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This Element takes a deep dive into Gödel's 1931 paper giving the first presentation of the Incompleteness Theorems, opening up completely passages in it that might possibly puzzle the student, such as the mysterious footnote 48a. It considers the main ingredients of Gödel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It samples some of the most important proofs of the Incompleteness Theorems, e.g. due to Kuratowski, Smullyan and Robinson, as well as newer proofs, also of other independent statements, due to H. Friedman, Weiermann and Paris-Harrington. It examines the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature.

Gödel's theorem. --- Incompleteness theorems. --- Logic, Symbolic and mathematical. --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Constructive mathematics --- Proof theory --- Theorems, Incompleteness --- Arithmetic --- Completeness theorem --- Incompleteness theorems --- Logic, Symbolic and mathematical --- Number theory --- Decidability (Mathematical logic) --- Gödel's incompleteness theorem --- Undecidable theories --- Foundations

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Dr Gregory Chaitin, one of the world's leading mathematicians, is best known for his discovery of the remarkable O number, a concrete example of irreducible complexity in pure mathematics which shows that mathematics is infinitely complex. In this volume, Chaitin discusses the evolution of these ideas, tracing them back to Leibniz and Borel as well as Gödel and Turing.This book contains 23 non-technical papers by Chaitin, his favorite tutorial and survey papers, including Chaitin's three Scientific American articles. These essays summarize a lifetime effort to use the notion of program-size co

Godel's theorem. --- Incompleteness theorems. --- Logic, Symbolic and mathematical. --- Metamathematics. --- Computational complexity. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Theorems, Incompleteness --- Constructive mathematics --- Proof theory --- Complexity, Computational --- Electronic data processing --- Machine theory --- Logic, Symbolic and mathematical --- Gödel's incompleteness theorem --- Undecidable theories --- Arithmetic --- Completeness theorem --- Incompleteness theorems --- Number theory --- Decidability (Mathematical logic) --- Philosophy --- Foundations --- Gödel's theorem.

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Provability, Computability and Reflection

Group theory --- Gödel's theorem --- Théorème de Gödel --- Congresses --- Congrès --- 510.6 --- Godel's theorem --- -Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Gödel's incompleteness theorem --- Undecidable theories --- Arithmetic --- Completeness theorem --- Incompleteness theorems --- Logic, Symbolic and mathematical --- Number theory --- Decidability (Mathematical logic) --- Mathematical logic --- Foundations --- Word problems (Mathematics) --- Congresses. --- -Mathematical logic --- Word problems (Mathematics). --- 510.6 Mathematical logic --- -510.6 Mathematical logic --- Groups, Theory of --- Gödel's theorem --- Théorème de Gödel --- Congrès --- ELSEVIER-B EPUB-LIV-FT --- Gödel, Kurt --- Décidabilité (logique mathématique) --- Group theory - Congresses --- Structures algebriques --- Probleme du mot

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Group theory --- Groupes, Théorie des --- Burnside, Problème de --- Burnside problem --- Groupes, Théorie des. --- Burnside, Problème de. --- Gödel's theorem --- Word problems (Mathematics) --- Théorème de Gödel --- Problèmes des mots (Mathématiques) --- Burnside, Problème de --- Congresses --- Congrès --- ELSEVIER-B EPUB-LIV-FT --- Burnside problem. --- Gödel's theorem --- Word problems (Mathematics). --- Congresses. --- 510.6 --- Groupes, Théorie des --- Décidabilité (logique mathématique) --- 510.6 Mathematical logic --- Mathematical logic --- Problems, Word (Mathematics) --- Story problems (Mathematics) --- Mathematics --- Gödel's incompleteness theorem --- Undecidable theories --- Arithmetic --- Completeness theorem --- Incompleteness theorems --- Logic, Symbolic and mathematical --- Number theory --- Decidability (Mathematical logic) --- Problem, Burnside --- Finite groups --- Foundations --- Generators --- Relations --- Group theory - Congresses --- Groupes (algebre) --- Structures algebriques --- Generateurs --- Probleme du mot --- Burnside, Problème de.

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A New York Times bestseller when it appeared in 1989, Roger Penrose's The Emperor's New Mind was universally hailed as a marvelous survey of modern physics as well as a brilliant reflection on the human mind, offering a new perspective on the scientific landscape and a visionary glimpse of the possible future of science. Now, in Shadows of the Mind, Penrose offers another exhilarating look at modern science as he mounts an even more powerful attack on artificial intelligence. But perhaps more important, in this volume he points the way to a new science, one that may eventually explain the physical basis of the human mind. Penrose contends that some aspects of the human mind lie beyond computation. This is not a religious argument (that the mind is something other than physical) nor is it based on the brain's vast complexity (the weather is immensely complex, says Penrose, but it is still a computable thing, at least in theory). Instead, he provides powerful arguments to support his conclusion that there is something in the conscious activity of the brain that transcends computation--and will find no explanation in terms of present-day science. To illuminate what he believes this "something" might be, and to suggest where a new physics must proceed so that we may understand it, Penrose cuts a wide swathe through modern science, providing penetrating looks at everything from Turing computability and Godel's incompleteness, via Schrodinger's Cat and the Elitzur-Vaidman bomb-testing problem, to detailed microbiology. Of particular interest is Penrose's extensive examination of quantum mechanics, which introduces some new ideas that differ markedly from those advanced in The Emperor's New Mind, especially concerning the mysterious interface where classical and quantum physics meet. But perhaps the most interesting wrinkle in Shadows of the Mind is Penrose's excursion into microbiology, where he examines cytoskeletons and microtubules, minute substructures lying de

Artificial intelligence --- Thought and thinking --- Gödel's theorem --- Quantum theory --- Physics --- Philosophy --- Gödel [Theorema van ] --- Gödel [Théorème de ] --- Gödel, Théorème de --- Godel's theorem --- Gödel's incompleteness theorem --- Undecidable theories --- Incompleteness theorems --- Decidability (Mathematical logic) --- Gödel's theorem. --- Gödel's theorem --- Gödel, Théorème de --- 510.21 --- 681.3*I2 --- 681.3*I2 Artificial intelligence. AI --- Artificial intelligence. AI --- Mind --- Thinking --- Thoughts --- Educational psychology --- Psychology --- Intellect --- Logic --- Perception --- Psycholinguistics --- Self --- 510.21 General philosophical considerations. Critical aspects. Logical antinomies --- General philosophical considerations. Critical aspects. Logical antinomies --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Mechanics --- Thermodynamics --- Arithmetic --- Completeness theorem --- Logic, Symbolic and mathematical --- Number theory --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Foundations --- Cognitive psychology --- Artificial intelligence. --- Intelligence artificielle --- Physique --- Théorie quantique --- Pensée --- Philosophie --- Quantum theory. --- Thought and thinking. --- Philosophy. --- Gèodel's theorem. --- Consciousness --- Conscience --- Pensée --- Théorie quantique --- Conscience. --- Intelligence artificielle. --- Pensée. --- Gödel, Théorème de. --- Théorie quantique. --- Philosophie. --- Physics - Philosophy --- Pensée. --- Gödel, Théorème de.