Review of Arai ’ s Some results on cut - elimination , provable well - orderings , induction , and reflection ∗ Jeremy Avigad

The fact that this paper was originally titled “From the Attic” is strong evidence that Arai’s attic is more interesting than most. The paper is a collection of results gathered over the course of a decade or so, spanning a wide range of topics in proof theory and tying up a number of loose ends. There are some new results here, but the general emphasis is on providing strengthenings and new proofs of previous results, many of them well known or folklore. Arai is always on the mark: while the paper does not break dramatically new ground, it is full of clever tricks, keen insights, satisfying observations, and nontrivial refinements, presented in a clean and elegant way. For the most part, each of the eight sections can be read independently. Arai is to be commended for providing detailed references, contextual notes, and explanatory remarks. In the following, therefore, I will provide only a brief synopsis, glossing over many of the details and omitting citations that can be found in the paper itself. Section 1 addresses the topic of provable well-orderings. Thanks to Gentzen, we know that any elementary recursive ordering that is provably well-ordered in Peano arithmetic has order-type less than ε0. Takeuti and, independently, Harrington, have shown that if R is an elementary relation that is provably wellordered in PA, there is, moreover, a <ε0-recursive comparison map between R and standard notation systems for ε0. Arai uses a clever coding trick to provide two nice strengthenings: one can, in fact, find a comparison map that is elementary, and even under the weaker assumption that R is just provably well-founded (not necessarily totally ordered). This analysis carries over to reasonable extensions of PA, but Arai notes that it is open as to whether one can prove the same result for fragments. In Section 2, Arai shows that one can extract elementary bounds on the increase in length when eliminating cuts from proofs in propositional sequent calculi. This result, which stands in sharp contrast to predicate logic, is wellknown, and the standard proofs are not difficult; but Arai uses instead a simple counting argument that highlights the difference between the propositional and ∗Annals of Pure and Applied Logic 95 (1998), 93–184.