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Kurt Gödel, mathematician and logician, was one of the most influential thinkers of the twentieth century. Gödel fled Nazi Germany, fearing for his Jewish wife and fed up with Nazi interference in the affairs of the mathematics institute at the University of Göttingen. In 1933 he settled at the Institute for Advanced Study in Princeton, where he joined the group of world-famous mathematicians who made up its original faculty. His 1940 book, better known by its short title, The Consistency of the Continuum Hypothesis, is a classic of modern mathematics. 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, famously delivered as a manifesto to the field of mathematics at the International Congress of Mathematicians in Paris in 1900. In The Consistency of the Continuum Hypothesis Gödel set forth his proof for this problem. In 1999, Time magazine ranked him higher than fellow scientists Edwin Hubble, Enrico Fermi, John Maynard Keynes, James Watson, Francis Crick, and Jonas Salk. He is most renowned for his proof in 1931 of the 'incompleteness theorem,' in which he demonstrated that there are problems that cannot be solved by any set of rules or procedures. His proof wrought fruitful havoc in mathematics, logic, and beyond.

Mathematical logic --- Mathematics --- Logic, Symbolic and mathematical --- Mathématiques --- Logique symbolique et mathématique --- Philosophy --- Philosophie --- Logic, Symbolic and mathematical. --- Philosophy. --- Set theory --- Théorie des ensembles --- Continuity --- Continu (philosophie) --- Algebra of logic --- Logic, Universal --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Syllogism --- Logic of mathematics --- Mathematics, Logic of --- Absoluteness. --- Addition. --- Axiom of choice. --- Axiom of extensionality. --- Axiom of infinity. --- Axiom. --- Axiomatic system. --- Boolean algebra (structure). --- Constructible set (topology). --- Continuum hypothesis. --- Existence theorem. --- Existential quantification. --- Integer. --- Mathematical induction. --- Mathematical logic. --- Mathematics. --- Metatheorem. --- Order by. --- Ordinal number. --- Propositional function. --- Quantifier (logic). --- Reductio ad absurdum. --- Requirement. --- Set theory. --- Theorem. --- Transfinite induction. --- Transfinite. --- Variable (mathematics). --- Well-order. --- Théorie des ensembles --- Logique mathématique --- Axiome du choix

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Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic. Ravenel's first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.

Homotopie --- Homotopy theory --- Homotopy theory. --- Deformations, Continuous --- Topology --- Abelian category. --- Abelian group. --- Adams spectral sequence. --- Additive category. --- Affine space. --- Algebra homomorphism. --- Algebraic closure. --- Algebraic structure. --- Algebraic topology (object). --- Algebraic topology. --- Algebraic variety. --- Algebraically closed field. --- Atiyah–Hirzebruch spectral sequence. --- Automorphism. --- Boolean algebra (structure). --- CW complex. --- Canonical map. --- Cantor set. --- Category of topological spaces. --- Category theory. --- Classification theorem. --- Classifying space. --- Cohomology operation. --- Cohomology. --- Cokernel. --- Commutative algebra. --- Commutative ring. --- Complex projective space. --- Complex vector bundle. --- Computation. --- Conjecture. --- Conjugacy class. --- Continuous function. --- Contractible space. --- Coproduct. --- Differentiable manifold. --- Disjoint union. --- Division algebra. --- Equation. --- Explicit formulae (L-function). --- Functor. --- G-module. --- Groupoid. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Hopf algebra. --- Hurewicz theorem. --- Inclusion map. --- Infinite product. --- Integer. --- Inverse limit. --- Irreducible representation. --- Isomorphism class. --- K-theory. --- Loop space. --- Mapping cone (homological algebra). --- Mathematical induction. --- Modular representation theory. --- Module (mathematics). --- Monomorphism. --- Moore space. --- Morava K-theory. --- Morphism. --- N-sphere. --- Noetherian ring. --- Noetherian. --- Noncommutative ring. --- Number theory. --- P-adic number. --- Piecewise linear manifold. --- Polynomial ring. --- Polynomial. --- Power series. --- Prime number. --- Principal ideal domain. --- Profinite group. --- Reduced homology. --- Ring (mathematics). --- Ring homomorphism. --- Ring spectrum. --- Simplicial complex. --- Simply connected space. --- Smash product. --- Special case. --- Spectral sequence. --- Steenrod algebra. --- Sub"ient. --- Subalgebra. --- Subcategory. --- Subring. --- Symmetric group. --- Tensor product. --- Theorem. --- Topological space. --- Topology. --- Vector bundle. --- Zariski topology.

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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.

Game theory. --- A priori probability. --- Accuracy and precision. --- Addition. --- Assignment problem. --- Basic solution (linear programming). --- Big O notation. --- Bilinear form. --- Binary game. --- Boolean algebra (structure). --- Brouwer fixed-point theorem. --- Calculation. --- Cartesian product. --- Cauchy sequence. --- Characteristic function (probability theory). --- Coefficient. --- Combination. --- Commutative property. --- Computation. --- Conditional probability distribution. --- Continuous function. --- Continuous game. --- Convex set. --- Cooperative game. --- Counterexample. --- Determinacy. --- Diagram (category theory). --- Dimension (vector space). --- Disjoint union. --- Duality (mathematics). --- Eigenfunction. --- Equilibrium point. --- Equivalence class. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- Expected value. --- Function (mathematics). --- Hyperplane. --- Idealization. --- Imputation (statistics). --- Inequality (mathematics). --- Infimum and supremum. --- Integral equation. --- Intersection (set theory). --- Interval (mathematics). --- Kakutani fixed-point theorem. --- Limit of a sequence. --- Limit point. --- Linear differential equation. --- Linear inequality. --- Matching Pennies. --- Mathematical induction. --- Mathematical optimization. --- Mathematics. --- Matrix (mathematics). --- Measure (mathematics). --- Minimax theorem. --- Monotonic function. --- Mutual exclusivity. --- Neumann series. --- Numerical analysis. --- Open problem. --- Open set. --- Outcome (probability). --- Partial derivative. --- Partially ordered set. --- Permutation matrix. --- Permutation. --- Polynomial. --- Positional notation. --- Preference (economics). --- Probability distribution. --- Probability measure. --- Probability. --- Property B. --- Proportionality (mathematics). --- Quantity. --- Ranking (information retrieval). --- Ring of sets. --- Saddle point. --- Scientific notation. --- Sign (mathematics). --- Solution set. --- Special case. --- Statistical hypothesis testing. --- Step function. --- Strategy (game theory). --- Subset. --- Summation. --- Symmetrization. --- Theorem. --- Theory of Games and Economic Behavior. --- Theory. --- Three-dimensional space (mathematics). --- Topology. --- Transcendental number. --- Transformation matrix. --- Unit interval. --- Units of measurement. --- Variable (mathematics).

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The description for this book, Automata Studies. (AM-34), Volume 34, will be forthcoming.

Machine theory. --- Conscious automata. --- A Mathematical Theory of Communication. --- Abstraction (software engineering). --- Accuracy and precision. --- Algebraic theory. --- Arithmetic. --- Automaton. --- Axiom. --- Binary number. --- Biophysics. --- Bit. --- Boolean algebra (structure). --- Calculation. --- Chaos theory. --- Circuit design. --- Classical conditioning. --- Combination lock. --- Combination. --- Computable number. --- Computation. --- Concepts (C++). --- Conditional probability. --- Connectivity (graph theory). --- Correlation and dependence. --- Counterexample. --- Counting. --- Decision problem. --- Description number. --- Detection. --- Determinism. --- Deterministic system (philosophy). --- Diagram (category theory). --- Diagram. --- Entscheidungsproblem. --- Enumeration. --- Equation. --- Estimation. --- Exponentiation. --- Finite-state machine. --- First principle. --- Fixed point (mathematics). --- Function (mathematics). --- Gödel numbering. --- Hardware random number generator. --- Heuristic argument. --- Instance (computer science). --- Interlacing (bitmaps). --- Intuitionistic logic. --- Lebesgue measure. --- Logarithm. --- Logic. --- Logical disjunction. --- Markov chain. --- Mathematical analysis. --- Mathematical induction. --- Mathematical logic. --- Mathematical notation. --- Mathematical theory. --- Mathematics. --- Measure (mathematics). --- Metamathematics. --- Modular arithmetic. --- Moment (mathematics). --- Moore machine. --- Notation. --- Number theory. --- OR gate. --- Parameter (computer programming). --- Parity (mathematics). --- Permutation. --- Pilot plant. --- Polynomial. --- Primitive recursive function. --- Probability measure. --- Probability. --- Proportionality (mathematics). --- Propositional function. --- Quantity. --- Recursion (computer science). --- Recursive set. --- Recursively enumerable set. --- Result. --- Schematic. --- Sequence. --- Series (mathematics). --- Set theory. --- Sheffer stroke. --- Sign (mathematics). --- Solver. --- Special case. --- Stochastic process. --- String (computer science). --- Subsequence. --- Subset. --- Theorem. --- Thermodynamic equilibrium. --- Trial and error. --- Turing machine. --- Typewriter. --- Universal Turing machine. --- Variable (mathematics).