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Ok, and now for the news! There's a newly published substantial book by Paolo Mancosu, Sergio Galvan and Richard Zach, An Introduction to Proof Theory (OUP).
"This book arose," they write, "out of the desire to enable students, especially those in philosophy, who have only a minimal background in mathematics and logic, to appreciate the most important results of classical proof theory. Most proof theory textbooks begin at a rather demanding level. The more accessible among them only cover structural proof theory, and those that also cover ordinal proof theory assume a strong mathematical background. Unfortunately, there is no good, elementary introduction to proof theory in English that does both. Our book is meant to fill this gap."
How well do they succeed in that ambition? I have been blogging about the book at Logic Matters, chapter by chapter, and the most recent post discusses their Chapter 6, on the Cut Elimination Theorem. Sad to say, I’m a bit disappointed. To read why, hit the button, and go down the thread of postings to the first in the series!
Kind regards, Peter S.