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Gödel Operation
In mathematical set theory Set theory is the branch of mathematical logic that studies 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 conce ..., a set of Gödel operations is a finite collection of operations on sets that can be used to construct the constructible sets from ordinals. introduced the original set of 8 Gödel operations 𝔉1,...,𝔉8 under the name fundamental operations. Other authors sometimes use a slightly different set of about 8 to 10 operations, usually denoted ''G''1, ''G''2,... Definition used the following eight operations as a set of Gödel operations (which he called fundamental operations): #\mathfrak_1(X,Y) = \ #\mathfrak_2(X,Y) = E\cdot X = \ #\mathfrak_3(X,Y) = X-Y #\mathfrak_4(X,Y) = X\upharpoonright Y= X\cdot (V\times Y) = \ #\mathfrak_5(X,Y) = X\cdot \mathfrak(Y) = \ #\mathfrak_6(X,Y) = X\cdot Y^= \ #\mathfr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies 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 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 theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result. What is can be thought of as being built in "stages" resembling the constr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jensen Hierarchy
In set theory, a mathematical discipline, the Jensen hierarchy or J-hierarchy is a modification of Gödel's constructible hierarchy, L, that circumvents certain technical difficulties that exist in the constructible hierarchy. The J-Hierarchy figures prominently in fine structure theory, a field pioneered by Ronald Jensen, for whom the Jensen hierarchy is named. Definition As in the definition of ''L'', let Def(''X'') be the collection of sets definable with parameters over ''X'': : \textrm(X) := \ The constructible hierarchy, L is defined by transfinite recursion. In particular, at successor ordinals, L_ = \textrm(L_\alpha). The difficulty with this construction is that each of the levels is not closed under the formation of unordered pairs; for a given x, y \in L_ \setminus L_\alpha, the set \ will not be an element of L_, since it is not a subset of L_\alpha. However, L_\alpha does have the desirable property of being closed under Σ0 separation. Jensen's modificat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
