1.
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

2.
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

3.
ZFC
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Zermelo–Fraenkel set theory with the historically controversial axiom of choice included is commonly abbreviated ZFC, where C stands for choice. Many authors use ZF to refer to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded, today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. ZFC is intended to formalize a single primitive notion, that of a hereditary well-founded set, thus the axioms of ZFC refer only to pure sets and prevent its models from containing urelements. Furthermore, proper classes can only be treated indirectly, specifically, ZFC does not allow for the existence of a universal set nor for unrestricted comprehension, thereby avoiding Russells paradox. Von Neumann–Bernays–Gödel set theory is a commonly used conservative extension of ZFC that does allow explicit treatment of proper classes, formally, ZFC is a one-sorted theory in first-order logic. The signature has equality and a primitive binary relation, set membership. The formula a ∈ b means that the set a is a member of the set b, there are many equivalent formulations of the ZFC axioms. Most of the ZFC axioms state the existence of particular sets defined from other sets, for example, the axiom of pairing says that given any two sets a and b there is a new set containing exactly a and b. Other axioms describe properties of set membership, a goal of the ZFC axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe. The metamathematics of ZFC has been extensively studied, landmark results in this area established the independence of the continuum hypothesis from ZFC, and of the axiom of choice from the remaining ZFC axioms. The consistency of a such as ZFC cannot be proved within the theory itself. In 1908, Ernst Zermelo proposed the first axiomatic set theory, moreover, one of Zermelos axioms invoked a concept, that of a definite property, whose operational meaning was not clear. They also independently proposed replacing the axiom schema of specification with the schema of replacement. Appending this schema, as well as the axiom of regularity, adding to ZF either the axiom of choice or a statement that is equivalent to it yields ZFC. There are many equivalent formulations of the ZFC axioms, for a discussion of this see Fraenkel, Bar-Hillel & Lévy 1973, the following particular axiom set is from Kunen. The axioms per se are expressed in the symbolism of first order logic, the associated English prose is only intended to aid the intuition. All formulations of ZFC imply that at least one set exists, Kunen includes an axiom that directly asserts the existence of a set, in addition to the axioms given below. Its omission here can be justified in two ways, first, in the standard semantics of first-order logic in which ZFC is typically formalized, the domain of discourse must be nonempty

4.
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

5.
Tamara Awerbuch-Friedlander
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Tamara Eugenia Awerbuch-Friedlander, PhD, is a biomathematician and public health scientist at Harvard School of Public Health in Boston, Massachusetts. Her primary research and publications focus on biosocial interactions that cause or contribute to disease and she also is believed to be the first female Harvard Faculty member to have had a jury trial for a lawsuit filed against Harvard University for sex discrimination. Currently, she is an instructor in the Department of Global Health, since the beginning of this century, she has organized and carried out research on conditions that lead to the emergence, maintenance, and spread of epidemics. Her research encompasses sexually-transmitted diseases such as HIV/AIDS, as well as diseases, such as Lyme disease, dengue. Dr. Awerbuch-Friedlander recently researched the spread and control of rabies based on an eco-historical analysis, ecology and evolutionary biology have traditionally been the dominant fields of mathematical biology, but public health fields are effectively contextualized within them. Conditions contributing to the emergence of epidemics are complex in nature, involving biological, ecological, behavioral, environmental, most of her research mathematically models these factors as systems that lend themselves to qualitative and quantitative analysis. These models can be used to explore the effect of each factor in the presence of the others as well as new interventions, some of her analytical mathematical models led to fundamental epidemiological discoveries, for example, that oscillations are an intrinsic property of tick dynamics. This means that a decrease in abundance in one year does not necessary imply that the same will happen in the next. She studied and completed two degrees at Hebrew University in Jerusalem and she studied chemistry and minored in biochemistry and completed the BSc degree in 1965. In 1967, she completed both the Master of Science in Physiology and the Master of Education degree from Hebrew University. She was certified to teach all grades, K–12, in Israel and she also served for two years in the Israeli army. Then, in Summer 1975, she matriculated as a full-time MIT student and she became a US citizen and has resided in the United States since that time. She enjoys her life as an international research academic and travels often to South America. She was recruited in 1983 to the Biostatistics Department of the Harvard T. H. Chan School of Public Health by Department Chair Marvin Zelen. In 1993, she began a career in the Department of Global Health. Her two sons, Danny and Ari, who were born in the 1980s and were reared in Brookline, Massachusetts, regularly accompanied her international travels for lectures, undergraduate study at Hebrew University in Israel. Awerbuch-Friedlander is a member of the New and Resurgent Disease Working Group. Within this context, she was involved in organizing a conference in Woods Hole on the emergence and resurgence of diseases, where she led the workshop on Mathematical Modeling

6.
International Standard Serial Number
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An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication. The ISSN is especially helpful in distinguishing between serials with the same title, ISSN are used in ordering, cataloging, interlibrary loans, and other practices in connection with serial literature. The ISSN system was first drafted as an International Organization for Standardization international standard in 1971, ISO subcommittee TC 46/SC9 is responsible for maintaining the standard. When a serial with the content is published in more than one media type. For example, many serials are published both in print and electronic media, the ISSN system refers to these types as print ISSN and electronic ISSN, respectively. 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 form of the ISSN code can be expressed as follows, NNNN-NNNC where N is in the set, a digit character. The ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the check digit, 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, the modulus 11 of the sum must be 0. There is an online ISSN checker that can validate an ISSN, ISSN codes are assigned by a network of ISSN National Centres, usually located at national libraries and coordinated by the ISSN International Centre based in Paris. The International Centre is an organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, at the end of 2016, the ISSN Register contained records for 1,943,572 items. ISSN and ISBN codes are similar in concept, where ISBNs are assigned to individual books, 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 identifier associated with a serial title. For this reason a new ISSN is assigned to a serial each time it undergoes a major title change, separate ISSNs are needed for serials in different media. Thus, the print and electronic versions of a serial need separate ISSNs. Also, 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

7.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker