Ordinal Analysis With An Introduction To Proof Theory
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Ordinal Analysis with an Introduction to Proof Theory
This book provides readers with a guide to both ordinal analysis, and to proof theory. It mainly focuses on ordinal analysis, a research topic in proof theory that is concerned with the ordinal theoretic content of formal theories. However, the book also addresses ordinal analysis and basic materials in proof theory of first-order or omega logic, presenting some new results and new proofs of known ones.Primarily intended for graduate students and researchers in mathematics, especially in mathematical logic, the book also includes numerous exercises and answers for selected exercises, designed to help readers grasp and apply the main results and techniques discussed.
Proof Theory
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. The "constructive" consequences of ordinal analysis are sketched out in the epilogue. The book provides a self-contained treatment assuming no prior knowledge of proof theory and almost none of logic. The author has, moreover, endeavoured not to use the "cabal language" of proof theory, but only a language familiar to most readers.
An Introduction to Proof Theory
"Proof theory is a central area of mathematical logic of special interest to philosophy . It has its roots in the foundational debate of the 1920s, in particular, in Hilbert's program in the philosophy of mathematics, which called for a formalization of mathematics, as well as for a proof, using philosophically unproblematic, "finitary" means, that these systems are free from contradiction. Structural proof theory investigates the structure and properties of proofs in different formal deductive systems, including axiomatic derivations, natural deduction, and the sequent calculus. Central results in structural proof theory are the normalization theorem for natural deduction, proved here for both intuitionistic and classical logic, and the cut-elimination theorem for the sequent calculus. In formal systems of number theory formulated in the sequent calculus, the induction rule plays a central role. It can be eliminated from proofs of sequents of a certain elementary form: every proof of an atomic sequent can be transformed into a "simple" proof. This is Hilbert's central idea for giving finitary consistency proofs. The proof requires a measure of proof complexity called an ordinal notation. The branch of proof theory dealing with mathematical systems such as arithmetic thus has come to be called ordinal proof theory. The theory of ordinal notations is developed here in purely combinatorial terms, and the consistency proof for arithmetic presented in detail"--