An Introduction to Proof Theory

16 x 2.54 x 23.88 cm, 432 pages

Published by Oxford University Press.

ISBN:: 978-0-19-289594-3

(1 review)

An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's …

An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics.

1 edition

gesang reviewed An Introduction to Proof Theory by Paolo Mancosu

Wordy

3 stars

Hard to tell. The book is accessible, but maybe too accessible; verbose. It literally starts with high-school-level induction, and devotes 50 pages to axiomatic calculi just to show that how difficult it is to work with. While more clearly presented, it used approximately 250 pages for what Negri and Plato's book presented with around 80 pages. Meanwhile it is self-contained; it even devoted an entire chapter to ordinal notations and transfinite inductions, which should be covered in set theory texts, and also contains quite some valuable and cleanly organized historical motivations.

In the preface the authors stated that the book is especially for those in philosophy who have only a minimal background in mathematics and logic. For those with relevant background the book is largely a waste of time. Troelstra & Schwichtenberg's Basic Proof Theory is the way to go.

Subjects

Mathematics
Logic
Proof Theory