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This book verifies with compelling evidence the author’s inclination to "write a book on proof theory which needs no previous knowledge of proof theory". Avoiding the cryptic terminology of proof as far as possible, the book starts at an elementary level and displays the connections between infinitary proof theory and generalized recursion theory, especially the theory of inductive definitions. As a "warm up" the classical analysis of Gentzen is presented in a more modern terminology to proceed with explaining and proving the famous result by Feferman and Schütte on the limits of predicativity. The author, too, provides an introduction to ordinal arithmetic, introduces the Veblen hierarchy and employs these functions to design an ordinal notation system for the ordinals below Epsilon 0 and Gamma 0, while emphasizing the first step into impredicativity, i.e., the first step beyond Gamma 0. An earlier version of this book was originally published in 1989 as volume 1407 of the Springer series "Lecture Notes in Mathematics".

Electronic books. -- local. --- Proof theory. --- Proof theory --- Mathematical Theory --- Mathematics --- Physical Sciences & Mathematics --- Mathematics. --- Mathematical logic. --- Operations research. --- Management science. --- Mathematical Logic and Foundations. --- Operations Research, Management Science. --- Quantitative business analysis --- Management --- Problem solving --- Operations research --- Statistical decision --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Math --- Science --- Logic, Symbolic and mathematical --- Logic, Symbolic and mathematical.

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Computer. Automation

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Mathematical logic --- 517 --- Analysis --- Induction (Mathematics) --- Mathematical analysis --- Proof theory. --- Foundations. --- Induction (Mathematics). --- 517 Analysis --- Logique mathématique

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A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction.

Proof theory. --- Mathematics. --- Logic, Symbolic and mathematical. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Math --- Science --- Logic, Symbolic and mathematical --- Mathematical Logic. --- Philosophy of Mathematics. --- Theoretical Computer Science.

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Philosophy