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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] |
Forcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. It was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. Forcing 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 genericity directly without mention of forcing. Intuition Intuitively, forcing consists of expanding the set theoretical universe V to a larger universe V^ . In this bigger universe, for example, one might have many new real numbers, identified with subsets of the set \mathbb of natural numbers, that were not there in the old ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Easton's Theorem
In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. (extending a result of Robert M. Solovay) showed via forcing that the only constraints on permissible values for 2''κ'' when ''κ'' is a regular cardinal are : \kappa < \operatorname(2^\kappa) (where cf(''α'') is the of ''α'') and : Statement If ''G'' is a whose domain consists of ordinals and whose range consists of ordinals such that # ''G'' is non-decreasing, ...[...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 a 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, Harvey Friedman, Sy Friedman, Leo Harrington, Richard Shore, Steve Simpson and Theodore Slaman Theodore Allen Slaman (born April 17, 1954) is a professor of mathematics at the University of California, Berkeley who works in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supercompact Cardinal
In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties. Formal definition If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary embedding ''j'' from the universe ''V'' into a transitive inner model ''M'' with critical point ''κ'', ''j''(''κ'')>''λ'' and :^\lambda M\subseteq M \,. That is, ''M'' contains all of its ''λ''-sequences. Then ''κ'' is supercompact means that it is ''λ''-supercompact for all ordinals ''λ''. Alternatively, an uncountable cardinal ''κ'' is supercompact if for every ''A'' such that , ''A'', ≥ ''κ'' there exists a normal measure over 'A''sup>< ''κ'' with the additional property that every function such that is constant on a set in . Here "constan ... [...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 on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivision of all of its subsets into large and small sets such that itself is large, and all singletons are small, complements of small sets are large and vice versa. The intersection of fewer than 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 Stanislaw 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 on the power set of ''κ''. (Here the term ''κ-additive'' means that, for any sequence ''A''''α'', α<λ of cardinality '' ... [...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 ( notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can be replaced with 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 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 ordinal is brought "from above" to give a name to that critical point. An example of how this works will be detailed below, for an o ... [...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_(mathematics), forcing notion List_of_forcing_notions#Mathias_forcing, Mathias forcing is named for him. Career Mathias was educated at Shrewsbury School, Shrewsbury and Trinity College, Cambridge, where he read mathematics and graduated in 1965. After graduation, he moved to University of Bonn, Bonn in Germany where he studied with Ronald Jensen, visiting UCLA, Stanford University, 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 University of Freiburg, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Menachem Magidor
Menachem Magidor (Hebrew: מנחם מגידור; 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 is currently the 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; 2016-2019). 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 succeed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Azriel Levy
Azriel, Asriel or Ezriel may refer to: People * Azriel of Gerona (c. 1160–c. 1238), Catalan kabbalist * Azriel Hildesheimer (1820–1899), German rabbi * Azriel Rabinowitz (1905–1941), Lithuanian rabbi and Holocaust victim * Azriel Rosenfeld (1931–2004), American professor and expert on computer image analysis * Azriel Graeber (born 1948), Talmudic Scholar and founder of the Jewish Scholarship Society * Azriel Lévy (born 1934), Logician, Hebrew University, Jerusalem * Ezriel Carlebach (1909–1956), Israeli journalist Fictional characters * Lord Asriel, a character in ''His Dark Materials'' by Philip Pullman * the title character's name in the Anne Rice novel ''Servant of the Bones'' * Asriel, a character in the 2015 indie game ''Undertale'' * Azrael, a character in the novel series ''No Game No Life'' * Azriel, a character in ''A Court of Thorns and Roses'' by Sarah J Maas * Azriel the father of Seraiah in the Bible, see Jeremiah 36#Verse 26 Other uses * Azrael, the tra ... [...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. 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, Understanding pres ... [...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 example of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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