“…AC, the axiom of choice, because of its non-constructive character, is the most controversial mathematical axiom, shunned by some, used indiscriminately by others. …”
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“…Following an overview of basic notions in combinatorics and first-order logic, the author outlines the main topics of classical set theory in the second part, including Ramsey theory and the axiom of choice. The revised edition contains new permutation models and recent results in set theory without the axiom of choice. …”
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“…Ernst Zermelo (1871-1953) is regarded as the founder of axiomatic set theory and best-known for the first formulation of the axiom of choice. This book presents a biography of Zermelo, and also covers set theory, the foundations of mathematics, and pure mathematics.…”
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“…Some emphasis is put on spaces of vector-valued functions and on the constructive viewpoint of the theory (without the axiom of choice). The reader should have basic knowledge in functional analysis and measure theory.…”
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“…The text briefly describes the theory of sets, binary relations, equivalence relations, partial ordering, minimum condition, and theorems equivalent to the axiom of choice. The text gives the definition of binary algebraic operation and the concepts of groups, groupoids, and semigroups. …”
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“…Set Theory includes elementary logic, sets, relations, functions, denumerable and non-denumerable sets, cardinal numbers, Cantor's theorem, axiom of choice, and order relations.…”
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“…This book is an introduction into modern cardinal arithmetic in the frame of the axioms of Zermelo-Fraenkel set theory together with the axiom of choice (ZFC). A first part describes the classical theory developed by Bernstein, Cantor, Hausdorff, König and Tarski between 1870 and 1930. …”
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“…Applications of the axiom of choice are also discussed, along with infinite games and the axiom of determinateness.Comprised of three chapters, this volume begins with an overview of naïve set theory and some important sets and notations. …”
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“…The covered topics include: set theory and cardinal arithmetic; axiom of choice and Zorn's lemma; topological spaces and continuous functions; connectedness and compactness; Alexandrov compactification; quotient topologies; countability and separation axioms; prebasis and Alexander's theorem; the Tychonoff theorem and paracompactness; complete metric spaces and function spaces; Baire spaces; homotopy of maps; the fundamental group; the van Kampen theorem; covering spaces; Brouwer and Borsuk's theorems; free groups and free product of groups; and basic category theory. …”
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“…At first glance, Robinson's original form of nonstandard analysis appears nonconstructive in essence, because it makes a rather unrestricted use of classical logic and set theory and, in particular, of the axiom of choice. Recent developments, however, have given rise to the hope that the distance between constructive and nonstandard mathematics is actually much smaller than it appears. …”
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“…Zermelo is best-known for the statement of the axiom of choice and his axiomatization of set theory. …”
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“…The authors start with an exposition of set theory and the axiom of choice as used in everyday mathematics. Proceeding at a gentle pace, they go on to present some of the first important results in model theory, followed by a careful exposition of Gentzen-style natural deduction and a detailed proof of Gödel’s completeness theorem for first-order logic. …”
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“…We prove that under a reasonable condition on the comparability of alternatives, there exists a unique collection of scores that produces accurate estimates of the overall performance of each alternative and satisfies a well-known axiom regarding choice probabilities. We apply the method to several problems in which varying choice sets and continuous outcomes may create problems for standard scoring methods. …”
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“…Moreover, we prove that under the Axiom of Choice, the existence of an empirical risk minimization (ERM) rule that has a vanishing learning capacity is equivalent to the assertion that the hypothesis space has a finite Vapnik–Chervonenkis (VC) dimension, thus establishing an equivalence relation between two of the most fundamental concepts in statistical learning theory and information theory. …”
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“…In 1963, the first author introduced a course in set theory at the Uni­ versity of Illinois whose main objectives were to cover G6del's work on the consistency of the axiom of choice (AC) and the generalized con­ tinuum hypothesis (GCH), and Cohen's work on the independence of AC and the GCH. …”
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