[FOM] comment on the video of the lecture by Voevodsky at IAS

joeshipman at aol.com joeshipman at aol.com
Mon May 16 21:46:43 EDT 2011

I must strongly disagree with Rodin and Messing here. Tennant's 
technical criticisms are not the main problem with Voevodsky's lecture. 
The real howler was Voevodsky's presentation of 3 alternatives as 
exhaustive, completely overlooking the obvious 4th alternative of a 
hierarchy of systems of increasing strength, such that the strongest 
system acceptable to him is stronger than Peano Arithmetic and proves 
its consistency.

Voevodsky acted like his failure to see the well-orderedness of 
epsilon_0 as an acceptably obvious axiom was all there was to say, 
ignoring the existence of second order arithmetic, even weak forms of 
which prove the consitency of PA. Even if Voevodsky's own work is 
entirely formalizable in PA, which I doubt, I am certain that he has 
read (and even refereed), and approved of as valid with no 
reservations, mathematical proofs which assume some principle of 
induction or set theory sufficient to prove the consistency of PA. If 
those papers had included as a corollary Con(PA) wouldhe have rejected 
them? if he wouldnot have, how can he say that the consistency of 
arithmetic cannot be proven?

Or does he not even understand the distinction between firstand second 
order arithmetic?

-- JS

-----Original Message-----
From: Andre.Rodin at ens.fr
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Mon, May 16, 2011 6:47 pm
Subject: Re: [FOM] comment on the video of the lecture by Voevodsky at 

I would like also to thank Juliette for posting this video.

> He stated the theorem as follows (written version, projected on the
> screen):
>    It is impossible to prove the consistency of any formal reasoning
>    system which is at least as strong as the standard axiomatization
>    of elementary number theory ("first order arithmetic").
> So he failed to inform his audience that the impossibility that Goedel
> actually established was the impossibility of proof-in-S of a sentence
> expressing the consistency of S, for any consistent and sufficiently
> strong system S.

He didn't mention this detail but I cannot see that it changed the 
sense of the
theorem. Do you seriously think Voevodsky is not aware about this 
detail or
that the detail is too subtle so he's unable to understand it?

> This cavalier inattention to detail marred the subsequent dialectic,

is "dialectic" a course word in your language?

> If a Fields Medallist working in algebraic geometry and homotopy 
> is able to give an account of GII at only such an amateurish level, 
> hope is there for the future of fom in Departments of Mathematics?

Perhaps this future is not so bright indeed (as far as one sticks to a 
meaning of fom) but I think that at least a part of the problem is that 
least a part of) fom community deliberately isolates itself from the 
rest of
mathematical community. This is harmful for the fom community at the 

Andrei Rodin
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FOM at cs.nyu.edu

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