[FOM] Voevodsky corr051711,11:12AM-052211,11:05AM

[Joe Shipman]

I hate repeating myself, but even after my previous posts there is so much point-missing going on (or, more charitably, so many posts which would have benefited from reading mine but could not because they were submitted before the moderator released mine) that I have to try this one more time. If my tone appears harsh and dismissive, please be aware that it has been edited and softened from what I originally intended.

1) The consistency of PA is a THEOREM.  It HAS BEEN PROVEN. Full Stop. Although philosophers may dispute this, professional mathematicians may ONLY do so if they publicly criticize Kruskal's theorem, Ulm's theorem, and many much more elementary results as dubious and not worthy of the status of "theorem". I have never heard such criticism from any mathematician who is willing to at the same time admit that the propositions whose proofs are being criticized are meaningful.

2) For a mathematician of the eminence of Voevodsky to be ignorant  of the simple technical facts Harvey informed him of regarding the relationship of Con(PA) to standard elementary theorems is embarrassing but not serious. But for such a person to presume to give a lecture displaying this ignorance in defense of a polemical position that a few minutes' discussion with a specialist would demolish is appallingly unprofessional.

3) One cannot escape this criticism by rejecting certain types of proofs of Con(PA) as philosophically insufficient, because "PA is inconsistent" is a statement of extreme concreteness which, if true, will have an extremely concrete and indubitable proof; to seriously entertain this as a possibility is to reject most of the last century and a half of mathematics, as well as to admit that statements like Kruskal's "theorem" will never be proved and to suggest that mathematicians therefore stop trying to. Such a rejection is indeed tenable, but people like Voevodsky ought to come right out and say it then and experience the kind of marginalization Nelson has, instead of emitting squid ink about alternative foundations which serves mainly to denigrate the impressive and valuable and technically impeccable work that has already been done in f.o.m.

4) None of this negates the philosophical arguments that theorems of PA have a privileged epistemological status compared to other mathematical propositions. But to preserve this status while maintaining  as a possibility that PA might be inconsistent is perverse, unless what is meant is that, although no actual inconsistencies might be found, a proof of the negation of the sentence Con(PA) might conceivably be found, in which case it would merely be bizarre, and ought, in any event, to be explicitly clarified as meaning such.

5) Everything above applies to PA and not to ZFC and no one should lump these two systems together as if the epistemological issues are the same for both. I have almost no objection to any of the remarks people have made here, if they are interpreted as concerning set theory rather than arithmetic.

--JS