The Association for Symbolic Logic has awarded its 2022 Shoenfield Logic Book and Article Prizes. The Shoenfield Prizes are “awarded for outstanding expository writing in the field of logic” and were established honor the late Joseph R. Shoenfield, a influential logician who died in 2000. The Shoenfield Book Prize was awarded to Paolo Mancosu (University of California, Berkeley), Sergio Galvan (Catholic University of the Sacred Heart), and Richard Zach (Calgary) for their book, An Introduction to Proof Theory—Normalization, Cut-Elimination, and Consistency Proofs (Oxford University Press, 2021). Here’s a summary of their book: Proof theory is a central area of mathematical logic of special interest to philosophy. It has its roots in the foundational debate of the 1920s, in particular, in Hilbert’s program in the philosophy of mathematics, which called for a formalization of mathematics, as well as for a proof, using philosophically unproblematic, “finitary” means, that these systems are free from contradiction. Structural proof theory investigates the structure and properties of proofs in different formal deductive systems, including axiomatic derivations, natural deduction, and the sequent calculus. Central results in structural proof theory are the normalization theorem for natural deduction, proved here for both intuitionistic and classical logic, and the cut-elimination theorem for the sequent calculus. In formal systems of number theory formulated in the sequent calculus, the induction rule plays a central role. It can be eliminated from proofs of sequents of a certain elementary form: every proof of an atomic sequent can be transformed into a “simple” proof. This is Hilbert’s central idea for giving finitary consistency proofs. The proof requires a measure of proof complexity called an ordinal notation. The branch of proof theory dealing with mathematical systems such as arithmetic thus has come to be called ordinal proof theory. The theory of ordinal notations is developed here in purely combinatorial terms, and the consistency proof for arithmetic presented in detail. The Shoenfield Article Prize was awarded to Vasco Brattka (Bundeswehr University Munich) for his article, “A Galois Connection between Turing Jumps and Limits”, published in Logical Methods in Computer Science in 2018. Here’s the abstract of his article: Limit computable functions can be characterized by Turing jumps on the input side or limits on the output side. As a..