"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.

Paolo Mancosu is Willis S. and Marion Slusser Professor of Philosophy at the University of California at Berkeley. He is the author of numerous articles and books in logic and philosophy of mathematics. During his career he has taught at Stanford, Oxford, and Yale. He was awarded a fellowship at the Wissenschaftskolleg zu Berlin in 1997-1998, a stipendiary position as Directeur de recherche invité; au CNRS in Paris in 2004-2005, a Guggenheim Fellowship in 2008-2009, a position as member at the Institute of Advanced Study in Princeton in 2009, a visiting professorship as LMU-UCB Research in the Humanities at LMU in 2014, and a Humboldt Research Award in 2017-2018.

Sergio Galvan is emeritus Professor of Logic at the Catholic University of Milan, Italy. His main areas of research are proof-theory and philosophical logic. In the first area he focuses on sequent calculi and natural deduction, the metamathematics of arithmetic systems (from Q to PA), Gödel's incompleteness theorems, and Gentzen's cut-elimination theorem. In the second area, his major interest is in the philosophical interpretations (deontic, epistemic and metaphysical) of modal logic. Recently, he has been working on the relationships between formal proof and intuition in mathematics, and on the metaphysics of essence and the ontology of possibilia.

Richard Zach is Professor of Philosophy at the University of Calgary, Canada. He works in logic, history of analytic philosophy, and the philosophy of mathematics. In logic, his main interests are non-classical logics and proof theory. He has also written on the development of formal logic and historical figures associated with this development such as Hilbert, Gödel, and Carnap. He has held visiting appointments at the University of California, Irvine, McGill University, and the University of Technology, Vienna.