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The continuum hypothesis, introduced by mathematician George Cantor in 1877, states that there is no set of numbers between the integers and real numbers. It was later included as the first of mathematician David Hilbert's twenty-three unsolved math problems. In this book, Kurt Godel sets forth his proof for this problem.

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Based on von Neumann's lecture notes, this book begins with the development of the axioms of continuous geometry, dimension theory, and - for the irreducible case - the function D(a). The properties of regular rings are then discussed, and a variety of results are presented for lattices that are continuous geometries.

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This is a study of the theory of models with truth values in a compact Hausdorff topological space.

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The description for this book, Contributions to the Theory of Games (AM-24), Volume I, will be forthcoming.

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These two new collections, numbers 28 and 29 respectively in the Annals of Mathematics Studies, continue the high standard set by the earlier Annals Studies 20 and 24 by bringing together important contributions to the theories of games and of nonlinear differential equations.

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A new group of contributions to the development of this theory by leading experts in the field. The contributors include L. D. Berkovitz, L. E. Dubins, H. Everett, W. H. Fleming, D. Gale, D. Gillette, S. Karlin, J. G. Kemeny, R. Restrepo, H. E. Scarf, M. Sion, G. L. Thompson, P. Wolfe, and others.

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The description for this book, Contributions to the Theory of Games (AM-40), Volume IV, will be forthcoming.

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The description for this book, Contributions to the Theory of Nonlinear Oscillations (AM-36), Volume III, will be forthcoming.

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Annals of Mathematics Studies: Number 41The present volume of the Contributions, fourth in the series, covers, like its predecessors, a great variety of topics in non-linear differential equations.

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The description for this book, Contributions to the Theory of Nonlinear Oscillations (AM-45), Volume V, will be forthcoming.

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The description for this book, Contributions to the Theory of Partial Differential Equations. (AM-33), will be forthcoming.

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The description for this book, Contributions to the Theory of Riemann Surfaces. (AM-30), will be forthcoming.

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Splines, both interpolatory and smoothing, have a long and rich history that has largely been application driven. This book unifies these constructions in a comprehensive and accessible way, drawing from various methods and applications to show how they arise naturally in the theory of linear control systems.

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The description for this book, Convergence and Uniformity in Topology. (AM-2), will be forthcoming.

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Focuses on the philosophical relationship between mathematics and the natural sciences. This title shows that the axioms needed for the mathematics used in physics can also generate practically various fields of contemporary pure mathematics. It also provides a foundation for investigations in string theory and other areas of physics.

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Presents readers with a coherent branch of nonlinear mathematical analysis that is suited to the study of optimization problems. This volume treats the topics such as: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, and more.

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Explores an important aspect of number theory - the theory of exponential sums over finite fields and their Mellin transforms - from a categorical point of view. This book presents fundamentally important results and a plethora of examples, opening up new directions in the subject.

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The description for this book, Curvature and Betti Numbers. (AM-32), will be forthcoming.

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Aims to construct "motivic cohomology theory," whose existence was conjectured by A Beilinson and S Lichtenbaum. This is achieved in the book's fourth paper, using results of the other papers whose additional role is to contribute to our understanding of various properties of algebraic cycles.

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The description for this book, Degree of Approximation by Polynomials in the Complex Domain. (AM-9), will be forthcoming.

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The description for this book, Degrees of Unsolvability. (AM-55), will be forthcoming.