[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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