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

Arnon Avron aa at tau.ac.il
Mon Mar 9 04:56:22 EDT 2015

On Fri, Mar 06, 2015 at 10:17:51PM -0500, Timothy Y. Chow wrote:
> 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 comparison is inadequate. Already Euclide distinguished between self-evident
propositions and those that are not, and realized that we can be sure about the 
truth of a proposition of the latter type  only
via a valid proof of it from the self-evident propositions.

Unilike TPC, derstanding the validity of the induction principle is inherent in
the understanding of the natural numbers. As Feferman has put it, it is a principle
that anyone that accepts the natural number should accept as well.

> 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.

This is a better point. Here, if I understand you correctly, you do not
compare the induction principle to the TPC, or doubt its validity,
but point out that applying it depends on what we accept as a 
"property" of the natural numbers. I agree. So the question boils
down to whether any formula in the language of PA
defines a definite property of the natural numbers. Assuming that nobody
questions this in the case of the atomic formulas of that language,
the answer is affirmative -  provided we agree that if A and B define
such properties then so do -A, A&B, and \exists x.A (say). Once one accepts 
this then BY INDUCTION(!) every formula of the language of PA
defines a definite property. (And there is a really no difference here 
between formulas with  5 alternations of quantifiers and formula 
with 6 alternations of quantifiers... If you think otherwise, can you point
out what is the minimal number of alternations that makes formulas doubtful,
and what is special about that number?). Now I do not think that one can
truly and honestly doubt this induction step. In particular: assuming that all
free variables in A are assigned values in N, "\exists x.A"
is true if by trying to substitute 0 for x, then 1 for x, then 2 for x etc
we eventually get a true sentence, and "\exists x.A" is false otherwise. 
If somebody really does not undersrtand this, then I can do nothing about it, 
but for me (and I strongly believe that deep inside by everyone else) this is crystal 
clear. Moreover: for this a *potential* understanding of the
quantifiers and the infinity of N suffices!

> 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.


I do not see the connection  between the two parts of this paragraph (before
and after the "Hence..."). Yes, I do know "many ways of weakening
the induction axiom to produce a system whose consistency doesn't
imply the consistency of PA". The simplest of them is to throw induction away 
altogether (like in Q). Still, I do not have any picture (let alone "fairly clear" one)
what it could mean for the *natural numbers* to exist, yet for PA to be inconsistent.


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