For those of you who might want to take a break from the Synthese affair :), there is an interesting discussion going on at the FOM mailing list (Foundations of Mathematics); you can read the messages here. It was prompted by videos of lectures by Fields medalist Vladimir Voevodsky (here and here), where (at least according to some) he seems to display a complete lack of understanding of Goedel's incompleteness results.

I wrote a slightly more technical post on this topic over at M-Phi, but let me try to summarize the main points here, assuming that people who do not have a background on the foundations of mathematics may also be interested. As is well known, Goedel's incompleteness results prove the impossibility of proof-in-S of a sentence expressing the consistency of S, for any consistent and sufficiently strong system S (containing Peano arithmetic, PA). But Goedel's results do not prove that the consistency of PA cannot be proved *simpliciter*, i.e. in absolute terms. And so Gentzen spent a good chunk of the 1930s formulating different proofs of the consistency of PA -- obviously not in PA itself or a system containing it, which Goedel had proved to be impossible, but rather in a different system, namely the theory obtained by adding quantifier free transfinite induction to primitive recursive arithmetic. The details don't matter much (for more, see von Plato's SEP entry on the development of proof-theory), but one important thing is that this system is not strictly stronger than PA, as there are theorems that PA can prove and this system cannot. So it was not just a matter of switching to a stronger theory to prove the consistency of PA. Gentzen's proofs are widely held to be cogent proofs of the consistency of PA (relative, of course, to the perceived reliability of the theory in which they are formulated).

However, in his lectures Voevodsky seems to suggest that the consistency of PA is an open problem in mathematics, and even to seriously entertain the possibility of PA being inconsistent. Obviously, he can only claim something like that if he thinks that Gentzen's results are not to be trusted, but he seems to dismiss them much too quickly. The foundations of mathematics community is utterly puzzled as to what is going on, given Voevodsky's attested brilliancy as a mathematician. Some seem pretty sure that Voevodsky is just goofing here, while others think that a more charitable interpretation of what he says should be given. It is a fact that most mathematicians these days do not care much about foundational issues, but Voevodsky' himself *is* involved in a foundational project, based on (according to his website) homotopy-theoretic semantics of Martin-Lof type theories.

So is this a misunderstanding of Goedel's and Gentzen's results, or is there something else going on? If there is something else going on and he is prepared to advance a bold, unconventional position on the basis of solid arguments (i.e. that there are reasons to entertain the possibility of PA being inconsistent), then this could of course have very important implications everywhere, not only for mathematics but also philosophically. So I for one eagerly await the next installments of this affair!

_____________________________________________________________________________________

UPDATE (May 19th): Voevodsky has replied to Friedman; I posted the reply and discussed it here.

## Recent Comments