Summary: Review of Arai's
Some results on cut-elimination, provable
well-orderings, induction, and reflection
December 22, 2000
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,