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Theorems and their proofs lie at the heart of mathematics. In speaking of the purely aesthetic qualities of theorems and proofs, G. H. Hardy wrote that in beautiful proofs 'there is a very high degree of unexpectedness, combined with inevitability and economy.' Charming Proofs present a collection of remarkable proofs in elementary mathematics that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, the proofs in this collection will invite readers to enjoy the beauty of mathematics, to share their discoveries with others, and to become involved in the process of creating new proofs. Charming Proofs is organized as follows. Following a short introduction about proofs and the process of creating proofs, the authors present, in twelve chapters, a wide and varied selection of proofs they consider charming, Topics include the integers, selected real numbers, points in the plane, triangles, squares, and other polygons, curves, inequalities, plane tilings, origami, colorful proofs, three-dimensional geometry, etc. At the end of each chapter are some challenges that will draw the reader into the process of creating charming proofs. There are over 130 such challenges. Charming Proofs concludes with solutions to all of the challenges, references, and a complete index. As in the authors’ previous books with the MAA (Math Made Visual and When Less Is More), secondary school and college and university teachers may wish to use some of the charming proofs in their classrooms to introduce their students to mathematical elegance. Some may wish to use the book as a supplement in an introductory course on proofs, mathematical reasoning, or problem solving.

Proof theory. --- Logic, Symbolic and mathematical --- Proof theory --- Théorie de la preuve

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Provability, Computability and Reflection

Mathematical logic --- Proof theory --- Théorie de la preuve --- ELSEVIER-B EPUB-LIV-FT --- Proof theory. --- 510.6 --- 510.6 Mathematical logic --- Logic, Symbolic and mathematical

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Modality (Logic) --- Proof theory --- Modalité (Logique) --- Théorie de la preuve --- Congresses. --- Congrès --- Modalité (Logique) --- Théorie de la preuve --- Congrès --- Modality (Logic) - Congresses. --- Proof theory - Congresses. --- Modalité (logique) --- Théorie de la démonstration

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This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (1979). Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency.

Mathematical logic --- Modalité (Logique) --- Théorie de la preuve --- Modalité (Logique) --- Modality (Logic) --- Proof theory. --- Modality (Logic). --- Proof theory --- Théorie de la preuve --- Modal logic --- Logic, Symbolic and mathematical --- Logic --- Nonclassical mathematical logic --- Bisimulation --- Arts and Humanities --- Philosophy

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Proof theory. --- Logic, Symbolic and mathematical. --- Logic, Symbolic and mathematical --- Proof theory --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Théorie de la preuve --- Logique symbolique et mathématique