By popular request, I’m continuing an informal lunchtime Mathematical Logic Reading Group with a number of grad students. This term, the plan is to do a ‘slow read’ of Gentzen’s two great papers on the consistency of arithmetic. But we started today with the lecture he wrote between the two papers, ‘The concept of infinity in mathematics’. This is short, very accessible, and gives a great sense of the conceptual problems that Gentzen sees as shaping his work. It is also very clearly sets out the headline news about the structure of his (first) consistency proof and about its supposed finitist/constructivist credentials.
The lecture has its shortcomings — there’s a general murkiness about the notion of a ‘constructivist’ view of infinity (why should a constructivist view of sets in the sense of the paradox-busting idea of a hiearchy in which sets at higher levels are formed from sets already constructed at lower levels go along with a constructivist rejection of excluded middle at the level of classical analysis?). But still it is wonderful, thought-provoking stuff.
I was moved to try editing the piece on Gentzen on Wikipedia in very modest ways (e.g. adding that he was Hilbert’s assistant, which you might have thought was a rather central fact about his intellectual trajectory). But twice my efforts were removed. And I wonder if that was because I’d over-written the claim that he was imprisoned after the war “due to his Nazi loyalties” (I’d put something less specific, but more detailed, i.e. the story as told by Szabo in his introduction to the Collected Papers). Is it true about Gentzen having Nazi sympathies? Regrettably it seems so. Discovering this was really rather depressing, as I’ve belatedly become a great admirer of Gentzen.