Cuts, consistency and axiomatized theories

In the Wednesday Logic Reading Group, where we are working through Sara Negri and Jan von Plato's Structural Proof Theory, I today introduced Chapter 6, 'Structural Proof Analysis of Axiomatic Theories'. In their commendable efforts to be brief, the authors are sometimes a bit brisk about motivation. So I thought it was worth trying to stand back a bit from the details of this action-packed chapter as far as I understood it in the few hours I had to prepare, and to try to give an overall sense of the project. These are the notes I wrote for myself. As often with such middle-of-term efforts dashed off in a couple of hours, I both would have liked to do better and do more justice to what we are reading, but I also just don't have time to do more now than make a few corrections to the first version. The logic enthusiasts at the seminar seemed to find the remarks useful, though, so for what they are worth here there are. The usual warning applies: caveat lector.