Proof Theory: Second Edition (Dover Books on Mathematics) by Gaisi Takeuti (PDF)

Description

Focusing on Gentzen-type proof theory, this volume presents a detailed overview of creative works by author Gaisi Takeuti and other twentieth-century logicians. The text explores applications of proof theory to logic as well as other areas of mathematics. Suitable for advanced undergraduates and graduate students of mathematics, this long-out-of-print monograph forms a cornerstone for any library in mathematical logic and related topics.The three-part treatment begins with an exploration of first order systems, including a treatment of predicate calculus involving Gentzen’s cut-elimination theorem and the theory of natural numbers in terms of Gödel’s incompleteness theorem and Gentzen’s consistency proof. The second part, which considers second order and finite order systems, covers simple type theory and infinitary logic. The final chapters address consistency problems with an examination of consistency proofs and their applications.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This classic proof theory text reprinted by Dover in early 2013 is one of the best yet. I’ve been reading this intensely detailed 1987 book in winter and spring of 2014, along with other books. In chapter 1, author Gaisi Takeuti of University of Illinois started the book right off with the Gentzen LK system with a bit of intuitionistic LJ as well. Further operations were also included.Chapter 2, an 83 page monster on Peano Arithmetic (PA) gets into extremely detailed and advanced work with ordinal mathematics as preparation for the following huge proof series to establish consistency of PA. My reading presently is in the long article 12 on those consistency proofs and they are fascinating. On Tue 29Apr14 I finished section 12 on p.120, after which the section goes too far afield of the main chapter subjects. Section 13 on ‘Provable Well Orderings’ is a terrifically interesting shorter section as long chapter 2 nearly ends. Finished my read of chapter 2 on Wed 30Apr14 afternoon.Part II on higher order proof theory starts with a 4-page introductory essay that is quite troubling. The author seems to think that set theory is some kind of mysterious unapproachable god in that essay, and it is not even clear which version of several versions of set theory he is talking about. This has me looking at several related books for guidance before continuing this book. Late day Wed 30Apr14. After reviewing many proof-oriented and newer set theory books, including Kunen’s 2011 set theory and 2009 foundations of math books, also including both of the late George Boolos’ two modal provability books, it still looks like Gaisi Takeuti was out of line in his bizarre Part II initial essay. In 2013, I also read papers on 2nd order logic vs. ZFC set theory and on higher order logics rather thoroughly. Those papers would have helped the author of this book in his intense confusion. One final point: In set theory books, the authors simply prove theorems in a normal mathematical way, so perhaps in 1987, a specifically proof theoretical attack on set theory was too difficult.That just might be an obsolete aspect of this book. Therefore, that essay is where my reading of Takeuti’s proof theory ends.It does seem as if the author might in Part II have taken some inspiration from Stephen Simpson’s reverse mathematics via Simpson’s paper in the appendix of this book. Please also see

⭐, which I own and have read most of great overview chapter I in that book.Paging thru Part II, Part III, and the appendices of this book on Thu 8May14 morning did not entice me into further reading of this proof theory book. First-order systems are the most used in mathematics anyway, so finishing a large majority of Part I on such first-order proof systems isn’t a bad place to stop reading for anybody.One small warning is that this book is definitely at postgraduate level. Also, Amazon has a ‘Look Inside’ for this book that is maximally open for viewing its contents, so definitely take a look at that.In spite of the highly open Amazon ‘Look Inside’, here is a basic list of contents for use in this review: Introduction-1 / PART I: FIRST-ORDER SYSTEMS-3 / Chapter 1 (Articles 1-8) First Order Predicate Calculus-5 / Chapter 2 (Articles 9-14) Peano Arithmetic-75 / PART II: SECOND-ORDER AND FINITE ORDER SYSTEMS-159 / Chapter 3 (Articles 15-21) Second-Order Systems and Simple Type Theory-165 / Chapter 4 (Articles 22-24) Infinitary Logic-208 / PART III: CONSISTENCY PROBLEMS-295 / Chapter 5 (Articles 25-28) Consistency Proofs-298 / Chapter 6 (Articles 29-31) Some Applications of Consistency Proofs-361 / Postscript-381 / APPENDIX-393 / Proof Theory (George Kreisel)-395 / Contributions of the Schutte School (Wolfram Pohlers)-406 / Subsystems of Z2 and Reverse Mathematics (Stephen Simpson)-432 / Proof Theory: A Personal Report (Solomon Feferman)-447 / Index-487

⭐This 1975/1987 book by Gaisi Takeuti (1926-2017), who apparently died just 3 weeks ago (2017-5-10 according to wikipedia), is a heavyweight book on proof theory at the graduate level (or higher).Two things stand out for me in this book.1. The use of the Gentzen sequent calculus approach to logic throughout the book, as opposed to the Hilbert approach which had been, and still is, fairly dominant in most formal logic textbooks.2. The proof of the consistency of integer arithmetic, which was famously shown by Gödel to be impossible within the system itself. But Gentzen did show, in a “finitist” system with triple recursion, that integer arithmetic is consistent. This theorem is very much less famous than Gödel’s earlier incompleteness theorem. The general lack of interest in Gentzen’s theorem is somewhat perplexing because even Hilbert and Bernays, in ”

⭐” in 1939, presented a proof of Gentzen’s integer consistency theorem.Whereas the Hilbert/Bernays presentation of Gentzen’s integer consistency theorem is very old-fashioned and difficult to understand, Takeuti’s presentation (pages 101-147) is very neat and tidy and modern. Unfortunately the proof is very long, and is all expressed in Gentzen’s proof-style, but this is to be expected for such a significant and difficult theorem.Since proof theory is not my professional focus, I will say no more about this book. Probably anyone who is up to the level of reading this book needs no review anyway. I’ll just say that it is very well printed indeed by Dover. It’s a little work of art. This book would be ideal for anyone who is well versed in Gentzen systems.I should mention that in Gentzen’s 1934/1935 papers, where he introduced his sequent calculus systems, he defined two kinds of systems, namely the natural deduction style (with single succedents) and the more general multi-succedent style LK (classical) and LJ (intuitionistic) logics. It is LK system which is the basis of most of this book after the introductory chapter.By the way, the final 2 pages of the appendix, pages 479-480, express some extreme skepticism about the whole effort to prove consistency of mathematics, apparently implying that even his own work might be of dubious real value. Personally, I’m one of those people who take it for granted that ZF is consistent, and if someone finds an inconsistency (as big as Russell’s paradox), it will be fixed up fairly quickly, and mathematicians can continue to ignore such questions. This book is intended, I think, for the people who do worry about consistency. I’m just too lazy to worry about it. Mathematics is too hard already!

⭐Spot on

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