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Moschovakis Coding Lemma
The Moschovakis coding lemma is a lemma from descriptive set theory involving sets of real numbers under the axiom of determinacy (the principle — incompatible with choice — that every two-player integer game is determined). The lemma was developed and named after the mathematician Yiannis N. Moschovakis. The lemma may be expressed generally as follows: :Let be a non-selfdual pointclass In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a ''point'' is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by ... closed under real quantification and , and a -well-founded relation on of rank . Let be such that . Then there is a -set which is a choice set for R , that is: # . # . A proof runs as follows: suppose for contradiction is a minimal counterexample, and fix , , and a good universal set for the -subsets of . Easily, must be a limit ordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemma (mathematics)
In mathematics, informal logic and argument mapping, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought. The word "lemma" derives from the Ancient Greek ("anything which is received", such as a gift, profit, or a bribe). Comparison with theorem There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof. Well-known lemmas A good stepping stone can lead to many others. Some powerful results in mathematics are known as lemmas, first named for their originally min ... [...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] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Determinacy
In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy. Steinhaus and Mycielski's motivation for AD was its interesting consequences, and suggested that AD could be true in the smallest natural model L(R) of a set theory, which accepts only a weak form of the axiom of choice (AC) but contains all real and all ordinal numbers. Some consequences of AD followed from theorems proved earlier by Stefan Banach and Stanisław Mazur, and Morton Davis. Mycielski and Stanisław Świerczkowski contributed another one: AD implies that all sets of real numbers are Lebesgue measurable. Later Donald A. Martin and others proved more important consequences, especially in descriptive set theory. In 1988, John R. Steel an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets, there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, a set arising from choosing elements arbitrarily can be made without invoking the axiom of choice; this is, in particular, the case if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available – some distinguishin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yiannis N
Yannis, Yiannis, or Giannis (Γιάννης) is a common Greek given name, a variant of ''John'' (Hebrew) meaning "God is gracious." In formal Greek (e.g. all government documents and birth certificates) the name exists only as Ioannis (Ιωάννης). Variants include ''Yannis'' (Also Janni), ''Iannis'', ''Yannakis'', ''Yanis'', and the rare ''Yannos'', usually found in the Peloponnese and Cyprus. Feminine forms are Γιάννα ( Yianna, Gianna) and Ιωάννα (Ioanna) which is the formal variant used in formal/government documents. Yannis may refer to: * Abu'l-Fath Yanis, Fatimid vizier * Giannis Agouris, Greek writer and journalist * Ioannis Amanatidis, Greek footballer *Yannis Anastasiou, Greek footballer *Yiannis Andrianopoulos, Greek footballer * Giannis Antetokounmpo, Greek basketball player *Giannis Apostolidis, Greek footballer * Yiannis Arabatzis, Greek goalkeeper *Yannis Bakos, economist *Ioannis Banias (1939–2012), Greek politician *Yannis Behrakis, Greek ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a ''point'' is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by some sort of ''definability property''; for example, the collection of all open sets in some fixed collection of Polish spaces is a pointclass. (An open set may be seen as in some sense definable because it cannot be a purely arbitrary collection of points; for any point in the set, all points sufficiently close to that point must also be in the set.) Pointclasses find application in formulating many important principles and theorems from set theory and real analysis. Strong set-theoretic principles may be stated in terms of the determinacy of various pointclasses, which in turn implies that sets in those pointclasses (or sometimes larger ones) have regularity properties such as Lebesgue measurability (and indeed universal measurability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantifier (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first order formula \forall x P(x) expresses that everything in the domain satisfies the property denoted by P. On the other hand, the existential quantifier \exists in the formula \exists x P(x) expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable. The mostly commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic, they are interdefinable using negation. They can also be used to define more complex quantifiers, as in the formula \neg \exists x P(x) which expresses that nothing has the property P. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Choice Set
A choice set is a finite collection of available options selected from a larger theoretical decision space. For example, a consumer has thousands of conceivable alternatives when purchasing a car, far more than they could reasonably be expected to evaluate. As such they will often narrow their search to only vehicles of a certain make, or within a specific price range. By reducing the choice set to a manageable number of alternatives, people are able to make complex decisions between theoretically infinite alternatives in a practical time frame. Choice sets are often used in psychological and market research to make data collection and evaluation more manageable, or to make direct comparisons between a specific set of choices. Choice task The respondent is asked a choice task. Usually this is which of the alternatives they prefer. In this example, the Choice task is ' forced'. An 'unforced' choice would allow the respondents to also select 'Neither'. The choice task is used as the d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
S-m-n Theorem
In computability theory the ' theorem, (also called the translation lemma, parameter theorem, and the parameterization theorem) is a basic result about programming languages (and, more generally, Gödel numberings of the computable functions) (Soare 1987, Rogers 1967). It was first proved by Stephen Cole Kleene (1943). The name ' comes from the occurrence of an ''S'' with subscript ''n'' and superscript ''m'' in the original formulation of the theorem (see below). In practical terms, the theorem says that for a given programming language and positive integers ''m'' and ''n'', there exists a particular algorithm that accepts as input the source code of a program with free variables, together with ''m'' values. This algorithm generates source code that effectively substitutes the values for the first ''m'' free variables, leaving the rest of the variables free. Details The basic form of the theorem applies to functions of two arguments (Nies 2009, p. 6). Given a Gödel nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axioms 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 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 system fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinacy
Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists. Determinacy was introduced by Gale and Stewart in 1950, under the name "determinateness". The games studied in set theory are usually Gale–Stewart games—two-player games of perfect information in which the players make an infinite sequence of moves and there are no draws. The field of game theory studies more general kinds of games, including games with draws such as tic-tac-toe, chess, or infinite chess, or games with imperfect information such as poker. Basic notions Games The first sort of game we shall consider is the two-player game of perfect information of length ω, in which the players play natural numbers. These games are often cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
