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This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis-finding the "right axioms" to prove fundamental theorems-and giving a novel approach to logic.Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it.By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

Reverse mathematics. --- Logic, Symbolic and mathematical

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This is a textbook for an undergraduate mathematics major transition course from technique-based mathematics (such as Algebra and Calculus) to proof-based mathematics. It motivates the introduction of the formal language of logic and set theory and develops the basics with examples, exercises with solutions and exercises without. It then moves to a discussion of proof structure and basic proof techniques, including proofs by induction with extensive examples. An in-depth treatment of relations, particularly equivalence and order relations completes the exposition of the basic language of mathematics. The last chapter treats infinite cardinalities. An appendix gives some complement on induction and order, and another provides full solutions of the in-text exercises. The primary audience is undergraduate mathematics major, but independent readers interested in mathematics can also use the book for self-study.

Proof theory. --- Structures and Proofs. --- Logic, Symbolic and mathematical

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This textbook provides a concise and self-contained introduction to mathematical logic, with a focus on the fundamental topics in first-order logic and model theory. Including examples from several areas of mathematics (algebra, linear algebra and analysis), the book illustrates the relevance and usefulness of logic in the study of these subject areas. 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. The book then explores the formal axiom system of Zermelo and Fraenkel before concluding with an extensive list of suggestions for further study. The present volume is primarily aimed at mathematics students who are already familiar with basic analysis, algebra and linear algebra. It contains numerous exercises of varying difficulty and can be used for self-study, though it is ideally suited as a text for a one-semester university course in the second or third year.

Proof theory. --- Algebra. --- Structures and Proofs. --- Mathematics --- Mathematical analysis --- Logic, Symbolic and mathematical --- Logic, Symbolic and mathematical.

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Pragmatics --- Pragmalinguistics --- General semantics --- Language and languages --- Logic, Symbolic and mathematical --- Semantics (Philosophy) --- Philosophy

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Machine learning is widely used for data analysis. Dynamic fuzzy data are one of the most difficult types of data to analyse in the field of big data, cloud computing, the Internet of Things, and quantum information. At present, the processing of this kind of data is not very mature. The authors carried out more than 20 years of research, and show in this book their most important results. The seven chapters of the book are devoted to key topics such as dynamic fuzzy machine learning models, dynamic fuzzy self-learning subspace algorithms, fuzzy decision tree learning, dynamic concepts based on dynamic fuzzy sets, semi-supervised multi-task learning based on dynamic fuzzy data, dynamic fuzzy hierarchy learning, examination of multi-agent learning model based on dynamic fuzzy logic. This book can be used as a reference book for senior college students and graduate students as well as college teachers and scientific and technical personnel involved in computer science, artificial intelligence, machine learning, automation, data analysis, mathematics, management, cognitive science, and finance. It can be also used as the basis for teaching the principles of dynamic fuzzy learning.

Fuzzy logic. --- Nonlinear logic --- Fuzzy mathematics --- Logic, Symbolic and mathematical --- Fuzzy systems