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Tree (descriptive Set Theory)
In descriptive set theory, a tree on a set X is a collection of finite sequences of elements of X such that every prefix of a sequence in the collection also belongs to the collection. Definitions Trees The collection of all finite sequences of elements of a set X is denoted X^. With this notation, a tree is a nonempty subset T of X^, such that if \langle x_0,x_1,\ldots,x_\rangle is a sequence of length n in T, and if 0\le m and called the ''body'' of the tree T. A tree that has no branches is called '' wellfounded''; a tree with at least one branch is ''illfounded''. By Kőnig's lemma, a tree on a finite set with an infinite number of sequences must necessarily be illfounded. Terminal nodes A finite sequence that belongs to a tree T is called a terminal node if it is not a prefix of a longer sequence in T. Equivalently, \langle x_0,x_1,\ldots,x_\rangle \in T is terminal if there is no element x of X such that that \langle x_0,x_1,\ldots,x_,x\rangle \in T. A tree that does no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descriptive Set Theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and Group action (mathematics), group actions, and mathematical logic. Polish spaces Descriptive set theory begins with the study of Polish spaces and their Borel sets. A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line \mathbb, the Baire space (set theory), Baire space \mathcal, the Cantor space \mathcal, and the Hilbert cube I^. Universality properties The class of Polish spaces has several universality properties, which show that there is no loss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Element
In mathematics, especially in order theory, a maximal element of a subset S of some preordered set is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some preordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset S of a preordered set is an element of S which is greater than or equal to any other element of S, and the minimum of S is again defined dually. In the particular case of a partially ordered set, while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements. Specializing further to totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide. As an example, in the collecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descriptive Set Theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and Group action (mathematics), group actions, and mathematical logic. Polish spaces Descriptive set theory begins with the study of Polish spaces and their Borel sets. A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line \mathbb, the Baire space (set theory), Baire space \mathcal, the Cantor space \mathcal, and the Hilbert cube I^. Universality properties The class of Polish spaces has several universality properties, which show that there is no loss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbach (1997, 2 ... [...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] |
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] |
Laver Tree
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] |
