1.
Set theory (music)
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Musical set theory provides concepts for categorizing musical objects and describing their relationships. The concepts of set theory are general and can be applied to tonal and atonal styles in any equally tempered tuning system. The methods of set theory are sometimes applied to the analysis of rhythm as well. Although musical set theory is thought to involve the application of mathematical set theory to music. For example, musicians use the terms transposition and inversion where mathematicians would use translation and reflection, furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences. In combinatorics, a subset of n objects, such as pitch classes, is called a combination. Musical set theory is best regarded as a field that is not so much related to mathematical set theory, the main connection to mathematical set theory is the use of the vocabulary of set theory to talk about finite sets. The fundamental concept of set theory is the set, which is an unordered collection of pitch classes. More exactly, a set is a numerical representation consisting of distinct integers. The elements of a set may be manifested in music as simultaneous chords, successive tones, notational conventions vary from author to author, but sets are typically enclosed in curly braces, or square brackets. Some theorists use angle brackets ⟨ ⟩ to denote ordered sequences, thus one might notate the unordered set of pitch classes 0,1, and 2 as. The ordered sequence C-C♯-D would be notated ⟨0,1,2 ⟩ or, although C is considered zero in this example, this is not always the case. For example, a piece with a pitch center of F might be most usefully analyzed with F set to zero (in which case would represent F, F♯. Though set theorists usually consider sets of equal-tempered pitch classes, it is possible to consider sets of pitches, non-equal-tempered pitch classes, rhythmic onsets, two-element sets are called dyads, three-element sets trichords. Sets of higher cardinalities are called tetrachords, pentachords, hexachords, heptachords, octachords, nonachords, decachords, undecachords, and, finally, the basic operations that may be performed on a set are transposition and inversion. Sets related by transposition or inversion are said to be related or inversionally related. Since transposition and inversion are isometries of space, they preserve the intervallic structure of a set. This can be considered the central postulate of musical set theory, in practice, set-theoretic musical analysis often consists in the identification of non-obvious transpositional or inversional relationships between sets found in a piece
2.
Venn diagram
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A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves, a Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. In Venn diagrams the curves are overlapped in every possible way and they are thus a special case of Euler diagrams, which do not necessarily show all relations. Venn diagrams were conceived around 1880 by John Venn and they are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram in which in addition the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram and this example involves two sets, A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures that are two-legged, the blue circle, set B, represents the living creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram, living creatures that both can fly and have two legs—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap. That region contains all such and only living creatures. Humans and penguins are bipedal, and so are then in the circle, but since they cannot fly they appear in the left part of the orange circle. Mosquitoes have six legs, and fly, so the point for mosquitoes is in the part of the circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly would all be represented by points outside both circles, the combined region of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that are either two-legged or that can fly, the region in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles. They are rightly associated with Venn, however, because he comprehensively surveyed and formalized their usage, Venn himself did not use the term Venn diagram and referred to his invention as Eulerian Circles. Of these schemes one only, viz. that commonly called Eulerian circles, has met with any general acceptance, the first to use the term Venn diagram was Clarence Irving Lewis in 1918, in his book A Survey of Symbolic Logic. Venn diagrams are similar to Euler diagrams, which were invented by Leonhard Euler in the 18th century. Baron has noted that Leibniz in the 17th century produced similar diagrams before Euler and she also observes even earlier Euler-like diagrams by Ramon Lull in the 13th Century. In the 20th century, Venn diagrams were further developed, D. W. Henderson showed in 1963 that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number
3.
Intersection (set theory)
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In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B, but no other elements. For explanation of the used in this article, refer to the table of mathematical symbols. The intersection of A and B is written A ∩ B, formally, A ∩ B = that is x ∈ A ∩ B if and only if x ∈ A and x ∈ B. For example, The intersection of the sets and is, the number 9 is not in the intersection of the set of prime numbers and the set of odd numbers. More generally, one can take the intersection of sets at once. The intersection of A, B, C, and D, Intersection is an associative operation, thus, A ∩ = ∩ C. Additionally, intersection is commutative, thus A ∩ B = B ∩ A, inside a universe U one may define the complement Ac of A to be the set of all elements of U not in A. We say that A intersects B if A intersects B at some element, a intersects B if their intersection is inhabited. We say that A and B are disjoint if A does not intersect B, in plain language, they have no elements in common. A and B are disjoint if their intersection is empty, denoted A ∩ B = ∅, for example, the sets and are disjoint, the set of even numbers intersects the set of multiples of 3 at 0,6,12,18 and other numbers. The most general notion is the intersection of a nonempty collection of sets. If M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if, the notation for this last concept can vary considerably. Set theorists will sometimes write ⋂M, while others will instead write ⋂A∈M A, the latter notation can be generalized to ⋂i∈I Ai, which refers to the intersection of the collection. Here I is a nonempty set, and Ai is a set for every i in I. In the case that the index set I is the set of numbers, notation analogous to that of an infinite series may be seen. When formatting is difficult, this can also be written A1 ∩ A2 ∩ A3 ∩, even though strictly speaking, A1 ∩ (A2 ∩ (A3 ∩. Finally, let us note that whenever the symbol ∩ is placed before other symbols instead of them, it should be of a larger size. Note that in the section we excluded the case where M was the empty set
4.
Set (mathematics)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
5.
Georg Cantor
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Georg Ferdinand Ludwig Philipp Cantor was a German mathematician. He invented set theory, which has become a theory in mathematics. In fact, Cantors method of proof of this theorem implies the existence of an infinity of infinities and he defined the cardinal and ordinal numbers and their arithmetic. Cantors work is of great philosophical interest, a fact of which he was well aware, E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Cantor, a devout Lutheran, believed the theory had been communicated to him by God, Kronecker objected to Cantors proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. The harsh criticism has been matched by later accolades, in 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics. David Hilbert defended it from its critics by declaring, From his paradise that Cantor with us unfolded, we hold our breath in awe, knowing, we shall not be expelled. Georg Cantor was born in the merchant colony in Saint Petersburg, Russia. Georg, the oldest of six children, was regarded as an outstanding violinist and his grandfather Franz Böhm was a well-known musician and soloist in a Russian imperial orchestra. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt, his skills in mathematics. In 1862, Cantor entered the Swiss Federal Polytechnic and he spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. Cantor submitted his dissertation on number theory at the University of Berlin in 1867, after teaching briefly in a Berlin girls school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the habilitation for his thesis, also on number theory. In 1874, Cantor married Vally Guttmann and they had six children, the last born in 1886. Cantor was able to support a family despite modest academic pay, during his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday. Cantor was promoted to Extraordinary Professor in 1872 and made full Professor in 1879, however, his work encountered too much opposition for that to be possible. Worse yet, Kronecker, a figure within the mathematical community and Cantors former professor. Cantor came to believe that Kroneckers stance would make it impossible for him ever to leave Halle, in 1881, Cantors Halle colleague Eduard Heine died, creating a vacant chair
6.
Richard Dedekind
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Julius Wilhelm Richard Dedekind was a German mathematician who made important contributions to abstract algebra, algebraic number theory and the definition of the real numbers. Dedekinds father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig, as an adult, he never used the names Julius Wilhelm. He was born, lived most of his life, and died in Braunschweig and he first attended the Collegium Carolinum in 1848 before transferring to the University of Göttingen in 1850. There, Dedekind was taught number theory by professor Moritz Stern, Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled Über die Theorie der Eulerschen Integrale and this thesis did not display the talent evident by Dedekinds subsequent publications. At that time, the University of Berlin, not Göttingen, was the facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and Bernhard Riemann were contemporaries, they were awarded the habilitation in 1854. Dedekind returned to Göttingen to teach as a Privatdozent, giving courses on probability and he studied for a while with Peter Gustav Lejeune Dirichlet, and they became good friends. Because of lingering weaknesses in his knowledge, he studied elliptic. Yet he was also the first at Göttingen to lecture concerning Galois theory, about this time, he became one of the first people to understand the importance of the notion of groups for algebra and arithmetic. In 1858, he began teaching at the Polytechnic school in Zürich, when the Collegium Carolinum was upgraded to a Technische Hochschule in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in 1894, but did occasional teaching and continued to publish and he never married, instead living with his sister Julia. Dedekind was elected to the Academies of Berlin and Rome, and he received honorary doctorates from the universities of Oslo, Zurich, and Braunschweig. While teaching calculus for the first time at the Polytechnic school, Dedekind developed the now known as a Dedekind cut. The idea of a cut is that an irrational number divides the rational numbers into two classes, with all the numbers of one class being strictly greater than all the numbers of the other class. Every location on the number line continuum contains either a rational or an irrational number, thus there are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet Stetigkeit und irrationale Zahlen, in modern terminology, Vollständigkeit, Dedekinds theorem states that if there existed a one-to-one correspondence between two sets, then Dedekind said that the two sets were similar. Thus the set N of natural numbers can be shown to be similar to the subset of N whose members are the squares of every member of N, N12345678910
7.
Naive set theory
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Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using a formal logic, naive set theory is defined informally and it describes the aspects of mathematical sets familiar in discrete mathematics, and suffices for the everyday usage of set theory concepts in contemporary mathematics. Sets are of importance in mathematics, in modern formal treatments. Naive set theory suffices for many purposes, while serving as a stepping-stone towards more formal treatments. A naive theory in the sense of set theory is a non-formalized theory, that is. Then, not, for some, for every are treated as in ordinary mathematics, as a matter of convenience, usage of naive set theory and its formalism prevails even in higher mathematics – including in more formal settings of set theory itself. The first development of set theory was a set theory. It was created at the end of the 19th century by Georg Cantor as part of his study of infinite sets, Naive set theory may refer to several very distinct notions. It may refer to Informal presentation of a set theory. Early or later versions of Georg Cantors theory and other informal systems, decidedly inconsistent theories, such as a theory of Gottlob Frege that yielded Russells paradox, and theories of Giuseppe Peano and Richard Dedekind. The assumption that any property may be used to form a set, without restriction, one common example is Russells paradox, there is no set consisting of all sets that do not contain themselves. Thus consistent systems of set theory must include some limitations on the principles which can be used to form sets. Some believe that Georg Cantors set theory was not actually implicated in the set-theoretic paradoxes, one difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system. Cantors paradox can actually be derived from the above assumption using for P x is a cardinal number, Axiomatic set theory was developed in response to these early attempts to understand sets, with the goal of determining precisely what operations were allowed and when. A naive set theory is not necessarily inconsistent, if it specifies the sets allowed to be considered. This can be done by the means of definitions, which are implicit axioms and it is possible to state all the axioms explicitly, as in the case of Halmos Naive Set Theory, which is actually an informal presentation of the usual axiomatic Zermelo–Fraenkel set theory. It is naive in that the language and notations are those of ordinary informal mathematics, likewise, an axiomatic set theory is not necessarily consistent, i. e. not necessarily free of paradoxes. However, the common systems are generally believed to be consistent
8.
Russell's paradox
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According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. Symbolically, Let R =, then R ∈ R ⟺ R ∉ R In 1908, two ways of avoiding the paradox were proposed, Russells type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelos axioms went well beyond Gottlob Freges axioms of extensionality and unlimited set abstraction, Let us call a set abnormal if it is a member of itself, and normal otherwise. For example, take the set of all squares in the plane and that set is not itself a square in the plane, and therefore is not a member of the set of all squares in the plane. On the other hand, if we take the set that contains all non-. Now we consider the set of all sets, R. This leads to the conclusion that R is neither normal nor abnormal, then by existential instantiation and universal instantiation we have y ∈ y ⟺ y ∉ y a contradiction. Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem and this theory became widely accepted once Zermelos axiom of choice ceased to be controversial, and ZFC has remained the canonical axiomatic set theory down to the present day. ZFC does not assume that, for property, there is a set of all things satisfying that property. Rather, it asserts that any set X, any subset of X definable using first-order logic exists. The object R discussed above cannot be constructed in this fashion, in some extensions of ZFC, objects like R are called proper classes. ZFC is silent about types, although the hierarchy has a notion of layers that resemble types. Zermelo himself never accepted Skolems formulation of ZFC using the language of first-order logic and this 2nd order ZFC preferred by Zermelo, including axiom of foundation, allowed a rich cumulative hierarchy. Ferreirós writes that Zermelos layers are essentially the same as the types in the versions of simple TT offered by Gödel. One can describe the cumulative hierarchy into which Zermelo developed his models as the universe of a cumulative TT in which types are allowed. Thus, simple TT and ZFC could now be regarded as systems that talk essentially about the same intended objects, the main difference is that TT relies on a strong higher-order logic, while Zermelo employed second-order logic, and ZFC can also be given a first-order formulation. The first-order description of the hierarchy is much weaker, as is shown by the existence of denumerable models
9.
Axiom of choice
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In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. It states that for every indexed family i ∈ I of nonempty sets there exists an indexed family i ∈ I of elements such that x i ∈ S i for every i ∈ I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. Informally put, the axiom of choice says that any collection of bins, each containing at least one object. One motivation for use is that a number of generally accepted mathematical results, such as Tychonoffs theorem. Contemporary set theorists also study axioms that are not compatible with the axiom of choice, the axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced. A choice function is a function f, defined on a collection X of nonempty sets, each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. The axiom of choice asserts the existence of elements, it is therefore equivalent to, Given any family of nonempty sets. In this article and other discussions of the Axiom of Choice the following abbreviations are common, ZF – Zermelo–Fraenkel set theory omitting the Axiom of Choice. ZFC – Zermelo–Fraenkel set theory, extended to include the Axiom of Choice, There are many other equivalent statements of the axiom of choice. These are equivalent in the sense that, in the presence of basic axioms of set theory. One variation avoids the use of functions by, in effect. Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains one element in common with each of the sets in X. This guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition. Another equivalent axiom only considers collections X that are essentially powersets of other sets, For any set A, authors who use this formulation often speak of the choice function on A, but be advised that this is a slightly different notion of choice function. With this alternate notion of function, the axiom of choice can be compactly stated as Every set has a choice function. Which is equivalent to For any set A there is a function f such that for any non-empty subset B of A, f lies in B. The negation of the axiom can thus be expressed as, There is a set A such that for all functions f, however, that particular case is a theorem of Zermelo–Fraenkel set theory without the axiom of choice, it is easily proved by mathematical induction
10.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
11.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
12.
Consistency
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In classical deductive logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms, the semantic definition states that a theory is consistent if and only if it has a model, i. e. there exists an interpretation under which all formulas in the theory are true. This is the used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory T is consistent if and only if there is no formula ϕ such that both ϕ and its negation ¬ ϕ are elements of the set T. Let A be set of closed sentences and ⟨ A ⟩ the set of closed sentences provable from A under some formal deductive system, the set of axioms A is consistent when ⟨ A ⟩ is. If there exists a system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic. Stronger logics, such as logic, are not complete. A consistency proof is a proof that a particular theory is consistent. The early development of mathematical theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilberts program. Hilberts program was impacted by incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency. Although consistency can be proved by means of theory, it is often done in a purely syntactical way. The cut-elimination implies the consistency of the calculus, since there is obviously no cut-free proof of falsity, in theories of arithmetic, such as Peano arithmetic, there is an intricate relationship between the consistency of the theory and its completeness. A theory is complete if, for every formula φ in its language, Presburger arithmetic is an axiom system for the natural numbers under addition. It is both consistent and complete, Gödels incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent. Gödels theorem applies to the theories of Peano arithmetic and Primitive recursive arithmetic, moreover, Gödels second incompleteness theorem shows that the consistency of sufficiently strong recursively enumerable theories of arithmetic can be tested in a particular way. Thus the consistency of a strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable theories that can describe a strong enough fragment of arithmetic—including set theories such as Zermelo–Fraenkel set theory and these set theories cannot prove their own Gödel sentence—provided that they are consistent, which is generally believed. Because consistency of ZF is not provable in ZF, the weaker notion relative consistency is interesting in set theory
13.
Georg Cantor's first set theory article
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Georg Cantors first set theory article was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is Cantors revolutionary discovery that the set of all numbers is uncountably, rather than countably. This theorem is proved using Cantors first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, On a Property of the Collection of All Real Algebraic Numbers, refers to its first theorem, Cantors article also contains a proof of the existence of transcendental numbers. As early as 1930, mathematicians have disagreed on whether this proof is constructive or non-constructive, books as recent as 2014 and 2015 indicate that this disagreement has not been resolved. Since Cantors proof either constructs transcendental numbers or does not, an analysis of his article can determine whether his proof is constructive or non-constructive, historians of mathematics have examined Cantors article and the circumstances in which it was written. For example, they have discovered that Cantor was advised to leave out his uncountability theorem in the article he submitted and they have traced this and other facts about the article to the influence of Karl Weierstrass and Leopold Kronecker. Historians have also studied Dedekinds contributions to the article, including his contributions to the theorem on the countability of the algebraic numbers. In addition, they have looked at the articles legacy—namely, the impact of the uncountability theorem, Cantors article is short, just 4 1/3 pages. Cantor restates this theorem in terms familiar to mathematicians of his time. Cantors second theorem works with an interval, which is the set of real numbers ≥ a and ≤ b. The theorem states, Given any sequence of real numbers x1, x2, x3, … and any interval, Hence, there are infinitely many such numbers. The first part of this theorem implies the Hence part, for example, let be the interval, and consider its pairwise disjoint subintervals, …. Applying the first part of the theorem to each subinterval produces infinitely many numbers in that are not contained in the given sequence, Cantor observes that combining his two theorems yields a new proof of the theorem that every interval contains infinitely many transcendental numbers. This theorem was first proved by Joseph Liouville and this remark contains Cantors uncountability theorem, which only states that an interval cannot be put into one-to-one correspondence with the set of positive integers. It does not state that this interval is a set of larger cardinality than the set of positive integers. Cardinality is defined in Cantors next article, which was published in 1878, Cantor does not explicitly prove his uncountability theorem, which follows easily from his second theorem. To prove it, we use proof by contradiction, applying Cantors second theorem to this sequence and produces a real number in that does not belong to the sequence
14.
Greek mathematics
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Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
15.
Zeno of Elea
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Zeno of Elea was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic and he is best known for his paradoxes, which Bertrand Russell has described as immeasurably subtle and profound. Little is known for certain about Zenos life, although written nearly a century after Zenos death, the primary source of biographical information about Zeno is Platos Parmenides and he is also mentioned in Aristotles Physics. In the dialogue of Parmenides, Plato describes a visit to Athens by Zeno and Parmenides, at a time when Parmenides is about 65, Zeno is nearly 40 and Socrates is a very young man. Assuming an age for Socrates of around 20, and taking the date of Socrates birth as 469 BC gives a date of birth for Zeno of 490 BC. Plato says that Zeno was tall and fair to look upon and was in the days of his youth … reported to have been beloved by Parmenides, according to Diogenes Laertius, Zeno conspired to overthrow Nearchus the tyrant. Eventually, Zeno was arrested and tortured, when Nearchus leaned in to listen to the secret, Zeno bit his ear. He did not let go until he lost his life and the tyrant lost that part of his body, within Men of the Same Name, Demetrius said it was the nose that was bit off instead. Zeno may have interacted with other tyrants. According to Laertius, Heraclides Lembus, within his Satyrus, said these events occurred against Diomedon instead of Nearchus, valerius Maximus recounts a conspiracy against the tyrant Phalaris, but this would be impossible as Phalaris had died before Zeno was even born. According to Plutarch, Zeno attempted to kill the tyrant Demylus, after failing, he had, with his own teeth bit off his tongue, he spit it in the tyrant’s face. Although many ancient writers refer to the writings of Zeno, none of his works survive intact, the main sources on the nature of Zenos arguments on motion, in fact, come from the writings of Aristotle and Simplicius of Cilicia. Plato says that Zenos writings were brought to Athens for the first time on the occasion of the visit of Zeno and Parmenides. Plato also has Zeno say that work, meant to protect the arguments of Parmenides, was written in Zenos youth, stolen. According to Proclus in his Commentary on Platos Parmenides, Zeno produced not less than forty arguments revealing contradictions, Zenos arguments are perhaps the first examples of a method of proof called reductio ad absurdum, literally meaning to reduce to the absurd. Parmenides is said to be the first individual to implement this style of argument and this form of argument soon became known as the epicheirema. In Book VII of his Topics, Aristotle says that an epicheirema is a dialectical syllogism and it is a connected piece of reasoning which an opponent has put forward as true. The disputant sets out to break down the dialectical syllogism, Zeno is also regarded as the first philosopher who dealt with the earliest attestable accounts of mathematical infinity
16.
Indian mathematics
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Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama, the decimal number system in worldwide use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, in addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China and this was followed by a second section consisting of a prose commentary that explained the problem in more detail and provided justification for the solution. In the prose section, the form was not considered so important as the ideas involved, all mathematical works were orally transmitted until approximately 500 BCE, thereafter, they were transmitted both orally and in manuscript form. A later landmark in Indian mathematics was the development of the series expansions for functions by mathematicians of the Kerala school in the 15th century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series. However, they did not formulate a theory of differentiation and integration. Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of practical mathematics. The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4,2,1, considered favourable for the stability of a brick structure. They used a system of weights based on the ratios, 1/20, 1/10, 1/5, 1/2,1,2,5,10,20,50,100,200. They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, the inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length was divided into ten equal parts, bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. The religious texts of the Vedic Period provide evidence for the use of large numbers, by the time of the Yajurvedasaṃhitā-, numbers as high as 1012 were being included in the texts. The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta, With three-fourths Puruṣa went up, the Satapatha Brahmana contains rules for ritual geometric constructions that are similar to the Sulba Sutras. The Śulba Sūtras list rules for the construction of fire altars. Most mathematical problems considered in the Śulba Sūtras spring from a single theological requirement, according to, the Śulba Sūtras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. The diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately and they contain lists of Pythagorean triples, which are particular cases of Diophantine equations
17.
Infinity
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Infinity is an abstract concept describing something without any bound or larger than any number. In mathematics, infinity is treated as a number but it is not the same sort of number as natural or real numbers. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th, in the theory he developed, there are infinite sets of different sizes. For example, the set of integers is countably infinite, while the set of real numbers is uncountable. Ancient cultures had various ideas about the nature of infinity, the ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus and he used the word apeiron which means infinite or limitless. However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea, aristotle called him the inventor of the dialectic. He is best known for his paradoxes, described by Bertrand Russell as immeasurably subtle, however, recent readings of the Archimedes Palimpsest have found that Archimedes had an understanding about actual infinite quantities. The Jain mathematical text Surya Prajnapti classifies all numbers into three sets, enumerable, innumerable, and infinite, on both physical and ontological grounds, a distinction was made between asaṃkhyāta and ananta, between rigidly bounded and loosely bounded infinities. European mathematicians started using numbers in a systematic fashion in the 17th century. John Wallis first used the notation ∞ for such a number, euler used the notation i for an infinite number, and exploited it by applying the binomial formula to the i th power, and infinite products of i factors. In 1699 Isaac Newton wrote about equations with an number of terms in his work De analysi per aequationes numero terminorum infinitas. The infinity symbol ∞ is a symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ infinity and in LaTeX as \infty and it was introduced in 1655 by John Wallis, and, since its introduction, has also been used outside mathematics in modern mysticism and literary symbology. Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers, in real analysis, the symbol ∞, called infinity, is used to denote an unbounded limit. X → ∞ means that x grows without bound, and x → − ∞ means the value of x is decreasing without bound. ∑ i =0 ∞ f = ∞ means that the sum of the series diverges in the specific sense that the partial sums grow without bound. Infinity can be used not only to define a limit but as a value in the real number system
18.
Bernard Bolzano
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Bernard Bolzano was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his antimilitarist views. Bolzano wrote in German, his mother tongue, for the most part, his work came to prominence posthumously. Bolzano was the son of two pious Catholics and his father, Bernard Pompeius Bolzano, was an Italian who had moved to Prague, where he married Maria Cecilia Maurer who came from Pragues German-speaking family Maurer. Only two of their children lived to adulthood. Bolzano entered the University of Prague in 1796 and studied mathematics, philosophy, starting in 1800, he also began studying theology, becoming a Catholic priest in 1804. He was appointed to the newly created chair of philosophy of religion at Prague University in 1805. He proved to be a popular lecturer not just in religion but also in philosophy, Bolzano alienated many faculty and church leaders with his teachings of the social waste of militarism and the needlessness of war. He urged a total reform of the educational, social, upon his refusal to recant his beliefs, Bolzano was dismissed from the university in 1819. His political convictions eventually proved to be too liberal for the Austrian authorities and he was exiled to the countryside and at that point devoted his energies to his writings on social, religious, philosophical, and mathematical matters. Although forbidden to publish in journals as a condition of his exile, Bolzano continued to develop his ideas. In 1842 he moved back to Prague, where he died in 1848, Bolzano made several original contributions to mathematics. His overall philosophical stance was that, contrary to much of the mathematics of the era, it was better not to introduce intuitive ideas such as time. These works presented. a sample of a new way of developing analysis, to the foundations of mathematical analysis he contributed the introduction of a fully rigorous ε–δ definition of a mathematical limit. Bolzano was the first to recognize the greatest lower bound property of the real numbers, like several others of his day, he was skeptical of the possibility of Gottfried Leibnizs infinitesimals, that had been the earliest putative foundation for differential calculus. Bolzano also gave the first purely analytic proof of the theorem of algebra. He also gave the first purely analytic proof of the intermediate value theorem, the logical theory that Bolzano developed in this work has come to be acknowledged as ground-breaking. Other works are a four-volume Lehrbuch der Religionswissenschaft and the metaphysical work Athanasia, Bolzano also did valuable work in mathematics, which remained virtually unknown until Otto Stolz rediscovered many of his lost journal articles and republished them in 1881. Bolzano begins his work by explaining what he means by theory of science, human knowledge, he states, is made of all truths that men know or have known
19.
Real analysis
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Real analysis is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. The theorems of real analysis rely intimately upon the structure of the number line. The real number system consists of a set, together with two operations and an order, and is, formally speaking, an ordered quadruple consisting of these objects, there are several ways of formalizing the definition of the real number system. The synthetic approach gives a list of axioms for the numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense there is a model for the axioms. Any one of these models must be constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. These constructions are described in detail in the main article. In addition to these notions, the real numbers, equipped with the absolute value function as a metric. Many important theorems in real analysis remain valid when they are restated as statements involving metric spaces and these theorems are frequently topological in nature, and placing them in the more abstract setting of metric spaces may lead to proofs that are shorter, more natural, or more elegant. The real numbers have several important lattice-theoretic properties that are absent in the complex numbers, most importantly, the real numbers form an ordered field, in which addition and multiplication preserve positivity. Moreover, the ordering of the numbers is total. These order-theoretic properties lead to a number of important results in analysis, such as the monotone convergence theorem, the intermediate value theorem. However, while the results in analysis are stated for real numbers. In particular, many ideas in analysis and operator theory generalize properties of the real numbers – such generalizations include the theories of Riesz spaces. Also, mathematicians consider real and imaginary parts of complex sequences, a sequence is a function whose domain is a countable, totally ordered set, usually taken to be the natural numbers or whole numbers. Occasionally, it is convenient to consider bidirectional sequences indexed by the set of all integers. Of interest in analysis, a real-valued sequence, here indexed by the natural numbers, is a map a, N → R, n ↦ a n. Each a = a n is referred to as a term of the sequence, a sequence that tends to a limit is said to be convergent, otherwise it is divergent
20.
Karl Weierstrass
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Karl Theodor Wilhelm Weierstrass was a German mathematician often cited as the father of modern analysis. Despite leaving university without a degree, he studied mathematics and trained as a teacher, eventually teaching mathematics, physics, botany, Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia. Weierstrass was the son of Wilhelm Weierstrass, a government official and his interest in mathematics began while he was a gymnasium student at the Theodorianum in Paderborn. He was sent to the University of Bonn upon graduation to prepare for a government position, because his studies were to be in the fields of law, economics, and finance, he was immediately in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his course of study. The outcome was to leave the university without a degree, after that he studied mathematics at the Münster Academy and his father was able to obtain a place for him in a teacher training school in Münster. Later he was certified as a teacher in that city, during this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions. In 1843 he taught in Deutsch Krone in West Prussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg, besides mathematics he also taught physics, botanics and gymnastics. Weierstrass may have had a child named Franz with the widow of his friend Carl Wilhelm Borchardt. After 1850 Weierstrass suffered from a period of illness, but was able to publish papers that brought him fame. The University of Königsberg conferred an honorary degree on him on 31 March 1854. In 1856 he took a chair at the Gewerbeinstitut, which became the Technical University of Berlin. In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which became the Humboldt Universität zu Berlin. He was immobile for the last three years of his life, and died in Berlin from pneumonia, delta-epsilon proofs are first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval, notably, in his 1821 Cours danalyse, Cauchy argued that the limit of continuous functions was itself continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the limit of continuous functions is continuous. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus, using this definition, he proved the Intermediate Value Theorem. He also proved the Bolzano–Weierstrass theorem and used it to study the properties of functions on closed and bounded intervals
21.
Leopold Kronecker
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Leopold Kronecker was a German mathematician who worked on number theory, algebra and logic. He criticized Cantors work on set theory, and was quoted by Weber as having said, Die ganzen Zahlen hat der liebe Gott gemacht, Kronecker was a student and lifelong friend of Ernst Kummer. Leopold Kronecker was born on 7 December 1823 in Liegnitz, Prussia in a wealthy Jewish family, Kronecker then went to the Liegnitz Gymnasium where he was interested in a wide range of topics including science, history and philosophy, while also practicing gymnastics and swimming. At the gymnasium he was taught by Ernst Kummer, who noticed and encouraged the boys interest in mathematics, in 1841 Kronecker became a student at the University of Berlin where his interest did not immediately focus on mathematics, but rather spread over several subjects including astronomy and philosophy. He spent the summer of 1843 at the University of Bonn studying astronomy, back in Berlin, Kronecker studied mathematics with Peter Gustav Lejeune Dirichlet and in 1845 defended his dissertation in algebraic number theory written under Dirichlets supervision. After obtaining his degree, Kronecker did not follow his interest in research on a career path. He went back to his hometown to manage a large farming estate built up by his mothers uncle, in 1848 he married his cousin Fanny Prausnitzer, and the couple had six children. For several years Kronecker focused on business, and although he continued to study mathematics as a hobby and corresponded with Kummer, in 1853 he wrote a memoir on the algebraic solvability of equations extending the work of Évariste Galois on the theory of equations. Due to his business activity, Kronecker was financially comfortable, Dirichlet, whose wife Rebecka came from the wealthy Mendelssohn family, had introduced Kronecker to the Berlin elite. He became a friend of Karl Weierstrass, who had recently joined the university. Over the following years Kronecker published numerous papers resulting from his previous years independent research, as a result of this published research, he was elected a member of the Berlin Academy in 1861. Although he held no official university position, Kronecker had the right as a member of the Academy to hold classes at the University of Berlin and he decided to do so, starting in 1862. In 1866, when Riemann died, Kronecker was offered the chair at the University of Göttingen. Only in 1883, when Kummer retired from the University, was Kronecker invited to succeed him, Kronecker was the supervisor of Kurt Hensel, Adolf Kneser, Mathias Lerch, and Franz Mertens, amongst others. Kronecker died on 29 December 1891 in Berlin, several months after the death of his wife, in the last year of his life, he converted to Christianity. He is buried in the Alter St Matthäus Kirchhof cemetery in Berlin-Schöneberg, an important part of Kroneckers research focused on number theory and algebra. In an 1853 paper on the theory of equations and Galois theory he formulated the Kronecker–Weber theorem and he also introduced the structure theorem for finitely-generated abelian groups. Kronecker studied elliptic functions and conjectured his liebster Jugendtraum, a generalization that was put forward by Hilbert in a modified form as his twelfth problem
22.
One-to-one correspondence
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In mathematical terms, a bijective function f, X → Y is a one-to-one and onto mapping of a set X to a set Y. A bijection from the set X to the set Y has a function from Y to X. If X and Y are finite sets, then the existence of a means they have the same number of elements. For infinite sets the picture is complicated, leading to the concept of cardinal number. A bijective function from a set to itself is called a permutation. Bijective functions are essential to many areas of including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group. Satisfying properties and means that a bijection is a function with domain X and it is more common to see properties and written as a single statement, Every element of X is paired with exactly one element of Y. Functions which satisfy property are said to be onto Y and are called surjections, Functions which satisfy property are said to be one-to-one functions and are called injections. With this terminology, a bijection is a function which is both a surjection and an injection, or using words, a bijection is a function which is both one-to-one and onto. Consider the batting line-up of a baseball or cricket team, the set X will be the players on the team and the set Y will be the positions in the batting order The pairing is given by which player is in what position in this order. Property is satisfied since each player is somewhere in the list, property is satisfied since no player bats in two positions in the order. Property says that for each position in the order, there is some player batting in that position, in a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks them all to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. The instructor was able to conclude there were just as many seats as there were students. For any set X, the identity function 1X, X → X, the function f, R → R, f = 2x +1 is bijective, since for each y there is a unique x = /2 such that f = y. In more generality, any linear function over the reals, f, R → R, f = ax + b is a bijection, each real number y is obtained from the real number x = /a. The function f, R →, given by f = arctan is bijective since each real x is paired with exactly one angle y in the interval so that tan = x
23.
Power set
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In mathematics, the power set of any set S is the set of all subsets of S, including the empty set and S itself. The power set of a set S is variously denoted as P, ℘, P, ℙ, or, in axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set. Any subset of P is called a family of sets over S, if S is the set, then the subsets of S are, and hence the power set of S is. If S is a set with |S| = n elements. This fact, which is the motivation for the notation 2S, may be demonstrated simply as follows, First and we write any subset of S in the format where γi,1 ≤ i ≤ n, can take the value of 0 or 1. If γi =1, the element of S is in the subset, otherwise. Clearly the number of subsets that can be constructed this way is 2n as γi ∈. Cantors diagonal argument shows that the set of a set always has strictly higher cardinality than the set itself. In particular, Cantors theorem shows that the set of a countably infinite set is uncountably infinite. The power set of the set of numbers can be put in a one-to-one correspondence with the set of real numbers. The power set of a set S, together with the operations of union, intersection, in fact, one can show that any finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set. For infinite Boolean algebras this is no true, but every infinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra. The power set of a set S forms a group when considered with the operation of symmetric difference. It can hence be shown that the power set considered together with both of these forms a Boolean ring. In set theory, XY is the set of all functions from Y to X, as 2 can be defined as, 2S is the set of all functions from S to. Hence 2S and P could be considered identical set-theoretically and this notion can be applied to the example above in which S = to see the isomorphism with the binary numbers from 0 to 2n −1 with n being the number of elements in the set. In S, a 1 in the corresponding to the location in the set indicates the presence of the element. The number of subsets with k elements in the set of a set with n elements is given by the number of combinations, C
24.
Logical paradox
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A paradox is a statement that, despite apparently sound reasoning from true premises, leads to a self-contradictory or a logically unacceptable conclusion. A paradox involves contradictory yet interrelated elements that exist simultaneously and persist over time, some logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking. Some paradoxes have revealed errors in definitions assumed to be rigorous, others, such as Currys paradox, are not yet resolved. Examples outside logic include the Ship of Theseus from philosophy, paradoxes can also take the form of images or other media. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, and staircases that appear to climb endlessly. In common usage, the word often refers to statements that may be both true and false i. e. ironic or unexpected, such as the paradox that standing is more tiring than walking. Common themes in paradoxes include self-reference, infinite regress, circular definitions, patrick Hughes outlines three laws of the paradox, Self-reference An example is This statement is false, a form of the liar paradox. The statement is referring to itself, another example of self-reference is the question of whether the barber shaves himself in the barber paradox. One more example would be Is the answer to this question No, contradiction This statement is false, the statement cannot be false and true at the same time. Another example of contradiction is if a man talking to a genie wishes that wishes couldnt come true, vicious circularity, or infinite regress This statement is false, if the statement is true, then the statement is false, thereby making the statement true. Another example of vicious circularity is the group of statements. Other paradoxes involve false statements or half-truths and the biased assumptions. This form is common in howlers, for example, consider a situation in which a father and his son are driving down the road. The car crashes into a tree and the father is killed, the boy is rushed to the nearest hospital where he is prepared for emergency surgery. On entering the suite, the surgeon says, I cant operate on this boy. The apparent paradox is caused by a hasty generalization, for if the surgeon is the boys father, the paradox is resolved if it is revealed that the surgeon is a woman — the boys mother. Paradoxes which are not based on a hidden error generally occur at the fringes of context or language, paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. Russells paradox, which shows that the notion of the set of all sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic
25.
Bertrand Russell
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Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, writer, social critic, political activist and Nobel laureate. At various points in his life he considered himself a liberal, a socialist, and a pacifist and he was born in Monmouthshire into one of the most prominent aristocratic families in the United Kingdom. In the early 20th century, Russell led the British revolt against idealism and he is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore, and protégé Ludwig Wittgenstein. He is widely held to be one of the 20th centurys premier logicians, with A. N. Whitehead he wrote Principia Mathematica, an attempt to create a logical basis for mathematics. His philosophical essay On Denoting has been considered a paradigm of philosophy, Russell mostly was a prominent anti-war activist, he championed anti-imperialism. Occasionally, he advocated preventive nuclear war, before the opportunity provided by the monopoly is gone. He went to prison for his pacifism during World War I, in 1950 Russell was awarded the Nobel Prize in Literature in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought. Bertrand Russell was born on 18 May 1872 at Ravenscroft, Trellech, Monmouthshire and his parents, Viscount and Viscountess Amberley, were radical for their times. Lord Amberley consented to his wifes affair with their childrens tutor, both were early advocates of birth control at a time when this was considered scandalous. Lord Amberley was an atheist and his atheism was evident when he asked the philosopher John Stuart Mill to act as Russells secular godfather, Mill died the year after Russells birth, but his writings had a great effect on Russells life. His paternal grandfather, the Earl Russell, had asked twice by Queen Victoria to form a government. The Russells had been prominent in England for several centuries before this, coming to power, Lady Amberley was the daughter of Lord and Lady Stanley of Alderley. Russell often feared the ridicule of his grandmother, one of the campaigners for education of women. Russell had two siblings, brother Frank, and sister Rachel, in June 1874 Russells mother died of diphtheria, followed shortly by Rachels death. In January 1876, his father died of bronchitis following a period of depression. Frank and Bertrand were placed in the care of their staunchly Victorian paternal grandparents and his grandfather, former Prime Minister Earl Russell, died in 1878, and was remembered by Russell as a kindly old man in a wheelchair. His grandmother, the Countess Russell, was the dominant family figure for the rest of Russells childhood, the countess was from a Scottish Presbyterian family, and successfully petitioned the Court of Chancery to set aside a provision in Amberleys will requiring the children to be raised as agnostics. Her favourite Bible verse, Thou shalt not follow a multitude to do evil, the atmosphere at Pembroke Lodge was one of frequent prayer, emotional repression, and formality, Frank reacted to this with open rebellion, but the young Bertrand learned to hide his feelings
26.
Ernst Zermelo
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Ernst Friedrich Ferdinand Zermelo 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, Ernst Zermelo graduated from Berlins Luisenstädtisches Gymnasium in 1889. He then studied mathematics, physics and philosophy at the universities of Berlin, Halle and he finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations. Zermelo remained at the University of Berlin, where he was appointed assistant to Planck, in 1897, Zermelo went to Göttingen, at that time the leading centre for mathematical research in the world, where he completed his habilitation thesis in 1899. In 1910, Zermelo left Göttingen upon being appointed to the chair of mathematics at Zurich University and he was appointed to an honorary chair at Freiburg im Breisgau in 1926, which he resigned in 1935 because he disapproved of Adolf Hitlers regime. At the end of World War II and at his request, Zermelo began to work on the problems of set theory under Hilberts influence and in 1902 published his first work concerning the addition of transfinite cardinals. By that time he had discovered the so-called Russell paradox. In 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved the well-ordering theorem and this result brought fame to Zermelo, who was appointed Professor in Göttingen, in 1905. Zermelo began to set theory in 1905, in 1908. See the article on Zermelo set theory for an outline of this paper, together with the original axioms, in 1922, Adolf Fraenkel and Thoralf Skolem independently improved Zermelos axiom system. The resulting 8 axiom system, now called Zermelo-Fraenkel axioms, is now the most commonly used system for set theory. Proposed in 1931, the Zermelos navigation problem is an optimal control problem. The problem deals with a boat navigating on a body of water, the boat is capable of a certain maximum speed, and we want to derive the best possible control to reach D in the least possible time. Without considering external forces such as current and wind, the control is for the boat to always head towards D. Its path then is a segment from O to D. With consideration of current and wind, if the force applied to the boat is non-zero the control for no current. Zermelo, Ernst, Ebbinghaus, Heinz-Dieter, Fraser, Craig G. Kanamori, Akihiro, from Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931. Proof that every set can be well-ordered, 139−41, a new proof of the possibility of well-ordering, 183–98
27.
Cardinal number
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In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a set is a natural number, the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets, cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, in the case of finite sets, this agrees with the intuitive notion of size. In the case of sets, the behavior is more complex. It is also possible for a subset of an infinite set to have the same cardinality as the original set. There is a sequence of cardinal numbers,0,1,2,3, …, n, …, ℵ0, ℵ1, ℵ2, …, ℵ α, …. This sequence starts with the natural numbers including zero, which are followed by the aleph numbers, the aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number, If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory and it is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra, and mathematical analysis. In category theory, the numbers form a skeleton of the category of sets. The notion of cardinality, as now understood, was formulated by Georg Cantor, cardinality can be used to compare an aspect of finite sets, e. g. the sets and are not equal, but have the same cardinality, namely three. Cantor applied his concept of bijection to infinite sets, e. g. the set of natural numbers N =, thus, all sets having a bijection with N he called denumerable sets and they all have the same cardinal number. This cardinal number is called ℵ0, aleph-null and he called the cardinal numbers of these infinite sets transfinite cardinal numbers. Cantor proved that any unbounded subset of N has the same cardinality as N and he later proved that the set of all real algebraic numbers is also denumerable. His proof used an argument with nested intervals, but in an 1891 paper he proved the result using his ingenious. The new cardinal number of the set of numbers is called the cardinality of the continuum. His continuum hypothesis is the proposition that c is the same as ℵ1 and this hypothesis has been found to be independent of the standard axioms of mathematical set theory, it can neither be proved nor disproved from the standard assumptions
28.
Grace Chisholm Young
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Grace Chisholm Young was an English mathematician. Her early writings were published under the name of her husband, William Henry Young, for her work on calculus, she was awarded the Gamble Prize for Mathematics by Girton College, University of Cambridge. She was the youngest of three surviving children and her father was a senior civil servant, with the title Warden of the Standards in charge of the Weights and Measures Department. The two girls were taught at home by their mother, father and a governess which was the custom during that time and her family encouraged her to become involved in social work, helping the poor in London. She had aspirations of studying medicine, but her family would not allow this, however, Chisholm wanted to continue her studies. She passed the examination for entrance into Cambridge University at the age of 17. Chisholm entered Girton College in 1889 aged 22, four years after she passed the entrance examination having been awarded the Sir Francis Goldsmid Scholarship by the college. At this time the college was associated with the University of Cambridge with men and women graded on separate. Although she wanted to study medicine, her mother would not permit this, so, supported by ther father, at the end of her first year, when the Mays list came out, top of the Second class immediately below Isabel Maddison. In 1893, Grace passed her final examinations with the equivalent of a first-class degree and she also took the exam for the Final Honours School in mathematics at the University of Oxford in 1892 in which she out-performed all the Oxford students. As a result, she became the first person to obtain a First class degree at both Oxford and Cambridge Universities in any subject, Chisholm remained at Cambridge for an additional year to complete Part II of the Mathematical Tripos, which was unusual for women. She wanted to continue her studies and since women were not yet admitted to schools in England she went to the University of Göttingen in Germany to study with Felix Klein. This was one of the major centres in the world. The decision to admit her had to be approved by the Berlin Ministry of Culture and was part of an experiment in admitting women to university studies. In 1895, at the age of 27, Chisholm became the first woman to be awarded a doctorate in any field in Germany. After returning to England in 1896 to marry, she resumed research she had initiated at Gӧttingen into an equation to determine the orbit of a comet and her husband continued his work coaching in mathematics. However, in 1897 they both returned to Gӧttingen, encouraged by Felix Klein, both attend advanced lectures and while she continued her mathematical research her husband started to work creatively for the first time. They visited Turin in Italy to study geometry and under Kleins guidance they becan to work in the new area of set theory
29.
Cambridge University Press
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Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the worlds oldest publishing house and it also holds letters patent as the Queens Printer. The Presss mission is To further the Universitys mission by disseminating knowledge in the pursuit of education, learning, Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global presence, publishing hubs, and offices in more than 40 countries. Its publishing includes journals, monographs, reference works, textbooks. Cambridge University Press is an enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest publishing house in the world and the oldest university press and it originated from Letters Patent granted to the University of Cambridge by Henry VIII in 1534, and has been producing books continuously since the first University Press book was printed. Cambridge is one of the two privileged presses, authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Bertrand Russell, and Stephen Hawking. In 1591, Thomass successor, John Legate, printed the first Cambridge Bible, the London Stationers objected strenuously, claiming that they had the monopoly on Bible printing. The universitys response was to point out the provision in its charter to print all manner of books. In July 1697 the Duke of Somerset made a loan of £200 to the university towards the house and presse and James Halman, Registrary of the University. It was in Bentleys time, in 1698, that a body of scholars was appointed to be responsible to the university for the Presss affairs. The Press Syndicates publishing committee still meets regularly, and its role still includes the review, John Baskerville became University Printer in the mid-eighteenth century. Baskervilles concern was the production of the finest possible books using his own type-design, a technological breakthrough was badly needed, and it came when Lord Stanhope perfected the making of stereotype plates. This involved making a mould of the surface of a page of type. The Press was the first to use this technique, and in 1805 produced the technically successful, under the stewardship of C. J. Clay, who was University Printer from 1854 to 1882, the Press increased the size and scale of its academic and educational publishing operation. An important factor in this increase was the inauguration of its list of schoolbooks, during Clays administration, the Press also undertook a sizable co-publishing venture with Oxford, the Revised Version of the Bible, which was begun in 1870 and completed in 1885. It was Wright who devised the plan for one of the most distinctive Cambridge contributions to publishing—the Cambridge Histories, the Cambridge Modern History was published between 1902 and 1912
30.
Zermelo
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Ernst Friedrich Ferdinand Zermelo 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, Ernst Zermelo graduated from Berlins Luisenstädtisches Gymnasium in 1889. He then studied mathematics, physics and philosophy at the universities of Berlin, Halle and he finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations. Zermelo remained at the University of Berlin, where he was appointed assistant to Planck, in 1897, Zermelo went to Göttingen, at that time the leading centre for mathematical research in the world, where he completed his habilitation thesis in 1899. In 1910, Zermelo left Göttingen upon being appointed to the chair of mathematics at Zurich University and he was appointed to an honorary chair at Freiburg im Breisgau in 1926, which he resigned in 1935 because he disapproved of Adolf Hitlers regime. At the end of World War II and at his request, Zermelo began to work on the problems of set theory under Hilberts influence and in 1902 published his first work concerning the addition of transfinite cardinals. By that time he had discovered the so-called Russell paradox. In 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved the well-ordering theorem and this result brought fame to Zermelo, who was appointed Professor in Göttingen, in 1905. Zermelo began to set theory in 1905, in 1908. See the article on Zermelo set theory for an outline of this paper, together with the original axioms, in 1922, Adolf Fraenkel and Thoralf Skolem independently improved Zermelos axiom system. The resulting 8 axiom system, now called Zermelo-Fraenkel axioms, is now the most commonly used system for set theory. Proposed in 1931, the Zermelos navigation problem is an optimal control problem. The problem deals with a boat navigating on a body of water, the boat is capable of a certain maximum speed, and we want to derive the best possible control to reach D in the least possible time. Without considering external forces such as current and wind, the control is for the boat to always head towards D. Its path then is a segment from O to D. With consideration of current and wind, if the force applied to the boat is non-zero the control for no current. Zermelo, Ernst, Ebbinghaus, Heinz-Dieter, Fraser, Craig G. Kanamori, Akihiro, from Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931. Proof that every set can be well-ordered, 139−41, a new proof of the possibility of well-ordering, 183–98
31.
Abraham Fraenkel
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Abraham Halevi Fraenkel, known as Abraham Fraenkel, was a German-born Israeli mathematician. He was an early Zionist and the first Dean of Mathematics at the Hebrew University of Jerusalem and he is known for his contributions to axiomatic set theory, especially his additions to Ernst Zermelos axioms, which resulted in the Zermelo–Fraenkel axioms. After leaving Marburg in 1928, Fraenkel taught at the University of Kiel for a year and he then made the fateful choice of accepting a position at the Hebrew University of Jerusalem, which had been founded four years earlier, where he spent the rest of his career. He became the first Dean of the Faculty of Mathematics, Fraenkel was a fervent Zionist and as such was a member of Jewish National Council and the Jewish Assembly of Representatives under the British mandate. He also belonged to the Mizrachi religious wing of Zionism, which promoted Jewish religious education and schools, fraenkels early work was on Kurt Hensels p-adic numbers and on the theory of rings. He is best known for his work on set theory. In 1922 and 1925, he published two papers that sought to improve Zermelos axiomatic system, the result is the Zermelo–Fraenkel axioms, Fraenkel worked in set theory and foundational mathematics. Fraenkel also was interested in the history of mathematics, writing in 1920 and 1930 about Gausss works in algebra, after retiring from the Hebrew University and being succeeded by his former student Abraham Robinson, Fraenkel continued teaching at the Bar Ilan University in Ramat Gan. In 1956, Fraenkel was awarded the Israel Prize, for exact sciences, bestimmung des Datums des jüdischen Osterfestes für die Zeitrechnung der Mohammedaner. In Zeitschrift für Mathematik und naturwissenschaft Unterricht, eine Formel zur Verwandlung jüdischer Daten in mohammedanische. In Monatsschrift für Geschichte und Wissenschaft des Judentums,53, journal für die reine und angewandte Mathematik, vol 138. Materialien für eine wissenschaftliche Biographie von Gauss, die neueren Ideen zur Grundlagung der Analysis und Mengenlehre. The notion of definite and the independence of the axiom of choice in Jean van Heijenoort,1967, from Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931. Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre, Mathematische Annalen 86, 230-7, die neueren Ideen zur Grundlegung der Analysis und Mengenlehre in Jahrsebericht Der Deutschen Mathematiker-Vereinigung, Vol 33, 97-103. 1924b The Jewish University in Jerusalem, zehn Vorlesungen über die Grundlegung der Mengenlehre. Georg Cantor, in Jahresbericht der Deutschen Mathematiker-Vereinigung 39, 189-266, also appeared separately as Georg Cantor Leipzig, B. G. Teubner and is abridged in Cantor’s Gesammelte Abhandlungen. Part 1 in ההד VI 16-19, part 2 in ההד VI, Reprinted together as a monograph by ההד in 1931 Reprinted in 1987-8. Translated by Mark Zelcer in Hakirah vol 12, die heutigen Gegensätze in der Grundlegung der Mathematik
32.
Thoralf Skolem
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Thoralf Albert Skolem was a Norwegian mathematician who worked in mathematical logic and set theory. Although Skolems father was a school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania, passing the university examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology, in 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled Investigations on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the zodiacal light, in 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science, Skolem did not at first formally enroll as a Ph. D. candidate, believing that the Ph. D. was unnecessary in Norway. He later changed his mind and submitted a thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations and his notional thesis advisor was Axel Thue, even though Thue had died in 1922. In 1927, he married Edith Wilhelmine Hasvold, Skolem continued to teach at Det kongelige Frederiks Universitet until 1930 when he became a Research Associate in Chr. This senior post allowed Skolem to conduct research free of administrative, in 1938, he returned to Oslo to assume the Professorship of Mathematics at the university. There he taught the courses in algebra and number theory. Skolems Ph. D. student Øystein Ore went on to a career in the USA, Skolem served as president of the Norwegian Mathematical Society, and edited the Norsk Matematisk Tidsskrift for many years. He was also the editor of Mathematica Scandinavica. After his 1957 retirement, he made trips to the United States. He remained intellectually active until his sudden and unexpected death, for more on Skolems academic life, see Fenstad. Skolem published around 180 papers on Diophantine equations, group theory, lattice theory and he mostly published in Norwegian journals with limited international circulation, so that his results were occasionally rediscovered by others. An example is the Skolem–Noether theorem, characterizing the automorphisms of simple algebras, Skolem published a proof in 1927, but Emmy Noether independently rediscovered it a few years later. Skolem was among the first to write on lattices, in 1912, he was the first to describe a free distributive lattice generated by n elements. In 1919, he showed that every lattice is distributive and, as a partial converse
33.
Henri Lebesgue
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His theory was published originally in his dissertation Intégrale, longueur, aire at the University of Nancy during 1902. Henri Lebesgue was born on 28 June 1875 in Beauvais, Oise, Lebesgues father was a typesetter and his mother was a school teacher. His parents assembled at home a library that the young Henri was able to use and his father died of tuberculosis when Lebesgue was still very young and his mother had to support him by herself. In 1894 Lebesgue was accepted at the École Normale Supérieure, where he continued to focus his energy on the study of mathematics, graduating in 1897. At the same time he started his studies at the Sorbonne. In 1899 he moved to a position at the Lycée Central in Nancy. In 1902 he earned his Ph. D. from the Sorbonne with the thesis on Integral, Length, Area, submitted with Borel, four years older. Lebesgue married the sister of one of his students, and he. After publishing his thesis, Lebesgue was offered in 1902 a position at the University of Rennes, lecturing there until 1906, in 1910 Lebesgue moved to the Sorbonne as a maître de conférences, being promoted to professor starting with 1919. In 1921 he left the Sorbonne to become professor of mathematics at the Collège de France, in 1922 he was elected a member of the Académie des Sciences. Henri Lebesgue died on 26 July 1941 in Paris, Lebesgues first paper was published in 1898 and was titled Sur lapproximation des fonctions. It dealt with Weierstrass theorem on approximation to continuous functions by polynomials, between March 1899 and April 1901 Lebesgue published six notes in Comptes Rendus. The first of these, unrelated to his development of Lebesgue integration, Lebesgues great thesis, Intégrale, longueur, aire, with the full account of this work, appeared in the Annali di Matematica in 1902. The first chapter develops the theory of measure, in the second chapter he defines the integral both geometrically and analytically. The next chapters expand the Comptes Rendus notes dealing with length, area, the final chapter deals mainly with Plateaus problem. This dissertation is considered to be one of the finest ever written by a mathematician and his lectures from 1902 to 1903 were collected into a Borel tract Leçons sur lintégration et la recherche des fonctions primitives. The problem of integration regarded as the search for a function is the keynote of the book. Lebesgue presents the problem of integration in its context, addressing Augustin-Louis Cauchy, Peter Gustav Lejeune Dirichlet
34.
Category theory
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Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows. A category has two properties, the ability to compose the arrows associatively and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, several terms used in category theory, including the term morphism, are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself, Category theory has practical applications in programming language theory, in particular for the study of monads in functional programming. Categories represent abstraction of other mathematical concepts, many areas of mathematics can be formalised by category theory as categories. Hence category theory uses abstraction to make it possible to state and prove many intricate, a basic example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However, the objects of a category need not be sets, any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category—and all the results of category theory apply to it. The arrows of category theory are said to represent a process connecting two objects, or in many cases a structure-preserving transformation connecting two objects. There are, however, many applications where more abstract concepts are represented by objects. The most important property of the arrows is that they can be composed, in other words, linear algebra can also be expressed in terms of categories of matrices. A systematic study of category theory allows us to prove general results about any of these types of mathematical structures from the axioms of a category. The class Grp of groups consists of all objects having a group structure, one can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proven from the axioms that the identity element of a group is unique, in the case of groups, the morphisms are the group homomorphisms. The study of group homomorphisms then provides a tool for studying properties of groups. Not all categories arise as structure preserving functions, however, the example is the category of homotopies between pointed topological spaces. If one axiomatizes relations instead of functions, one obtains the theory of allegories, a category is itself a type of mathematical structure, so we can look for processes which preserve this structure in some sense, such a process is called a functor. Diagram chasing is a method of arguing with abstract arrows joined in diagrams. Functors are represented by arrows between categories, subject to specific defining commutativity conditions, functors can define categorical diagrams and sequences
