[FOM] 461: Reflections on Vienna Meeting

Alex Simpson Alex.Simpson at ed.ac.uk
Tue Jun 14 09:48:15 EDT 2011

Thanks to Anon Barfod for posting the link to Angus Macintyre's
April talk in Vienna:


In his posting of 11th May, Harvey Friedman wrote of this talk:

> I asked very specifically whether
> The consistency of Peano Arithmetic
> is a legitimate mathematical problem in present day mathematical  culture.
> I asked this because the Fields Medalist Voevodsky has explicitly   
> considered this to be an open problem.
> To my shock, Angus declared, unequivocally, yes.

I was surprised when I first read this last month, because Harvey's
account did not correspond to my own memory of the talk, which I
attended in person. Memory being fallible, however, I was not
entirely confident of my position. But now that the talk is publicly
available, I wonder if Harvey could point us to the place at which
Angus makes his supposed unequivocal claim that the consistency of PA
is a "legitimate mathematical problem".

As far as I can see, the relevant exchange takes place around 32:40 and

   HF: [Voevodsky] is extremely confused about whether PA might be

   AM: How is he confused? It *might* be inconsistent. I mean, it would
       be a shock. That would be a genuine crisis. I think the inconsistency
       of ZFC would not be. ... None of us know that PA is consistent.

I don't read Angus here as saying that he believes the consistency of PA
to be a "legitimate mathematical problem". For example, what Angus says
is wholly compatible with the view, expressed by William Tait in his
posting of 22nd May, that "the search for nontrivial consistency proofs
is off the board" (since, roughly, any consistency proof is at least as
doubtful as the system whose consistency it establishes). Also, given the
tenor of the rest of his talk, I imagine that Angus would very
much not view consistency investigations as the concern of legitimate

Perhaps I should add that my own view about the consistency of PA
is that it is as much a theorem of mathematics as other theorems of
mainstream mathematics (some of which are demonstrably interreducible
with it). I believe this is similar to Harvey's view.

Nevertheless, I think that the record should be set straight about
what Angus actually said in his talk. Or perhaps Harvey can clarify
where Angus made the claim he attributes to him.


Alex Simpson, LFCS, School of Informatics, Univ. of Edinburgh, UK
Email: Alex.Simpson at ed.ac.uk             Tel: +44 (0)131 650 5113
Web: http://homepages.inf.ed.ac.uk/als   Fax: +44 (0)131 651 1426

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