Next up in Kurt Gödel and the Foundations of Mathematics is Ulrich Kohlenbach, writing on ‘Gödel’s Functional Interpretation and Its Use in Current Mathematics’. This rachets up the technical level radically, and will be pretty inaccessible to most readers (certainly, to most philosophers). The author has done significant work in this area: but as an effort towards making this available and/or explaining its importance to a slightly wider readership than researchers in one corner of proof theory, this over-brisk paper surely quite misses the mark. (I guess enthusiasts who want to know more about recent developments will just have to go for the long haul and try Kohlenbach’s 2008 book on Applied Proof Theory, but that too is very hard going.)
Then, for the eighteenth paper, we have Harvey Friedman, aiming to discuss a ‘sample of research progjects that are suggested by some of Gödel’s most famous contributions’ — a prospectus that immediately alerts the reader to the likelihood that the paper will cover too much too fast. The piece has the remarkably self-regarding title ‘My Forty Years on His Shoulders’ and ends with the usual Friedmanesque announcements of results about the equivalence of the provability-in-various-arithmetics of certain combinatorial claims with the consistency of certain set theories with large cardinals. The style and content will be very familiar to readers of the FOM list, and probably pretty baffling to others.
One place where Friedman’s paper goes a bit slower is in discussing the Second Incompleteness Theorem, and there are intimations by the author that he has found a neater, more insightful way of developing the result. But with his customary academic incivility, Friedman doesn’t bother to explain this in accordance with the normal standards of exchange between colleagues, but refers to online unpublications … where things remain equally unexplained. This is, to put it mildly, irritating: and I know I’m not the only person who has long since lost patience with this mode of proceding. Humphhhh!