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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] |
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] |
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] |
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] |
Iterated Forcing
In mathematics, iterated forcing is a method for constructing models of set theory by repeating Cohen's forcing method a transfinite number of times. Iterated forcing was introduced by in their construction of a model of set theory with no Suslin tree. They also showed that iterated forcing can construct models where Martin's axiom holds and the continuum is any given regular cardinal. In iterated forcing, one has a transfinite sequence ''P''α of forcing notions indexed by some ordinals α, which give a family of Boolean-valued models ''V''''P''α. If α+1 is a successor ordinal then ''P''α+1 is often constructed from ''P''α using a forcing notion in ''V''''P''α, while if α is a limit ordinal then ''P''α is often constructed as some sort of limit (such as the direct limit) of the ''P''β for β<α. A key consideration is that, typically, it is necessary that is not collapsed. This is often accomplished by the use of a preservation theorem such as: * Fi ... [...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] |
