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Product Description: "Necessity is the mother of invention. " Part I: What is in this book - details. There are several different types of formal proof procedures that logicians have invented. The ones we consider are: 1) tableau systems, 2) Gentzen sequent calculi, 3) natural deduction systems, and 4) axiom systems...read more

This volume is the first in a series which will, in the future, be published jointly by Bibliopolis, Edizioni di Filosofia e Scienze spa, Napoli, and under the North-Holland imprint by Elsevier Science Publishers, Amsterdam. This is the first of a two-part work covering the main ideas and techniques of proof theory. This volume is introductory, starting with Hilbert (whose second problem in the famous list of problems of 1900 was the consistency of number theory), describing Hilbert's program and its demise at the hands of Godel. It then proceeds with Gentzen's result (the Hauptsatz is the main result of Part I) and improvements and generalizations to omega-logic. Volume II will cover more advanced logics.

Product Description: This book develops and demonstrates efficient matrix proof methods for automated deduction within an important and comprehensive class of first order and intuitionistic logics. Traditional techniques for the design of efficient proof systems are abstracted from their original setting which allows their application to a wider class of mathematical logic...read more

Product Description: This book deals with the presentation and systematic design of mathematical proofs, including correctness proofs of algorithms. Its purpose is to show how completeness of argument, an important constraint especially for the correctness of algorithms, can be combined with brevity...read more

The mathematical proof is the most important form of justification in mathematics. It is not, however, the only kind of justification for mathematical propositions. The existence of other forms, some of very significant strength, places a question mark over the prominence given to proof within mathematics. This collection of essays, by leading figures working within the philosophy of mathematics, is a response to the challenge of understanding the nature and role of the proof.

Product Description: Gentzen's cut-elimination theorem is widely used as a tool for meta-mathematical investigations. It is sometimes claimed however that the theorem and its proof have interest which is independent of these applications and derives from the information they supply about the structure of proofs in general...read more

Product Description: The Power of Interaction presents a new algebraic technique for constructing interactive proof systems and demonstrates the immense power of randomization and interaction in proving statements efficiently. Lund shows that two provers can interact with a randomized verifier to provide proofs that are exponentially more efficient than traditional proofs, and that one prover can interact with an efficient randomized verifier to prove statements that have no known efficient traditional proofs...read more

Product Description: This book principally concerns the rapidly growing area of what might be termed "Logical Complexity Theory": the study of bounded arithmetic, propositional proof systems, length of proof, and similar themes, and the relations of these topics to computational complexity theory...read more

Product Description: This work is derived from the SERC "Logic for IT" Summer School Conference on Proof Theory held at Leeds University. The contributions come from acknowledged experts and comprise expository and research articles which form an invaluable introduction to proof theory aimed at both mathematicians and computer scientists...read more (view table of contents, read Amazon.com's description)

Product Description: The aim of this work is to develop the tool of logical deduction schemata and use it to establish upper and lower bounds on the complexity of proofs and their transformations in axiomatized theories. The main results are establishment of upper bounds on the elongation of deductions in cut eliminations; a proof that the length of a direct deduction of an existence theorem in the predicate calculus cannot be bounded above by an elementary function of the length of an indirect deduction of the same theorem; a complexity version of the existence property of the constructive predicate calculus; and, for certain formal systems of arithmetic, restrictions on the complexity of deductions that guarantee that the deducibility of a formula for all natural numbers in some finite set implies the deducibility of the same formula with a universal quantifier over all sufficiently large numbers...read more

Product Description: This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (1979). Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians...read more (view table of contents, read Amazon.com's description)

Product Description: This text includes an eclectic blend of math: number theory, analysis, and algebra, with logic as an extra.

Product Description: Although this is an introductory text on proof theory, most of its contents is not found in a unified form elsewhere in the literature, except at a very advanced level. The heart of the book is the ordinal analysis of axiom systems, with particular emphasis on that of the impredicative theory of elementary inductive definitions on the natural numbers...read more

Product Description: "Proof" has been and remains one of the concepts which characterises mathematics. Covering basic propositional and predicate logic as well as discussing axiom systems and formal proofs, the book seeks to explain what mathematicians understand by proofs and how they are communicated...read more (view table of contents, read Amazon.com's description)

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