[FOM] "Proof" of the consistency of PA published by Oxford UP
Richard Heck
richard_heck at brown.edu
Wed Mar 4 15:33:52 EST 2015
On 03/02/2015 09:51 PM, Timothy Y. Chow wrote:
> Gyorgy Sereny wrote:
>> I would like to inform you about a strange publication.
>> I have just come across a book newly published by
>> Oxford University Press:
>>
>> The Consistency of Arithmetic: And Other Essays Hardcover
>> 24 Jul 2014 by Storrs McCall (Author)
>
> I have to agree with Gyorgy Sereny that the first article in this book
> is strange. Here I mainly want to point out that the full text of the
> article is linked from the author's website:
>
> http://www.mcgill.ca/philosophy/people/faculty/mccall
>
> Or you can go directly to the Word document:
>
> http://www.mcgill.ca/philosophy/files/philosophy/the_consistency_of_arithmetic_feb_10_2011.doc
>
>
> Skimming through the paper, I do not see any interesting mathematical
> insight. I would characterize it as an argument for the consistency
> of PA based on physical intuition. Perhaps there is an audience for
> this sort of thing, but I'm having trouble imagining one.
What I find hard to understand is what contrast exactly he thinks there
is between the kind of argument he is giving and other sorts of proofs
of Con(PA). He claims "no semantic consistency proof of Peano arithmetic
has yet been constructed", but that is pretty obviously false. So the
fact that the proof is semantic rather than syntactic isn't, by itself,
all that significant.
The fact that the proof, as stated, isn't fomalized isn't all that
interesting, either. Very few actual proofs are formalized, and there is
no obvious bar to formalizing this one.
So we're being given an informal semantic consistency proof. But exactly
what the cash value of the proof is, it seems to me, isn't obvious until
we know exactly what sorts of assumptions it is employing. Somewhere,
obviously, there are some strong assumptions being deployed. The fact
that the presentation obscures where they are is not a virtue.
Here again, an early remark seems to me at best misleading: " But to
deduce [PA's] consistency in some stronger system PA+ that includes PA
is self-defeating, since if PA+ is itself inconsistent the proof of PA's
consistency is worthless". That isn't the only option, and it isn't the
usual reaction, it seems to me, to Gentzen's proof of Con(PA), either.
If there's something interesting here, it's the way the semantics he
develops doesn't require there to be a single infinite model, but only a
succession of every-larger finite models. There are antecedents to that
sort of idea in modal structuralist views, I believe, of the sort
developed by Hellman, and perhaps more than antecedents. Maybe there are
more developed forms of this idea, too, and if so I'd be interested to
know where.
Richard Heck
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