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Axiom Of Limitation Of Size
In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes.; English translation: . It formalizes the limitation of size principle, which avoids the paradoxes encountered in earlier formulations of set theory by recognizing that some classes are too big to be sets. Von Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class.. A class that is a member of a class is a set; a class that is not a set is a proper class. Every class is a subclass of '' V'', the class of all sets. The axiom of limitation of size says that a class is a set if and only if it is smaller than ''V''—that is, there is no function mapping it onto ''V''. Usually, this axiom is stated in the equivalent form: A class is a proper class if and only if there is a function that maps it onto ''V''. Von Neumann's axiom implies the axioms of replacement, separation, union, and global choice. It is equi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse–Kelley Set Theory
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG). While von Neumann–Bernays–Gödel set theory restricts the bound variables in the schematic formula appearing in the axiom schema of Class Comprehension to range over sets alone, Morse–Kelley set theory allows these bound variables to range over proper classes as well as sets, as first suggested by Quine in 1940 for his system ML. Morse–Kelley set theory is named after mathematicians John L. Kelley and Anthony Morse and was first set out by and later in an appendix to Kelley's textbook ''General Topology'' (1955), a graduate level introduction to topology. Kelley said the system in his book was a variant of the systems due to Thoralf Skolem and Morse. Morse's own versi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-ordered
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-order relation is then called a well-ordered set. In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element. Every element ''s'' of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than ''s''. There may be elements besides the least element which have no predecessor (see below for an example). A well-ordered set ''S'' contains for every subset ''T'' with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of ''T'' in ''S''. If ≤ is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called non-empty. In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). The empty set may also be called the void set. Notation Common notations for the empty set include "", "\emptyset", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabets. In the past, "0" was occasionally used as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF. The axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is "small enough" to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining formulas. Statement Suppose P is a definable binary relation (which may be a proper class) such that for every set x there is a unique set y such that P(x,y) holds. There is a corresponding definable function F_P, where F_P(x)=y if and only if P(x,y). Consider ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Function
In mathematics, function composition is an operation that takes two function (mathematics), functions and , and produces a function such that . In this operation, the function is function application, applied to the result of applying the function to . That is, the functions and are composed to yield a function that maps in domain of a function, domain to in codomain . Intuitively, if is a function of , and is a function of , then is a function of . The resulting ''composite'' function is denoted , defined by for all in . The notation is read as " of ", " after ", " circle ", " round ", " about ", " composed with ", " following ", " then ", or " on ", or "the composition of and ". Intuitively, composing functions is a chaining process in which the output of function feeds the input of function . The composition of functions is a special case of the composition of relations, sometimes also denoted by \circ. As a result, all properties of composit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Contradiction
In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and ''reductio ad impossibile''. It is an example of the weaker logical refutation ''reductio ad absurdum''. A mathematical proof employing proof by contradiction usually proceeds as follows: #The proposition to be proved is ''P''. #We assume ''P'' to be false, i.e., we assume ''¬P''. #It is then shown that ''¬P'' implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, ''Q'' and ''¬Q'', and appealing to the Law of noncontradiction. #Since assuming ''P'' to be false leads to a contradiction, it is concluded that ''P'' is in fact true. An important special case is the existence proof by contradiction: in order to demonstrate the existence of an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings of the language. Each string concatenated from symbols of this alphabet is called a word, and the words that belong to a particular formal language are sometimes called ''well-formed words'' or ''well-formed formulas''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar, which consists of its formation rules. In computer science, formal languages are used among others as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages in which the words of the language represent concepts that are associated with particular meanings or semantics. In computational complexity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Power Set
In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :\forall x \, \exists y \, \forall z \, \in y \iff \forall w \, (w \in z \Rightarrow w \in x)/math> where ''y'' is the Power set of ''x'', \mathcal(x). In English, this says: :Given any set ''x'', there is a set \mathcal(x) such that, given any set ''z'', this set ''z'' is a member of \mathcal(x) if and only if every element of ''z'' is also an element of ''x''. More succinctly: ''for every set x, there is a set \mathcal(x) consisting precisely of the subsets of x.'' Note the subset relation \subseteq is not used in the formal definition as subset is not a primitive relation in formal set theory; rather, subset is defined in terms of set membership, \in. By the axiom of extensionality, the set \mathcal(x) is unique. The axiom of power set appears in most axiomatizations of set theory. It is g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Azriel Lévy
Azriel Lévy (Hebrew: עזריאל לוי; born c. 1934) is an Israeli mathematician, logician, and a professor emeritus at the Hebrew University of Jerusalem. Biography Lévy obtained his Ph.D. at the Hebrew University of Jerusalem in 1958, under the supervision of Abraham Fraenkel and Abraham Robinson. Using Cohen's method of forcing, he proved several results on the consistency of various statements contradicting the axiom of choice. For example, with J. D. Halpern he proved that the Boolean prime ideal theorem does not imply the axiom of choice. He discovered the models ''L'' 'x''used in inner model theory. He also introduced the notions of Lévy hierarchy of the formulas of set theory, Levy collapse and the Feferman–Levy model. His students include Dov Gabbay, Moti Gitik, and Menachem Magidor Menachem Magidor (Hebrew: מנחם מגידור; born January 24, 1946) is an Israeli mathematician who specializes in mathematical logic, in particular set theory. He serve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abraham Fraenkel
Abraham Fraenkel ( he, אברהם הלוי (אדולף) פרנקל; February 17, 1891 – October 15, 1965) was a German-born Israeli mathematician. He was an early Zionist and the first Dean of Mathematics at the Hebrew University of Jerusalem. He is known for his contributions to axiomatic set theory, especially his additions to Ernst Zermelo's axioms, which resulted in the Zermelo–Fraenkel set theory. Biography Abraham Adolf Halevi Fraenkel studied mathematics at the Universities of Munich, Berlin, Marburg and Breslau. After graduating, he lectured at the University of Marburg from 1916, and was promoted to professor in 1922. In 1919 he married Wilhelmina Malka A. Prins (1892–1983). Due to the severe housing shortage in post-war Germany, for a few years the couple lived as subtenants at professor Hensel's place. After leaving Marburg in 1928, Fraenkel taught at the University of Kiel for a year. He then made the fateful choice of accepting a position at the Hebrew Un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem. Furthermore, his 1929 work on ranking chess players is the first description of a model for pairwise comparison that continues to have a profound impact on various applied fields utilizing this method. Life Ernst Zermelo graduated from Berlin's Luisenstädtisches Gymnasium (now ) in 1889. He then studied mathematics, physics and philosophy at the University of Berlin, the University of Halle, and the University of Freiburg. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations (''Untersuchungen zur Variationsrechnung''). Zermelo remained at the University of Berlin, where he was appointed assistant to Planck, under whose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
