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Set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions among many researchers, Set theory, however, was founded by a single paper in 1874 by Georg Cantor, On a Property of the Collection of All Real Algebraic Numbers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day, while Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined

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Robert M. Solovay
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Robert Martin Solovay is an American mathematician specializing in set theory. Solovay earned his Ph. D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, Solovay has spent his career at the University of California at Berkeley, where his Ph. D. students include W. Hugh Woodin and Matthew Foreman. Outside of set theory, developing the Solovay–Strassen primality test, used to large natural numbers that are prime with high probability. This method has had implications for cryptography, proving that GL completely axiomatizes the logic of the provability predicate of Peano Arithmetic. With Alexei Kitaev, proving that a set of quantum gates can efficiently approximate an arbitrary unitary operator on one qubit. A model of set-theory in which set of reals is Lebesgue measurable. A nonconstructible Δ13 set of integers, transactions of the American Mathematical Society. Solovay, Robert M. and Volker Strassen, a fast Monte-Carlo test for primality. Provability logic Robert M. Solovay at the Mathematics Genealogy Project Robert Solovay at DBLP Bibliography Server

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Dagger (typography)
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A dagger or obelisk is a typographical symbol usually used to indicate a footnote if an asterisk has already been used. It is present in Unicode as U+2020 † dagger, the term obelisk derives from the Greek ὀβελίσκος, which means little obelus, from ὀβελός meaning roasting spit. It was originally represented by the subtraction and division symbols by Ancient Greek scholars as critical marks in manuscripts, a double dagger or diesis is a variant with two handles that is usually used for a third footnote after the asterisk and dagger. In Unicode, it is encoded as U+2021 ‡ double dagger, the triple dagger, a variant with three handles, was accepted by the Unicode Technical Committee in 2016. The dagger symbol originated from a variant of the obelus, originally depicted by a line or a line with one or two dots. It represented an iron roasting spit, a dart, or the end of a javelin. The obelus is believed to have been invented by the Homeric scholar Zenodotus as one of a system of editorial symbols and they were used to mark questionable or corrupt words or passages in manuscripts of the Homeric epics. While the asterisk was used for additions, the obelus was used for corrective deletions of invalid reconstructions. It was used when non-attested words are reconstructed for the sake of argument only and they were used to indicate the end of a marked passage. It was used much in the way by later scholars to mark differences between various translations or versions of the Bible and other manuscripts. The early Christian Alexandrian scholar Origen used it as a method of indicating differences between different versions of the Old Testament in his Hexapla, epiphanius of Salamis used both a horizontal slash or hook and an upright and slightly slanting dagger to represent an obelus. St. Jerome used a horizontal slash for an obelus. He describes the use of the asterisk and the dagger as, an asterisk makes a light shine, the obelus accompanied by points is used when we do not know whether a passage should be suppressed or not. Medieval scribes used the symbols extensively for critical markings of manuscripts, in addition to this, the dagger was also used in notations in early Christianity, to indicate a minor intermediate pause in the chanting of Psalms, equivalent to the quaver rest notation. It is also used to indicate a breath mark when reciting, along with the asterisk, in the sixteenth century, the printer and scholar Robert Estienne used it to mark differences in the words or passages between different printed versions of the Greek New Testament. The obelus was also used as a mathematical symbol for subtraction. It was first used as a symbol for division by the Swiss mathematician Johann Rahn in his book Teutsche Algebra in 1659 and this gave rise to the modern mathematical symbol ÷. Due to the variations as to the different uses of the different forms of obeli, the lemniscus and its variant, the hypolemniscus, is sometimes considered to be different from other obeli