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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 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] |
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] |
Low Basis Theorem
The low basis theorem is one of several basis theorems in computability theory, each of which showing that, given an infinite subtree of the binary tree 2^, it is possible to find an infinite path through the tree with particular computability properties. The low basis theorem, in particular, shows that there must be a path which is low; that is, the Turing jump of the path is Turing equivalent to the halting problem \emptyset'. Statement and proof The low basis theorem states that every nonempty \Pi^0_1 class in 2^\omega (see arithmetical hierarchy In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define th ...) contains a set of low degree (Soare 1987:109). This is equivalent, by definition, to the statement that each infinite computable subtree of the binary tree 2^ has an infinite path o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sacks Property
In mathematical set theory, the Sacks property holds between two models of Zermelo–Fraenkel set theory if they are not "too dissimilar" in the following sense. For M and N transitive models of set theory, N is said to have the Sacks property over M if and only if for every function g\in M mapping \omega to \omega\setminus\ such that g diverges to infinity, and every function f\in N mapping \omega to \omega there is a tree T\in M such that for every n the n^ level of T has cardinality at most g(n) and f is a branch of T. The Sacks property is used to control the value of certain cardinal invariants in forcing arguments. It is named for Gerald Enoch Sacks. A forcing notion is said to have the Sacks property if and only if the forcing extension has the Sacks property over the ground model. Examples include Sacks forcing and Silver forcing. Shelah Shelah may refer to: * Shelah (son of Judah), a son of Judah according to the Bible * Shelah (name), a Hebrew personal name * Shl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerald Sacks
Gerald Enoch Sacks (1933 – October 4, 2019) was an American logician whose most important contributions were in recursion theory. Named after him is Sacks forcing, a forcing notion based on perfect sets and the Sacks Density Theorem, which asserts that the partial order of the recursively enumerable Turing degrees is dense. Sacks had a joint appointment as a professor at the Massachusetts Institute of Technology and at Harvard University starting in 1972 and became emeritus at M.I.T. in 2006 and at Harvard in 2012. Sacks was born in Brooklyn in 1933. He earned his Ph.D. in 1961 from Cornell University under the direction of J. Barkley Rosser, with his dissertation ''On Suborderings of Degrees of Recursive Insolvability''. Among his notable students are Lenore Blum Lenore Carol Blum (née Epstein, born December 18, 1942) is an American computer scientist and mathematician who has made contributions to the theories of real number computation, cryptography, and pseudorandom n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Cardinal
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure (mathematics), measure on a cardinal ''κ'', or more generally on any set. For a cardinal ''κ'', it can be described as a subdivision of all of its subsets into large and small sets such that ''κ'' itself is large, ∅ and all singleton (mathematics), singletons (with ''α'' ∈ ''κ'') are small, set complement, complements of small sets are large and vice versa. The intersection of fewer than ''κ'' large sets is again large. It turns out that uncountable cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from ZFC. The concept of a measurable cardinal was introduced by Stanisław Ulam in 1930. Definition Formally, a measurable cardinal is an uncountable cardinal number ''κ'' such that there exists a ''κ''-additive, non-trivial, 0-1-valued measure (mathematics), measure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adrian Mathias
Adrian Richard David Mathias (born 12 February 1944) is a British mathematician working in set theory. The forcing notion Mathias forcing is named for him. Career Mathias was educated at Shrewsbury and Trinity College, Cambridge, where he read mathematics and graduated in 1965. After graduation, he moved to Bonn in Germany where he studied with Ronald Jensen, visiting UCLA, Stanford, the University of Wisconsin, and Monash University during that period. In 1969, he returned to Cambridge as a research fellow at Peterhouse and was admitted to the Ph.D. at Cambridge University in 1970. From 1969 to 1990, Mathias was a fellow of Peterhouse; during this period, he was the editor of the Mathematical Proceedings of the Cambridge Philosophical Society from 1972 to 1974, spent one academic year (1978/79) as ''Hochschulassistent'' to Jensen in Freiburg and another year (1989/90) at the MSRI in Berkeley. After leaving Peterhouse in 1990, Mathias had visiting positions in Warsaw, at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Menachem Magidor
Menachem Magidor (; born January 24, 1946) is an Israeli mathematician who specializes in mathematical logic, in particular set theory. He served as president of the Hebrew University of Jerusalem, was president of the Association for Symbolic Logic from 1996 to 1998 and as president of the Division for Logic, Methodology and Philosophy of Science and Technology of the International Union for History and Philosophy of Science (DLMPST/IUHPS) from 2016 to 2019. In 2016 he was elected an honorary foreign member of the American Academy of Arts and Sciences. In 2018 he received the Solomon Bublick Award. Biography Menachem Magidor was born in Petah Tikva, Israel. He received his Ph.D. in 1973 from the Hebrew University of Jerusalem. His thesis, ''On Super Compact Cardinals'', was written under the supervision of Azriel Lévy. He served as president of the Hebrew University of Jerusalem from 1997 to 2009, following Hanoch Gutfreund and succeeded by Menachem Ben-Sasson. The Ox ... [...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] |
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 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] |
Supercompact Cardinal
In set theory, a supercompact cardinal is a type of large cardinal independently introduced by Solovay and Reinhardt. They display a variety of reflection properties. Formal definition If \lambda is any ordinal, \kappa is \lambda-supercompact means that there exists an elementary embedding j from the universe V into a transitive inner model M with critical point \kappa, j(\kappa)>\lambda and :^\lambda M\subseteq M \,. That is, M contains all of its \lambda-sequences. Then \kappa is supercompact means that it is \lambda-supercompact for all ordinals \lambda. Alternatively, an uncountable cardinal \kappa is supercompact if for every A such that \vert A\vert\geq\kappa there exists a normal measure over , in the following sense. is defined as follows: : := \. An ultrafilter U over is ''fine'' if it is \kappa-complete and \ \in U, for every a \in A. A normal measure over is a fine ultrafilter U over with the additional property that every function f: \to A such th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laver Property
In mathematical set theory, the Laver property holds between two models if they are not "too dissimilar", in the following sense. For M and N transitive models of set theory, N is said to have the Laver property over M if and only if for every function g\in M mapping \omega to \omega\setminus\ such that g diverges to infinity, and every function f\in N mapping \omega to \omega and every function h\in M which bounds f, there is a tree T\in M such that each branch of T is bounded by h and for every n the n^\text level of T has cardinality at most g(n) and f is a branch of T. A forcing notion is said to have the Laver property if and only if the forcing extension has the Laver property over the ground model. Examples include Laver forcing. The concept is named after Richard Laver. Saharon Shelah proved that when proper forcings with the Laver property are iterated using countable supports, the resulting forcing notion will have the Laver property as well.C. Schlindwein, Understan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Measure Zero Set
In mathematical analysis, a strong measure zero set is a subset ''A'' of the real line with the following property: :for every sequence (ε''n'') of positive reals there exists a sequence (''In'') of intervals such that , ''I''''n'', < ε''n'' for all ''n'' and ''A'' is contained in the union of the ''I''''n''. (Here , ''I''''n'', denotes the length of the interval ''I''''n''.) Every is a strong measure zero set, and so is every union of countably many strong measure zero sets. Every strong measure zero set has 0. The is an exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |