The next chapter of my Beginning Mathematical Logic: A Study Guide is announced as covering proof theory. I’ve been quietly bemoaning the fact that there hasn’t been a good, well-written, up-to-date introduction that can readily be recommended as a teach-yourself book at the right sort of level. So I thought I was going to have to put together a rather complicated set of suggestions (“read this chapter from this book, then look at those excerpts from that book, and follow this up with some of that other book”), and that would require a lot of homework — especially as I was finding that my memories of what is covered where are pretty fallible.
So I’m just delighted to discover by accident that there is book forthcoming from OUP next month by Paolo Mancosu, Sergio Galvan, and Richard Zach with the very promising title An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs. How did I miss that before? Here’s the blurb:
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.
Surely, especially given these authors, exactly what we need! There are a few more details here. I really look forward to reading this. And a bonus point to OUP, by the way, for bringing out this 432 page paperback for a very reasonable price.