|
Indestructibility
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 a ... [...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] |
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] |
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] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Halpern–Läuchli Theorem
In mathematics, the Halpern–Läuchli theorem is a partition result about finite products of infinite trees. Its original purpose was to give a model for set theory in which the Boolean prime ideal theorem is true but the axiom of choice is false. It is often called the Halpern–Läuchli theorem, but the proper attribution for the theorem as it is formulated below is to Halpern–Läuchli–Laver–Pincus or HLLP (named after James D. Halpern, Hans Läuchli, Richard Laver, and David Pincus), following . Let ''d'',''r'' < ω, \langle T_i: i \in d \rangle be a sequence of finitely splitting trees of height ω. Let :\bigcup_ \left(\prod_T_i(n)\right) = C_1 \cup \cdots \cup C_r, then there exists a sequence of subtrees \langle S_i: i \in d \rangle strongly embedded in \langle T_i: i \in d \rangle such that :\bigcup_ \left(\prod_S_i(n)\right) \subset C_k\textk \le r. Alternatively, let : S^d_ = \bigcup_ \left(\prod_T_i(n)\right) and : \mathbb^d=\bigcup_ S^d_.. The HLLP the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57–5 ... [...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 at most 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. See also * ''Bulletin of the American Mathematical Society'' * '' Journal of the American Mathematical Society'' * ''Memoirs of the American Mathematical Society'' * ''Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... American Mathematical Society academic journals Mathematics journals Publications established in 1900 {{math-journal-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Forcing Axiom
In the mathematical field of set theory, the proper forcing axiom (''PFA'') is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings. Statement A forcing or partially ordered set P is proper if for all regular uncountable cardinals \lambda , forcing with P preserves stationary subsets of lambda\omega . The proper forcing axiom asserts that if P is proper and Dα is a dense subset of P for each α<ω1, then there is a filter G P such that Dα ∩ G is nonempty for all α<ω1. The class of proper forcings, to which PFA can be applied, is rather large. For example, standard arguments show that if P is or [...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] |
Chain Condition
In order theory, a partially ordered set ''X'' is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in ''X'' is countable. Overview There are really two conditions: the ''upwards'' and ''downwards'' countable chain conditions. These are not equivalent. The countable chain condition means the downwards countable chain condition, in other words no two elements have a common lower bound. This is called the "countable chain condition" rather than the more logical term "countable antichain condition" for historical reasons related to certain chains of open sets in topological spaces and chains in complete Boolean algebras, where chain conditions sometimes happen to be equivalent to antichain conditions. For example, if κ is a cardinal, then in a complete Boolean algebra every antichain has size less than κ if and only if there is no descending κ-sequence of elements, so chain conditions are equivalent to antichain conditions. Partial orders and s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laver Function
In set theory, a Laver function (or Laver diamond, named after its inventor, Richard Laver) is a function connected with supercompact cardinals. Definition If κ is a supercompact cardinal, a Laver function is a function ''Æ’'':κ → ''V''κ such that for every set ''x'' and every cardinal λ ≥ , TC(''x''), + κ there is a supercompact measure ''U'' on »sup><κ such that if ''j'' ''U'' is the associated elementary embedding then ''j'' ''U''(''Æ’'')(κ) = ''x''. (Here ''V''κ denotes the κ-th level of the , TC(''x'') is the of ''x'') Applications The original application of Laver funct ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
