Axiom of choice
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. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of one object from each bin if the collection is infinite. Formally, 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. In many cases, such a selection can be made without invoking the axiom of choice. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. in the smallest elements are. In this case, "select the smallest number" is a choice function. If infinitely many sets were collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set.
That is, the choice function provides the set of chosen elements. However, no choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked. Bertrand Russell coined an analogy: for any collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection. For an infinite collection of pairs of socks, there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice. Although controversial, the axiom of choice is now used without reservation by most mathematicians, it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice. One motivation for this use is that a number of accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy.
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, such that for every set A in X, f is an element of A. With this concept, the axiom can be stated: Formally, this may be expressed as follows: ∀ X. Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no choice function; each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; the axiom of choice asserts the existence of such elements. In this article and other discussions of the Axiom of Choice the following abbreviations are common: AC – the Axiom of Choice. 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 other basic axioms of set theory, they imply the axiom of choice and are implied by it. One variation avoids the use of choice functions by, in effect, replacing each choice function with its range. 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 one element from each part of the partition. Another equivalent axiom only considers collections X that are powersets of other sets: For any set A, the power set of A has a choice function. Authors who use this formulation speak of the choice function on A, but be advised that this is a different notion of choice function, its domain is the powerset of A, and
In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting and bending, but not tearing or gluing. An n-dimensional topological space is a space with certain properties of connectedness and compactness; the space discrete. It can be closed. Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space and transformation; such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems; the term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics. Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.
Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realised the importance of the edges of a polyhedron; this led to his polyhedron formula, V − E + F = 2. Some authorities regard this analysis as the first theorem. Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print; the English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". The term "topologist" in the sense of a specialist in topology was used in 1905 in the magazine Spectator.
Their work was corrected and extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space. A topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Modern topology depends on the ideas of set theory, developed by Georg Cantor in the part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series.
For further developments, see point-set topology and algebraic topology. Topology can be formally defined as "the study of qualitative properties of certain objects that are invariant under a certain kind of transformation those properties that are invariant under a certain kind of invertible transformation." Topology is used to refer to a structure imposed upon a set X, a structure that characterizes the set X as a topological space by taking proper care of properties such as convergence and continuity, upon transformation. Topological spaces show up in every branch of mathematics; this has made topology one of the great unifying ideas of mathematics. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside.
In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg that would cross each of its seven bridges once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks; this Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory. The hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is convincing to most people though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of t
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Charles Ehresmann was a German born French mathematician who worked in differential topology and category theory. He was an early member of the Bourbaki group, is known for his work on the differential geometry of smooth fiber bundles, notably the Ehresmann connection, the concept of jets of a smooth map, his seminar on category theory. Ehresmann was born in Strasbourg to a family, he attended school in Strasbourg in 1924 went to university at the École Normale Supérieure in Paris. On graduating in 1927 he did one year of military service, taught at a French school at Rabat in Morocco, he studied further at the University of Göttingen during the years 1930–31, at Princeton University in 1932–34. He completed his Ph. D. thesis entitled Sur la topologie de certains espaces homogènes at ENS in 1934 under the supervision of Élie Cartan, became a researcher with the Centre national de la recherche scientifique. From 1935 to 1937 he contributed to the seminar of Gaston Julia, a forerunner of the Bourbaki seminar.
Ehresmann was a lecturer at the French University of Strasbourg in 1939, when the German occupation of France irrupted and the whole faculty was evacuated to Clermont-Ferrand. When Germany withdrew in 1945, he returned to Strasbourg. From 1955 he was Professor of Topology in Paris, his post was at the Sorbonne, but after the reorganization of Parisian universities in 1969 he moved to Paris Diderot University. He held visiting chairs at Yale University, Princeton University, in Brazil, Buenos Aires, Mexico City and the Tata Institute of Fundamental Research in Bombay, he was President of the Société Mathématique de France in 1965. He retired in 1975. After retirement and until 1978 he gave lectures at the University of Picardy at Amiens, to which he moved because his second wife, Andrée Charles-Ehresmann, was a professor of mathematics there, he died at Amiens in 1979. Ehresmann first investigated the topology and homology of manifolds associated with classical Lie groups, such as Grassmann manifolds and other homogeneous spaces.
He developed the concept of fiber bundle, building on work by Hassler Whitney. Norman Steenrod was working in the same direction in the USA, but Ehresmann was interested in differentiable fiber bundles, in differential-geometric aspects of these, he was a pioneer of differential topology. By 1957, having become a leading proponent of categorical methods, he founded the mathematical journal Cahiers de Topologie et Géométrie Différentielle Catégoriques. Jean Dieudonné described Ehresmann's personality as "... distinguished by forthrightness and total absence of conceit or careerism. As a teacher he was outstanding, not so much for the brilliance of his lectures as for the inspiration and tireless guidance he generously gave to his research students... " He had 76 PhD students, including Georges Reeb, Wu Wenjun, André Haefliger, Valentin Poénaru, Daniel Tanré. His first student was Jacques Feldbau, his publications include the books Algèbre. His collected works, edited by his wife, appeared in seven volumes in 1980–1983.
Ehresmann's theorem Ehresmann connection International Conference "Charles Ehresmann: 100 ans" Université de Picardie Jules Verne à Amiens, 7-8-9 October 2005. Http://pagesperso-orange.fr/vbm-ehr/ChEh/indexAng.htm'The mathematical legacy of Charles Ehresmann', Proceedings of the 7th Conference on the Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann, Będlewo, 8.05.2005–15.05.2005, Edited by J. Kubarski, J. Pradines, T. Rybicki, R. Wolak, Banach Center Publications, vol. 76, Institute of Mathematics of the Polish Academy of Sciences, Warsaw, 2007. Https://www.impan.pl/pl/wydawnictwa/banach-center-publications/all/76 O'Connor, John J.. Charles Ehresmann at the Mathematics Genealogy Project Michèle Audin. "Publier sous l'Occupation. Autour du cas de Jacques Feldbau et de l'Académie des Sciences"
Steve Vickers (computer scientist)
Steve Vickers is a British mathematician and computer scientist. In the early 1980s, he wrote ROM firmware and manuals for three home computers, the Sinclair ZX81 and ZX Spectrum and the Jupiter Ace; the latter was produced by Jupiter Cantab, a short-lived company Vickers formed together with Richard Altwasser, after the two had left Sinclair Research. Since the late 1980s, Vickers has been an academic in the field of geometric logic, writing over 30 papers in scholarly journals on mathematical aspects of computer science, his book Topology via Logic has been influential over a range of fields. In October 2018, he retired as senior lecturer at the University of Birmingham; as announced on his University homepage, he continues to supervise PhD students at the university and focus on his research. Vickers graduated from King's College, Cambridge with a degree in mathematics and completed a PhD at Leeds University in mathematics. In 1980 he started working for Nine Tiles, which had written the Sinclair BASIC for the ZX80.
He was responsible for the adaptation of the 4K ZX80 ROM into the 8K ROM used in the ZX81 and wrote the ZX81 manual. He wrote most of the ZX Spectrum ROM, assisted with the user documentation. Vickers left in 1982 to form "Rainbow Computing Co." with Richard Altwasser. The company became Jupiter Cantab and they were together responsible for the development of the commercially unsuccessful Jupiter ACE, a competitor to the similar Sinclair ZX Spectrum. At the Department of Computing at Imperial College London, Vickers joined the Department of Pure Mathematics at the Open University before moving to the School of Computer Science at the University of Birmingham, where he is a senior lecturer and the research student tutor of the School of Computer Science. Vickers' main interest lies within geometric logic, his book Topology via Logic introduces topology from the point of view of some computational insights developed by Samson Abramsky and Mike Smyth. It stresses the point-free approach and can be understood as dealing with theories in the so-called geometric logic, known from topos theory and is a more stringent form of intuitionistic logic.
However, the book was written in the language of classical mathematics. Extending the ideas to toposes he found himself channelled into constructive mathematics in a geometric form and in Topical Categories of Domains he set out a geometrisation programme of, where possible, using this geometric mathematics as a tool for treating point-free spaces as though they had "enough points". Much of his subsequent work has been in case studies to show that, with suitable techniques, it was indeed possible to do useful mathematics geometrically. In particular, a notion of "geometric transformation of points to spaces" gives a natural fibrewise treatment of topological bundles. A recent project of his has been to connect this with the topos approaches to physics as developed by Chris Isham and others at Imperial College, Klaas Landsman's group at Radboud University Nijmegen. Steven Vickers, "An induction principle for consequence in arithmetic universes", Journal of Pure and Applied Algebra 216, ISSN 0022-4049, pp. 1705 – 2068, 2012.
Jung and Moshier, M. Andrew and Vickers, Steven, "Presenting dcpos and dcpo algebras", in Bauer, A. and Mislove, M. Proceedings of the 24th Conference on the Mathematical Foundations of Programming Semantics, pp. 209–229, Electronic Notes in Theoretical Computer Science, Elsevier, 2008. Steven Vickers, "Cosheaves and connectedness in formal topology", Annals of Pure and Applied Logic, ISSN 0168-0072, 2009. Steven Vickers, "A localic theory of lower and upper integrals", Mathematical Logic Quarterly, 54, pp. 109–103, 2008. Steven Vickers, "Locales and toposes as spaces", in Aiello and Pratt-Hartmann, Ian E. and van Benthem, Johan F. A. K. Springer, Handbook of Spatial Logics, Springer, 2007, ISBN 978-1-4020-5586-7, Chapter 8, pp. 429–496. Palmgren and Vickers, Steven, "Partial Horn logic and cartesian categories", Annals of Pure and Applied Logic, 145, pp. 314–353, ISSN 0168-0072, 2007. Steven Vickers, "Localic completion of generalized metric spaces I, Theory and Applications of Categories", ISSN 1201-561X, 14, pp. 328–356, 2005.
Steven Vickers, "Localic completion of generalized metric spaces II: Powerlocales, Journal of Logic and Analysis", ISSN 1759-9008, 1, pp. 1–48, 2009. Steven Vickers, "The double powerlocale and exponentiation: a case study in geometric logic", Theoretical Computer Science, ISSN 0304-3975, vol. 316, pp. 297–321, 2004. Steven Vickers, "Topical Categories of Domains", in Winskel, Proceedings of the CLICS workshop, Computer Science Department, Aarhus University, 1992. Vickers, S. J. "Topology via Constructive Logic", in Moss and Ginzburg and de Rijke, Logic and Computation Vol II, Proceedings of conference on Information-Theoretic Approaches to Logic and Computation, 1996, ISBN 1575861801, 157586181X, CSLI Publications, Stanford, pp. 336–345, 1999. Vickers, S. J. "Toposes pour les vraiment nuls", in Edalat, A. and Jourdan, S. and McCusker, G. Advances in Theory and Formal Methods of Computing 1996, ISBN 1-86094-031-5, Imperial College Press, London, pp. 1–12, 1996. Vickers, S. J. "Toposes pour les nuls", Techreport Doc96/4, Department of Computing, Imperial College, London.
Broda, K. and Eisenbach
International Standard Serial Number
An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication, such as a magazine. The ISSN is helpful in distinguishing between serials with the same title. ISSN are used in ordering, interlibrary loans, other practices in connection with serial literature; the ISSN system was first drafted as an International Organization for Standardization international standard in 1971 and published as ISO 3297 in 1975. ISO subcommittee TC 46/SC 9 is responsible for maintaining the standard; when a serial with the same content is published in more than one media type, a different ISSN is assigned to each media type. For example, many serials are published both in electronic media; the ISSN system refers to these types as electronic ISSN, respectively. Conversely, as defined in ISO 3297:2007, every serial in the ISSN system is assigned a linking ISSN the same as the ISSN assigned to the serial in its first published medium, which links together all ISSNs assigned to the serial in every medium.
The format of the ISSN is an eight digit code, divided by a hyphen into two four-digit numbers. As an integer number, it can be represented by the first seven digits; the last code digit, which may be 0-9 or an X, is a check digit. Formally, the general form of the ISSN code can be expressed as follows: NNNN-NNNC where N is in the set, a digit character, C is in; the ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the check digit, C=5. To calculate the check digit, the following algorithm may be used: Calculate the sum of the first seven digits of the ISSN multiplied by its position in the number, counting from the right—that is, 8, 7, 6, 5, 4, 3, 2, respectively: 0 ⋅ 8 + 3 ⋅ 7 + 7 ⋅ 6 + 8 ⋅ 5 + 5 ⋅ 4 + 9 ⋅ 3 + 5 ⋅ 2 = 0 + 21 + 42 + 40 + 20 + 27 + 10 = 160 The modulus 11 of this sum is calculated. For calculations, an upper case X in the check digit position indicates a check digit of 10. To confirm the check digit, calculate the sum of all eight digits of the ISSN multiplied by its position in the number, counting from the right.
The modulus 11 of the sum must be 0. There is an online ISSN checker. ISSN codes are assigned by a network of ISSN National Centres located at national libraries and coordinated by the ISSN International Centre based in Paris; the International Centre is an intergovernmental organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, the ISDS Register otherwise known as the ISSN Register. At the end of 2016, the ISSN Register contained records for 1,943,572 items. ISSN and ISBN codes are similar in concept. An ISBN might be assigned for particular issues of a serial, in addition to the ISSN code for the serial as a whole. An ISSN, unlike the ISBN code, is an anonymous identifier associated with a serial title, containing no information as to the publisher or its location. For this reason a new ISSN is assigned to a serial each time it undergoes a major title change. Since the ISSN applies to an entire serial a new identifier, the Serial Item and Contribution Identifier, was built on top of it to allow references to specific volumes, articles, or other identifiable components.
Separate ISSNs are needed for serials in different media. Thus, the print and electronic media versions of a serial need separate ISSNs. A CD-ROM version and a web version of a serial require different ISSNs since two different media are involved. However, the same ISSN can be used for different file formats of the same online serial; this "media-oriented identification" of serials made sense in the 1970s. In the 1990s and onward, with personal computers, better screens, the Web, it makes sense to consider only content, independent of media; this "content-oriented identification" of serials was a repressed demand during a decade, but no ISSN update or initiative occurred. A natural extension for ISSN, the unique-identification of the articles in the serials, was the main demand application. An alternative serials' contents model arrived with the indecs Content Model and its application, the digital object identifier, as ISSN-independent initiative, consolidated in the 2000s. Only in 2007, ISSN-L was defined in the