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Axiom Of Countable Choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ... of 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, i ... that states that every countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... collection of non-empty #REDIRECT Empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty se ...
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Axiom Of Countable Choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ... of 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, i ... that states that every countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... collection of non-empty #REDIRECT Empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty se ...
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Zermelo–Fraenkel Set Theory
In set theory illustrating the intersection of two sets 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 ..., Zermelo–Fraenkel set theory, named after mathematicians 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 set theory, Zer ... and Abraham Fraenkel Abraham Fraenkel ( he, אברהם הלוי (אדולף) פרנקל; February 17, 1891 – October 15, 1965) was a German-born Israel Israel (; he, יִשְׂרָאֵל; ar, إِسْرَائِيل), officially known as the State of Israe ..., is an axiomatic system In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...
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Metric Space
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a metric space is a non empty set together with a metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ... on the set. The metric is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... that defines a concept of ''distance'' between any two members Member may refer to: * Military jury, referred to as "Memb ...
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Sequence
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a sequence is an enumerated collection of objects in which repetitions are allowed and order Order or ORDER or Orders may refer to: * Orderliness Orderliness is associated with other qualities such as cleanliness Cleanliness is both the abstract state of being clean and free from germs, dirt, trash, or waste, and the habit of achieving a ... matters. Like a set, it contains members Member may refer to: * Military jury, referred to as "Members" in military jargon * Element (mathematics), an object that belongs to a mathematical set * In object-oriented programming, a member of a class ** Field (computer science), entries in ... (also called ''elements'', or ''terms''). The number of elements (possibly infinite ...
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Limit (mathematics)
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a limit is the value that a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... (or sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...) approaches as the input (or index) approaches some value Value or values may refer to: * Value (ethics) In ethics Ethics or moral philosophy is a branch of phil ...
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Accumulation Point
In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ... X is a point x that can be "approximated" by points of S in the sense that every neighbourhood A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ... of x with respect to the topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... ...
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Real Number
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ..., a real number is a value of a continuous quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measu ... that can represent a distance along a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ... (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective ''real'' in this co ...
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Mathematical Analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early da ...
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Dedekind-infinite Set
In mathematics, a set ''A'' is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset ''B'' of ''A'' is equinumerous to ''A''. Explicitly, this means that there exists a bijective function from ''A'' onto some proper subset ''B'' of ''A''. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers. A simple example is \mathbb, the set of natural numbers. From Galileo's paradox, there exists a bijection that maps every natural number ''n'' to its square number, square ''n''2. Since the set of squares is a proper subset of \mathbb, \mathbb is Dedekind-infinite. Until the foundational crisis of mathematics showed the need for a more careful treatment of set theory, most mathematicians tacit assumption, assumed that a set is infinite set, infinite if and only if it i ...
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Infinite Set
In 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, i ..., an infinite set is a set that is not a finite set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t .... Infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ... sets may be countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formul ...
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Solovay Model
In the mathematical field of set theory, the Solovay model is a model theory, model constructed by in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all set (mathematics), sets of real numbers are Lebesgue measurable. The construction relies on the existence of an inaccessible cardinal. In this way Solovay showed that the axiom of choice is essential to the proof of the existence of a non-measurable set, at least granted that the existence of an inaccessible cardinal is consistent with ZFC, the axioms of Zermelo–Fraenkel set theory including the axiom of choice. Statement ZF stands for Zermelo–Fraenkel set theory, and DC for the axiom of dependent choice. Solovay's theorem is as follows. Assuming the existence of an inaccessible cardinal, there is an inner model of ZF + DC of a suitable forcing extension ''V''[''G''] such that every set of reals is Lebesgue measurable, has the perfect set property, and has t ...
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Paul Cohen (mathematician)
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independence (mathematical logic), independent from Zermelo–Fraenkel set theory, for which he was awarded a Fields Medal. Early life and education Cohen was born in Long Branch, New Jersey, into a Jewish family that had immigrated to the United States from what is now Poland; he grew up in Brooklyn.. He graduated in 1950, at age 16, from Stuyvesant High School in New York City. Cohen next studied at the Brooklyn College from 1950 to 1953, but he left without earning his bachelor's degree when he learned that he could start his graduate studies at the University of Chicago with just two years of college. At Chicago, Cohen completed his master's degree in mathematics in 1954 and his Doctor of Philosophy degree in 1958, under supervision of Antoni Zygmund. The title of his doctoral thesis was ''Topics in the ...
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