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

Arnon Avron aa at tau.ac.il
Fri Mar 6 05:29:37 EST 2015

I truly do not understand the point here.

I should admit a terrible sin: I am 100%  certain of the
consistency of full PA (not only Q). Worse: the reason
I am sure of it no less than I am sure about any theorem of Q is that
the axioms of PA are indeed *obviously true* in the natural
numbers. What better and more convincing proof can one want?

I would add that I *do not believe* people who pretend to
doubt the consistency of PA. As I have pointed out 
on FOM in the past, people who do not understand the 
natural numbers cannot understand the notions of formulas
and of formal proofs either (both being recursively defined).
Accordingly, they cannot even understand what PA is and what 
its consistency mean - let alone doubt its truth.

Arnon Avron

On Wed, Mar 04, 2015 at 12:34:03PM -0800, Charlie wrote:
> 	Consider the axioms of Peano Arithmetic minus the induction schema, plus Robinson???s axiom, which says that if x differs from 0, then it???s a successor.  Every one of these axioms is ???obviously true??? for the natural numbers.   One might say ??? I???m *not* saying this ???  that since every single axiom is obviously true of the natural numbers, plus since they do not *seem* to interfere with each other, the entire system (Q) *must be* consistent.   If we wished to dignify this reasoning by calling it a real proof, we could say it???s a proof of the consistency of Q ???by intuitive inspection???.
> > On Mar 2, 2015, at 6:51 PM, Timothy Y. Chow <tchow at alum.mit.edu> 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.
> > 
> > Tim
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