|
Vopěnka's Alternative Set Theory
In set theory, a semiset is a proper class that is a subclass of a set. In the typical foundations of Zermelo–Fraenkel set theory, semisets are impossible due to the axiom schema of specification. The theory of semisets was proposed and developed by Czech mathematicians Petr Vopěnka and Petr Hájek (1972). It is based on a modification of the von Neumann–Bernays–Gödel set theory; in standard NBG, the existence of semisets is precluded by the axiom of separation. The concept of semisets opens the way for a formulation of an alternative set theory. In particular, Vopěnka's Alternative Set Theory (1979) axiomatizes the concept of semiset, supplemented with several additional principles. Semisets can be used to represent sets with imprecise boundaries. Novák (1984) studied approximation of semisets by fuzzy sets, which are often more suitable for practical applications of the modeling of imprecision. Vopěnka's alternative set theory Vopěnka's "Alternative Set Theory ... [...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] |
Alternative Set Theory
In mathematical logic, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set theory. Alternative set theories Alternative set theories include: * Vopěnka's alternative set theory *Von Neumann–Bernays–Gödel set theory * Morse–Kelley set theory * Tarski–Grothendieck set theory * Ackermann set theory *Type theory *New Foundations *Positive set theory *Internal set theory * Pocket set theory *Naive set theory *S (set theory) * Double extension set theory *Kripke–Platek set theory *Kripke–Platek set theory with urelements * Scott–Potter set theory *Constructive set theory * Zermelo set theory * General set theory *Mac Lane set theory Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (Z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Stanford Encyclopedia Of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication. Each entry is written and maintained by an expert in the field, including professors from many academic institutions worldwide. Authors contributing to the encyclopedia give Stanford University the permission to publish the articles, but retain the copyright to those articles. Approach and history As of August 5, 2022, the ''SEP'' has 1,774 published entries. Apart from its online status, the encyclopedia uses the traditional academic approach of most encyclopedias and academic journal An academic journal (or scholarly journal or scientific journal) is a periodical publication in which Scholarly method, scholarship relating to a particular academic discipline is published. They serve as permanent and transparent forums for the ...s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edward N
Edward is an English male name. It is derived from the Anglo-Saxon name ''Ēadweard'', composed of the elements '' ēad'' "wealth, fortunate; prosperous" and '' weard'' "guardian, protector”. History The name Edward was very popular in Anglo-Saxon England, but the rule of the Norman and Plantagenet dynasties had effectively ended its use amongst the upper classes. The popularity of the name was revived when Henry III named his firstborn son, the future Edward I, as part of his efforts to promote a cult around Edward the Confessor, for whom Henry had a deep admiration. Variant forms The name has been adopted in the Iberian peninsula since the 15th century, due to Edward, King of Portugal, whose mother was English. The Spanish/Portuguese forms of the name are Eduardo and Duarte. Other variant forms include French Édouard, Italian Edoardo and Odoardo, German, Dutch, Czech and Romanian Eduard and Scandinavian Edvard. Short forms include Ed, Eddy, Eddie, Ted, Teddy a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite set, infinite and well-order, well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal number, cardinal and ordinal number, ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Wey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908. Formal statement Using first-order logic primitive symbols, the axiom can be expressed as follows: \exist \mathrm \ (\exist o \ (o \in \mathrm \ \land \lnot \exist n \ (n \in o)) \ \land \ \forall x \ (x \in \mathrm \Rightarrow \exist y \ (y \in \mathrm \ \land \ \forall a \ (a \in y \Leftrightarrow (a \in x \ \lor \ a = x))))). If the notations of both set-builder and empty set are allowed: \exists \mathrm \, ( \varnothing \in \mathrm \, \land \, \forall x \, (x \in \mathrm \Rightarrow \, ( x \cup \ ) \in \mathrm ) ). Some mathematicians may call a set built this way an inductive set. Hint: In English, it reads: " There exists a set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axioms
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an ''axiom'' may be a "logical axiom" or a " non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example ''a'' + 0 = ''a'' in integer arithmetic. Non- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formula (mathematical Logic)
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbol (formal), symbols from a given alphabet (computer science), alphabet that is part of a formal language. The abbreviation wff is pronounced "woof", or sometimes "wiff", "weff", or "whiff". A formal language can be identified with the set of formulas in the language. A formula is a syntax (logic), syntactic object that can be given a semantic Formal semantics (logic), meaning by means of an interpretation (logic), interpretation. Two key uses of formulas are in propositional logic and predicate logic. Introduction A key use of formulas is in propositional logic and Higher-order logic, predicate logic such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, Mathematical proo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case n = k, ''then'' it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the trut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Set
Fuzzy or Fuzzies may refer to: Music * Fuzzy (band), a 1990s Boston indie pop band * Fuzzy (composer), Danish composer Jens Vilhelm Pedersen (born 1939) * Fuzzy (album), ''Fuzzy'' (album), 1993 debut album of American rock band Grant Lee Buffalo * "Fuzzy", a song from the 2009 ''Collective Soul (2009 album), Collective Soul'' album by Collective Soul * "Fuzzy", a song from ''Poppy.Computer'', the debut 2017 album by Poppy * Fuzzy, an Australian events company that organises Listen Out, a multi-city Australian music festival Nickname * Faustina Agolley (born 1984), Australian television presenter, host of the Australian television show ''Video Hits'' * Fuzzy Haskins (1941–2023), American singer and guitarist with the doo-wop group Parliament-Funkadelic * Fuzzy Hufft (1901−1973), American baseball player * Fuzzy Knight (1901−1976), American actor * Andrew Levane (1920−2012), American National Basketball Association player and coach * Robert Alfred Theobald (1884−1957), Uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternative Set Theory
In mathematical logic, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set theory. Alternative set theories Alternative set theories include: * Vopěnka's alternative set theory *Von Neumann–Bernays–Gödel set theory * Morse–Kelley set theory * Tarski–Grothendieck set theory * Ackermann set theory *Type theory *New Foundations *Positive set theory *Internal set theory * Pocket set theory *Naive set theory *S (set theory) * Double extension set theory *Kripke–Platek set theory *Kripke–Platek set theory with urelements * Scott–Potter set theory *Constructive set theory * Zermelo set theory * General set theory *Mac Lane set theory Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (Z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
