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Double Extension Set Theory
In mathematics, the Double extension set theory (DEST) is an axiomatic set theory proposed by Andrzej Kisielewicz consisting of two separate membership relations on the universe of sets, denoted here by \in and \varepsilon, and a set of axioms relating the two. The intention behind defining the two membership relations is to avoid the usual paradoxes of 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 ..., without substantially weakening the axiom of unrestricted comprehension. Intuitively, in DEST, comprehension is used to define the elements of a set under one membership relation using formulas that involve only the other membership relation. Let \phi(x) be a first-order formula with free variable x in the language of DEST not involving the membership relation \varepsi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiomatic 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 theory is co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paradoxes Of Set Theory
This article contains a discussion of paradoxes of set theory. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory. Basics Cardinal numbers Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be proved from first principles it has been introduced into axiomatic set theory by the axiom of infinity, which asserts the existence of the set N of natural numbers. Every infinite set which can be enumerated by natural numbers is the same size (cardinality) as N, and is said to be countable. Examples of countably infinite sets are the natural numbers, the even numbers, the prime numbers, and also all the rational numbers, i.e., the fractions. These sets have in common the cardinal number , N, = \aleph_0 (aleph-nought), a number greater than every natural number. Cardinal numbers can be defined a ... [...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] |
Axiom Of Comprehension
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set. Some mathematicians call it the axiom schema of comprehension, although others use that term for ''unrestricted'' comprehension, discussed below. Because restricting comprehension avoided Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory. Statement One instance of the schema is included for each formula φ in the language of set theory with free variables among ''x'', ''w''1, ..., ''w''''n'', ''A''. So ''B'' does not occur free in φ. In the formal language of set theory, the axiom schema is: :\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow x \in A \land \varphi(x, w_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russell's Paradox
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter. According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let ''R'' be the set of all sets that are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
