Logic, mathematics, and computer science: modern foundations with practical applications

This text for the first or second year undergraduate in mathematics, logic, computer science, or social sciences, introduces the reader to logic, proofs, sets, and number theory. It also serves as an excellent independent study reference and resource for instructors. Adapted from Foundations of Logic and Mathematics: Applications to Science and Cryptography © 2002 Birkhӓuser, this second edition provides a modern introduction to the foundations of logic, mathematics, and computers science, developing the theory that demonstrates construction of all mathematics and theoretical computer science from logic and set theory.  The focus is on foundations, with specific statements of all the associated axioms and rules of logic and set theory, and  provides complete details and derivations of formal proofs. Copious references to literature that document historical development is also provided. Answers are found to many questions that usually remain unanswered: Why is the truth table for logical implication so unintuitive? Why are there no recipes to design proofs? Where do these numerous mathematical rules come from? What issues in logic, mathematics, and computer science still remain unresolved? And the perennial question: In what ways are we going to use this material?   Additionally, the selection of topics presented reflects many major accomplishments from the twentieth century and includes applications in game theory and Nash's e quilibrium, Gale and Shapley's match making algorithms, Arrow's Impossibility Theorem in voting, to name a few. From the reviews of the first edition: "...All the results are proved in full detail from first principles...remarkably, the arithmetic laws on the rational numbers are proved, step after step, starting from the very definitions!...This is a valuable reference text and a useful companion for anybody wondering how basic mathematical concepts can be rigorously developed within set theory." —MATHEMATICAL REVIEWS "Rigorous and modern in its theoretical aspect, attractive as a detective novel in its applied aspects, this paper book deserves the attention of both beginners and advanced students in mathematics, logic and computer sciences as well as in social sciences." —Zentralblatt MATH  

Bisection Hardly Ever Converges Linearly

Introduction According to the mathematical folklore, if a continuous and monotone function crosses an interval in its domain, then the method of bisection converges "approximately " linearly, in a loose sense; see Burden & Faires (1989), Kelly (1967), and Pennington (1970). Yet the present work shows that a strictly linear convergence occurs if, but only if, the zero belongs to a countable subset of that interval. Because the concept of order of convergence remains invariant under affine transformations, the present discussion restricts itself to functions on the unit interval, with a negative value at the origin and a positive value at one. There, call a number "diadic" if, but only if, it is a sum of finitely many integral powers of two. To avoid ambiguities, normalize the binary expansions of all diadic numbers to end with an infinite sequence of zeroes. Under the conditions just described, for the method of bisection to converge to a limit, it suffices that ?<F43