later Emeritus Professor at the University of Colorado at Boulder. Richard Laver died in Boulder, CO, on September 19, 2012 after a long illness. Research contributions Among Laver's notable achievements some are the following. * Using the theory of better-quasi-orders, introduced by Nash-Williams, (an extension of the notion of well-quasi-ordering), he proved Fraïssé's conjecture (now Laver's theorem): if (''A''0,≤),(''A''1,≤),...,(''A''''i'',≤), are countable ordered sets, then for some ''i''<''j'' (''A''i,≤) isomorphically embeds into (''A''''j'',≤). This also holds if the ordered sets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Earl Baumgartner
James Earl Baumgartner (March 23, 1943 – December 28, 2011) was an American mathematician who worked in set theory, mathematical logic and foundations, and topology. Baumgartner was born in Wichita, Kansas, began his undergraduate study at the California Institute of Technology in 1960, then transferred to the University of California, Berkeley, from which he received his PhD in 1970 from for a dissertation entitled ''Results and Independence Proofs in Combinatorial Set Theory''. His advisor was Robert Vaught. He became a professor at Dartmouth College in 1969, and spent his entire career there. One of Baumgartner's results is the consistency of the statement that any two \aleph_1-dense sets of reals are order isomorphic (a set of reals is \aleph_1-dense if it has exactly \aleph_1 points in every open interval). With András Hajnal he proved the Baumgartner–Hajnal theorem, which states that the partition relation \omega_1\to(\alpha)^2_n holds for \alpha<\omega_1 and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Forcing Notions
In mathematics, forcing (mathematics), forcing is a method of constructing new models ''M''[''G''] of set theory by adding a generic subset ''G'' of a poset ''P'' to a model ''M''. The poset ''P'' used will determine what statements hold in the new universe (the 'extension'); to force a statement of interest thus requires construction of a suitable ''P''. This article lists some of the posets ''P'' that have been used in this construction. Notation *''P'' is a poset with order < *''V'' is the universe of all sets *''M'' is a countable transitive model of set theory *''G'' is a generic subset of ''P'' over ''M''. Definitions *''P'' satisfies the countable chain condition if every antichain in ''P'' is at most countable. This implies that ''V'' and ''V''[''G''] have the same cardinals (and the same cofinalities). *A subset ''D'' of ''P'' is called dense if for every there is some with . *A filter on ''P'' is a nonempty subset ''F'' of ''P'' such that if and th ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countable Support Iteration
In 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'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are quite comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 and measure theory. He also worked on non-associative algebraic systems, such as loops, and used computer software, such as the Otter theorem prover, to derive theorems in these areas. Personal life Kunen was born in New York City in 1943 and died in 2020. He lived in Madison, Wisconsin, with his wife Anne, with whom he had two sons, Isaac and Adam. Education Kunen completed his undergraduate degree at the California Institute of Technology and received his Ph.D. in 1968 from Stanford University, where he was supervised by Dana Scott. Career and research Kunen showed that if there exists a nontrivial elementary embedding ''j'' : ''L'' → ''L'' of the constructible universe, then 0# exists. He proved the consistency o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrafilter (set Theory)
In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X (such filters are called ) and that is also "maximal" in that there does not exist any other proper filter on X that contains it as a proper subset. Said differently, a proper filter U is called an ultrafilter if there exists proper filter that contains it as a subset, that proper filter (necessarily) being U itself. More formally, an ultrafilter U on X is a proper filter that is also a maximal filter on X with respect to set inclusion, meaning that there does not exist any proper filter on X that contains U as a proper subset. Ultrafilters on sets are an important special instance of ultrafilters on partially ordered sets, where the partially ordered set consists of the power set \wp(X) and the partial order is subset inclusion \,\subsete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
