[FOM] "Proof" of the consistency of PA published by Oxford UP

[Michael Kremer]

The following is related to Richard's points...

On a quick skim I find the final section of the paper confused. He wants to ward off the claim that his argument requires an infinite set. He does this by claiming that his argument does not require an infinite model; and that he does not have an unrestricted comprehension scheme which would allow one to construct the set of all models. But his argument requires that there is no upper bound on the size of his models. Whether this requires an infinite set or not, it surely requires in some sense that there are infinitely many blocks. (There is no finite bound on the number of blocks.)

Moreover, the argument makes use of an operation of concatenation which yield larger arrays of blocks out of smaller arrays of blocks. And this operation needs to satisfy some set of axioms with sufficient similarities to the axioms of arithmetic to make one suspicious that the theory of concatenation is at least as strong as arithmetic. If there are only finitely many blocks in the universe we would be in a position in which the concatenation operation could not always be applied. (If there are 1000 blocks in the universe, we could make an array of 600 blocks, and an array of 601 blocks but we could not concatenate them as they would always have overlap.)

Put it another way: Russell and Whitehead's Axiom of Infinity in Principia Mathematics would seem to be satisfied by McCall's blocks (iirc, it does not state that there is an infinite set; it states that for every inductive cardinal number n, there are n things).

Michael Kremer
________________________________
From: fom-bounces at cs.nyu.edu [fom-bounces at cs.nyu.edu] on behalf of Richard Heck [richard_heck at brown.edu]
Sent: Wednesday, March 04, 2015 2:33 PM
To: tchow at alum.mit.edu; Foundations of Mathematics
Subject: Re: [FOM] "Proof" of the consistency of PA published by Oxford UP

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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