Kurepa Tree

	Kurepa Tree In set theory, a Kurepa tree is a tree (''T'', <) of height ω₁, each of whose levels is countable , and has at least ℵ₂ many branches. This concept was introduced by . The existence of a Kurepa tree (known as the Kurepa hypothesis, though Kurepa originally conjectured that this was false) is consistent with the axioms of ZFC: Solovay ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Set Theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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 of set theory, 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 the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Jech–Kunen Tree A Jech–Kunen tree is a set-theoretic tree with properties that are incompatible with the generalized continuum hypothesis. It is named after Thomas Jech and Kenneth Kunen, both of whom studied the possibility and consequences of its existence. Definition In set theory, a tree is a partially ordered set in which the predecessors of any element form a well-ordering. The height of any element is the order type of this well-ordering, and the height of the tree is the least ordinal number that exceeds the height of all elements. A ''branch'' of a tree is a maximal well-ordered subset. A ''ω''1-tree is a tree with cardinality \aleph_1 and height ''ω''1, where ''ω''1 is the first uncountable ordinal and \aleph_1 is the associated cardinal number. A Jech–Kunen tree is a ''ω''1-tree in which the number of branches is greater than \aleph_1 and less than 2^. Existence The generalized continuum hypothesis implies that there is no cardinal number between \aleph_1 and 2^; when this is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Journal Of Symbolic Logic The '' Journal of Symbolic Logic'' is a peer-reviewed mathematics journal published quarterly by Association for Symbolic Logic. It was established in 1936 and covers mathematical logic. The journal is indexed by '' Mathematical Reviews'', Zentralblatt MATH, and Scopus. Its 2009 MCQ was 0.28, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... was 0.631. External links * Mathematical logic journals Academic journals established in 1936 Multilingual journals Quarterly journals Association for Symbolic Logic academic journals Logic journals Cambridge University Press academic journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Suslin Tree In mathematics, a Suslin tree is a tree of height ω1 such that every branch and every antichain is countable. They are named after Mikhail Yakovlevich Suslin. Every Suslin tree is an Aronszajn tree. The existence of a Suslin tree is independent of ZFC, and is equivalent to the existence of a Suslin line (shown by ) or a Suslin algebra. The diamond principle, a consequence of V=L, implies that there is a Suslin tree, and Martin's axiom MA(ℵ1) implies that there are no Suslin trees. More generally, for any infinite cardinal κ, a κ-Suslin tree is a tree of height κ such that every branch and antichain has cardinality less than κ. In particular a Suslin tree is the same as a ω1-Suslin tree. showed that if V=L then there is a κ-Suslin tree for every infinite successor cardinal κ. Whether the Generalized Continuum Hypothesis implies the existence of an ℵ2-Suslin tree, is a longstanding open problem. See also * Glossary of set theory * Kurepa tree * List of statem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Aronszajn Tree In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal ''κ'', a ''κ''-Aronszajn tree is a tree of height ''κ'' in which all levels have size less than ''κ'' and all branches have height less than ''κ'' (so Aronszajn trees are the same as \aleph_1-Aronszajn trees). They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by . A cardinal ''κ'' for which no ''κ''-Aronszajn trees exist is said to have the tree property (sometimes the condition that ''κ'' is regular and uncountable is included). Existence of κ-Aronszajn trees Kőnig's lemma states that \aleph_0-Aronszajn trees do not exist. The existence of Aronszajn trees (=\aleph_1-Aronszajn trees) was proven by Nachman Aronszajn, and implies that the analogue of Kőnig's lemma does not hold for uncountable trees ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ordinal Collapsing Function In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (Ordinal notation, notations for) certain Recursive ordinal, recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even Large cardinal property, large cardinals (though they can be replaced with Large countable ordinal#Beyond admissible ordinals, recursively large ordinals at the cost of extra technical difficulty), and then "collapse" them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an Impredicativity, impredicative manner of naming ordinals. The details of the definition of ordinal collapsing functions vary, and get more complicated as greater ordinals are being defined, but the typical idea is that whenever the notation system "runs out of fuel" and cannot name a certain ordinal, a much larger ordin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Forcing (mathematics) In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence (mathematical logic), independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe (mathematics), universe V to a larger universe V[G] by introducing a new "generic" object G. Forcing was first used by Paul Cohen (mathematician), Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. It has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of mathematical logic such as recursion theory. Descriptive set theory uses the notions of forcing from both recursion theory and set theory. Forcing has also been used in model theory, but it is common in model theory to define generic filter, genericity directly without mention of forcing. Intuition Forcing is ... [...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]
	Ronald Jensen Ronald Björn Jensen (born April 1, 1936) is an American mathematician who lives in Germany, primarily known for his work in mathematical logic and set theory. Career Jensen completed a BA in economics at American University in 1959, and a Ph.D. in mathematics at the University of Bonn in 1964. His supervisor was Gisbert Hasenjaeger. Jensen taught at Rockefeller University, 1969–71, and the University of California, Berkeley, 1971–73. The balance of his academic career was spent in Europe at the University of Bonn, the University of Oslo, the University of Freiburg, the University of Oxford, and the Humboldt-Universität zu Berlin, from which he retired in 2001. He now resides in Berlin. Jensen was honored by the Association for Symbolic Logic as the first Gödel Lecturer in 1990. In 2015, the European Set Theory Society awarded him and John R. Steel the Hausdorff Medal for their paper "K without the measurable". Results Jensen's better-known results include the: * Axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ineffable Cardinal In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by . In the following definitions, \kappa will always be a regular uncountable cardinal number. A cardinal number \kappa is called almost ineffable if for every f: \kappa \to \mathcal(\kappa) (where \mathcal(\kappa) is the powerset of \kappa) with the property that f(\delta) is a subset of \delta for all ordinals \delta < \kappa, there is a subset $S$ of $\kappa$ having cardinality $\kappa$ and homogeneous for $f$ , in the sense that for any $\delta_1 < \delta_2$ in $S$ , $f(\delta_1) = f(\delta_2) \cap \delta_1$ . A [...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 using linearly ordered greek letter variables that include the natural numbers and have the property that every set of ordinals has a least or "smallest" 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 (omega) to be the least element that is greater than every natural number, along with ordinal numbers , , etc., which are even greater than . A linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-orde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	List Of Forcing Notions In 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 sam ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]

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