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This book presents a comprehensive treatment of basic mathematical logic. The author's aim is to make exact the vague, intuitive notions of natural number, preciseness, and correctness, and to invent a method whereby these notions can be communicated to others and stored in the memory. He adopts a symbolic language in which ideas about natural numbers can be stated precisely and meaningfully, and then investigates the properties and limitations of this language. The treatment of mathematical concepts in the main body of the text is rigorous, but, a section of 'historical remarks' traces the evolution of the ideas presented in each chapter. Sources of the original accounts of these developments are listed in the bibliography.

Logic, symbolic and mathematical --- Numbers, Natural --- Logic, Symbolic and mathematical --- Natural numbers --- Numbers, Whole --- Whole numbers --- Numbers, Rational --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Logic, Symbolic and mathematical. --- Numbers, Natural. --- Logique mathématique --- Mathematical Sciences --- General and Others

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Prime numbers beckon to the beginner, as the basic notion of primality is accessible even to children. Yet, some of the simplest questions about primes have confounded humankind for millennia. In the new edition of this highly successful book, Richard Crandall and Carl Pomerance have provided updated material on theoretical, computational, and algorithmic fronts. New results discussed include the AKS test for recognizing primes, computational evidence for the Riemann hypothesis, a fast binary algorithm for the greatest common divisor, nonuniform fast Fourier transforms, and more. The authors also list new computational records and survey new developments in the theory of prime numbers, including the magnificent proof that there are arbitrarily long arithmetic progressions of primes, and the final resolution of the Catalan problem. Numerous exercises have been added. Richard Crandall currently holds the title of Apple Distinguished Scientist, having previously been Apple's Chief Cryptographer, the Chief Scientist at NeXT, Inc., and recipient of the Vollum Chair of Science at Reed College. Though he publishes in quantum physics, biology, mathematics, and chemistry, and holds various engineering patents, his primary interest is interdisciplinary scientific computation. Carl Pomerance is the recipient of the Chauvenet and Conant Prizes for expository mathematical writing. He is currently a mathematics professor at Dartmouth College, having previously been at the University of Georgia and Bell Labs. A popular lecturer, he is well known for his research in computational number theory, his efforts having produced important algorithms now in use. From the reviews of the first edition: "Destined to become a definitive textbook conveying the most modern computational ideas about prime numbers and factoring, this book will stand as an excellent reference for this kind of computation, and thus be of interest to both educators and researchers." ^ L'Enseignement Mathématique "...Prime Numbers is a welcome addition to the literature of number theory---comprehensive, up-to-date and written with style." - American Scientist "It's rare to say this of a math book, but open Prime Numbers to a random page and it's hard to put down. Crandall and Pomerance have written a terrific book." - Bulletin of the AMS.

Numbers, Prime. --- Numbers, Natural. --- Natural numbers --- Numbers, Whole --- Whole numbers --- Numbers, Rational --- Prime numbers --- Numbers, Natural --- Number theory. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- 511.3 --- Numbers, Prime --- 681.3*F22 --- 511.3 Analytical, additive and other number-theory problems. Diophantine approximations --- Analytical, additive and other number-theory problems. Diophantine approximations --- 681.3*F22 Nonnumerical algorithms and problems: complexity of proof procedures; computations on discrete structures; geometrical problems and computations; pattern matching --See also {?681.3*E2-5}; {681.3*G2}; {?681.3*H2-3} --- Nonnumerical algorithms and problems: complexity of proof procedures; computations on discrete structures; geometrical problems and computations; pattern matching --See also {?681.3*E2-5}; {681.3*G2}; {?681.3*H2-3}

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The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic.

Research & information: general --- Mathematics & science --- quine --- logic --- ontology --- multiple conclusion rule --- disjunction property --- metadisjunction --- axiomatizations of arithmetic of natural and integers numbers --- second-order theories --- Peano’s axioms --- Wilkosz’s axioms --- axioms of integer arithmetic modeled on Peano and Wilkosz axioms --- equivalent axiomatizations --- metalogic --- categoricity --- independence --- consistency --- logic of typical and atypical instances (LTA) --- logic of determination of objects (LDO) --- quasi topology structure (QTS) --- concept --- object --- typical object --- atypical object --- lattice --- filter --- ideal --- discussive logics --- the smallest discussive logic --- discussive operators --- seriality --- accessibility relation --- Kotas’ method --- modal logic --- deontic logic --- ontology of situations --- semantics of law --- formal theory of law --- Wittgenstein --- Wolniewicz --- non-Fregean logic --- identity connective --- sentential calculus with identity --- situational semantics --- deduction --- (dual) tableau --- Gentzen system --- deductive refutability --- refutation systems --- hybrid deduction–refutation rules --- derivative hybrid rules --- soundness --- completeness --- natural deduction --- meta-proof theory --- synthetic tableaux --- principle of bivalence --- cut --- first-order theory --- universal axiom --- Peano’s axiomatics of natural numbers --- Leśniewski’s elementary ontology --- Frege’s predication scheme --- Frege’s Zahl-Anzahl distinction --- term logic --- Franz Brentano --- Lewis Carroll --- logic trees --- logic diagrams --- paraconsistent logic --- paraconsistency --- Sette’s calculus --- the law of explosion --- the principle of ex contradictione sequitur quodlibet --- semantic tree --- distribution --- Aristotle’s logic --- syllogistic --- Jan Łukasiewicz --- axiomatic system --- axiomatic refutation --- temporal logic --- intuitionistic logic --- minimal system --- knowledge --- sequent-type calculi --- nonmonotonic logics --- default logic --- rejection systems --- Kripke models --- logics of evidence and truth --- n/a --- Peano's axioms --- Wilkosz's axioms --- Kotas' method --- hybrid deduction-refutation rules --- Peano's axiomatics of natural numbers --- Leśniewski's elementary ontology --- Frege's predication scheme --- Frege's Zahl-Anzahl distinction --- Sette's calculus --- Aristotle's logic

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The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic.

Research & information: general --- Mathematics & science --- quine --- logic --- ontology --- multiple conclusion rule --- disjunction property --- metadisjunction --- axiomatizations of arithmetic of natural and integers numbers --- second-order theories --- Peano's axioms --- Wilkosz's axioms --- axioms of integer arithmetic modeled on Peano and Wilkosz axioms --- equivalent axiomatizations --- metalogic --- categoricity --- independence --- consistency --- logic of typical and atypical instances (LTA) --- logic of determination of objects (LDO) --- quasi topology structure (QTS) --- concept --- object --- typical object --- atypical object --- lattice --- filter --- ideal --- discussive logics --- the smallest discussive logic --- discussive operators --- seriality --- accessibility relation --- Kotas' method --- modal logic --- deontic logic --- ontology of situations --- semantics of law --- formal theory of law --- Wittgenstein --- Wolniewicz --- non-Fregean logic --- identity connective --- sentential calculus with identity --- situational semantics --- deduction --- (dual) tableau --- Gentzen system --- deductive refutability --- refutation systems --- hybrid deduction-refutation rules --- derivative hybrid rules --- soundness --- completeness --- natural deduction --- meta-proof theory --- synthetic tableaux --- principle of bivalence --- cut --- first-order theory --- universal axiom --- Peano's axiomatics of natural numbers --- Leśniewski's elementary ontology --- Frege's predication scheme --- Frege's Zahl-Anzahl distinction --- term logic --- Franz Brentano --- Lewis Carroll --- logic trees --- logic diagrams --- paraconsistent logic --- paraconsistency --- Sette's calculus --- the law of explosion --- the principle of ex contradictione sequitur quodlibet --- semantic tree --- distribution --- Aristotle's logic --- syllogistic --- Jan Łukasiewicz --- axiomatic system --- axiomatic refutation --- temporal logic --- intuitionistic logic --- minimal system --- knowledge --- sequent-type calculi --- nonmonotonic logics --- default logic --- rejection systems --- Kripke models --- logics of evidence and truth --- quine --- logic --- ontology --- multiple conclusion rule --- disjunction property --- metadisjunction --- axiomatizations of arithmetic of natural and integers numbers --- second-order theories --- Peano's axioms --- Wilkosz's axioms --- axioms of integer arithmetic modeled on Peano and Wilkosz axioms --- equivalent axiomatizations --- metalogic --- categoricity --- independence --- consistency --- logic of typical and atypical instances (LTA) --- logic of determination of objects (LDO) --- quasi topology structure (QTS) --- concept --- object --- typical object --- atypical object --- lattice --- filter --- ideal --- discussive logics --- the smallest discussive logic --- discussive operators --- seriality --- accessibility relation --- Kotas' method --- modal logic --- deontic logic --- ontology of situations --- semantics of law --- formal theory of law --- Wittgenstein --- Wolniewicz --- non-Fregean logic --- identity connective --- sentential calculus with identity --- situational semantics --- deduction --- (dual) tableau --- Gentzen system --- deductive refutability --- refutation systems --- hybrid deduction-refutation rules --- derivative hybrid rules --- soundness --- completeness --- natural deduction --- meta-proof theory --- synthetic tableaux --- principle of bivalence --- cut --- first-order theory --- universal axiom --- Peano's axiomatics of natural numbers --- Leśniewski's elementary ontology --- Frege's predication scheme --- Frege's Zahl-Anzahl distinction --- term logic --- Franz Brentano --- Lewis Carroll --- logic trees --- logic diagrams --- paraconsistent logic --- paraconsistency --- Sette's calculus --- the law of explosion --- the principle of ex contradictione sequitur quodlibet --- semantic tree --- distribution --- Aristotle's logic --- syllogistic --- Jan Łukasiewicz --- axiomatic system --- axiomatic refutation --- temporal logic --- intuitionistic logic --- minimal system --- knowledge --- sequent-type calculi --- nonmonotonic logics --- default logic --- rejection systems --- Kripke models --- logics of evidence and truth