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

Timothy Y. Chow tchow at alum.mit.edu
Fri Mar 6 22:17:51 EST 2015

Arnon Avron wrote:

> 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.

I agree with your criticism of those who claim to understand what PA is 
but also claim not to understand what the natural numbers are.

However, there is another possible avenue to doubting the consistency of 
PA.  Namely, having some grasp of the natural numbers is not the same as 
understanding every *property* of the natural numbers.  Certainly none of 
us is able to directly apprehend the *truth* of all first-order statements 
about the integers.  If we can be uncertain about the truth of (for 
example) the twin prime conjecture, then why can't we be uncertain about 
the truth of instances of the induction schema with arbitrarily many 
alternations of quantifiers?  The doubts here would not be about the 
natural numbers per se but about whether such incomprehensibly complex 
formulas coherently define a "property" of the natural numbers to which 
induction can be applied.

Although I don't have such doubts myself, I don't see anything incoherent 
about them.  After all, we know many ways of weakening the induction axiom 
to produce a system whose consistency doesn't imply the consistency of PA. 
Hence we have a fairly clear picture of what it could mean for the natural 
numbers to exist and to satisfy a weaker induction schema, yet for PA to 
be inconsistent.


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