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Kurt Gödel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently 'undecidable.' His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Gödel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the text will appeal to mathematicians, philosophers, and computer scientists.

Gödel's theorem. --- Gödel's theorem. --- Gödel, Kurt. --- Gödel's incompleteness theorem --- Undecidable theories --- Arithmetic --- Completeness theorem --- Incompleteness theorems --- Logic, Symbolic and mathematical --- Number theory --- Decidability (Mathematical logic) --- Foundations --- Gkentel, Kourt --- גדל

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Completeness is one of the most important notions in logic and the foundations of mathematics. Many variants of the notion have been de?ned in literature. We shallconcentrateonthesevariants,andaspects,of completenesswhicharede?ned in propositional logic. Completeness means the possibility of getting all correct and reliable sc- mata of inference by use of logical methods. The word ‘all’, seemingly neutral, is here a crucial point of distinction. Assuming the de?nition as given by E. Post we get, say, a global notion of completeness in which the reliability refers only to syntactic means of logic and outside the correct schemata of inference there are only inconsistent ones. It is impossible, however, to leave aside local aspects of the notion when we want to make it relative to some given or invented notion of truth. Completeness understood in this sense is the adequacy of logic in relation to some semantics, and the change of the logic is accompanied by the change of its semantics. Such completeness was e?ectively used by J. ?ukasiewicz and investigated in general terms by A. Tarski and A. Lindenbaum, which gave strong foundations for research in logic and, in particular, for the notion of consequence operation determined by a logical system. The choice of logical means, by use of which we intend to represent logical inferences, is also important. Most of the de?nitions and results in completeness theory were originally developed in terms of propositional logic. Propositional formal systems ?nd many applications in logic and theoretical computer science.

Mathematics. --- Mathematical logic. --- Mathematical Logic and Foundations. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Math --- Science --- Completeness theorem. --- Logic, Symbolic and mathematical --- Model theory --- Logic, Symbolic and mathematical.

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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

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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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Cryptography lies at the heart of most technologies deployed today for secure communications. At the same time, mathematics lies at the heart of cryptography, as cryptographic constructions are based on algebraic scenarios ruled by group or number theoretical laws. Understanding the involved algebraic structures is, thus, essential to design robust cryptographic schemes. This Special Issue is concerned with the interplay between group theory, symmetry and cryptography. The book highlights four exciting areas of research in which these fields intertwine: post-quantum cryptography, coding theory, computational group theory and symmetric cryptography. The articles presented demonstrate the relevance of rigorously analyzing the computational hardness of the mathematical problems used as a base for cryptographic constructions. For instance, decoding problems related to algebraic codes and rewriting problems in non-abelian groups are explored with cryptographic applications in mind. New results on the algebraic properties or symmetric cryptographic tools are also presented, moving ahead in the understanding of their security properties. In addition, post-quantum constructions for digital signatures and key exchange are explored in this Special Issue, exemplifying how (and how not) group theory may be used for developing robust cryptographic tools to withstand quantum attacks.

NP-Completeness --- protocol compiler --- post-quantum cryptography --- Reed–Solomon codes --- key equation --- euclidean algorithm --- permutation group --- t-modified self-shrinking generator --- ideal cipher model --- algorithms in groups --- lightweight cryptography --- generalized self-shrinking generator --- numerical semigroup --- pseudo-random number generator --- symmetry --- pseudorandom permutation --- Berlekamp–Massey algorithm --- semigroup ideal --- algebraic-geometry code --- non-commutative cryptography --- provable security --- Engel words --- block cipher --- cryptography --- beyond birthday bound --- Weierstrass semigroup --- group theory --- braid groups --- statistical randomness tests --- group-based cryptography --- alternating group --- WalnutDSA --- Sugiyama et al. algorithm --- cryptanalysis --- digital signatures --- one-way functions --- key agreement protocol --- error-correcting code --- group key establishment