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

[Arnon Avron]

Tim,

You recently raise more issues and questions than I can answer
immediately. So for the time being this is my reply only (or mainly) 
to the first of three postings of yours that were explicitly
connected to things I said. 

> Again to play devil's advocate, suppose that one accepts the
> induction schema outright only for quantifier-free formulas?

This is indeed the only "reasonable" choice for a devil's advocate,
because my argument is based on the principle that
if one adds one quantifier on N to a formula which 
expresses a meaningful property of tuples of natural
numbers, then one gets another formula with the same
property. Everyone who accepts this principle,
and accepts induction on quantifier-free formulas of PA
necessarily accepts  it for each formula of PA. She
might pretend to do it on a "case-by-case basis", refusing
to admit that she accepts it "for all", but both of us
would know that she would accept it for each specific
formula I bring her, and this is the same for me
(on the basis of seeing N as a potentially infinite set).
However, if one refuses to accept the above principle
then one is really stuck with induction on quantifier-free formulas.

Now I am unable to understand someone who says that A(x,y)
expresses a meaningful property which is  meaningful for every pair
of natural numbers (or for each pair of natural numbers: 
for me these are the same -  in Hebrew there is no such
distinction...), but maintains that the existence
of a natural number  y such that A(x,y)
is not a property of x. I have no way to convince
such a person that he is  wrong. I can only think
that he is deceiving himself, and wait till I catch him
in a case he says something which is in direct conflict
with his official beliefs (this is almost sure to happen...).

> Specific formulas with quantifiers could also be accepted on a
> case-by-case basis, but only after one has gone through the mental
> process of thinking about the formula and convincing oneself that
> induction for that property is "obvious."

I have already hinted above what I think about this scenario.
 
> >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.
> 
> "Fairly clear" might be an overstatement, but as an analogy, let's
> consider the power set axiom P of ZF.  Someone might feel that they
> can grasp the set-theoretic universe (including of course the set of
> natural numbers) clearly enough to be convinced that ZF - P is
> consistent, yet not grasp the concept of an "arbitrary subset" with
> enough clarity to rule out the possibility that ZF is inconsistent.

The difference is that induction is an essential part of our
mental construction of the natural numbers (1,2,3, *and so on*...).
Therefore the validity of induction is inherent in the very
concept of natural numbers - a concept which  is crystal clear to 
everyone. In contrast, the concept of a set has never been
so clear (originally it meant, I believe, only definable sets,
though "definable" is of course a fuzzy notion). Accordingly,
I find this last scenario you describe as not impossible, but the 
analogy is in my opinion is very weak.

> Do you agree with that?  If so, then returning to PA, someone might
> feel that they understand what is meant by "1, 2, 3, 4, and so on
> indefinitely" yet not understand what an "arbitrary first-order
> property" of the natural numbers is with enough clarity to rule out
> the possibility that PA is inconsistent.  

I am not assuming the notion of an "arbitrary first-order property
of the natural numbers". I am assuming the absolutely clear notion
of an "arbitrary first-order formula in the language of PA",
and the in-general fuzzy property (of formulas in some language)
of "expressing property of natural numbers".
I have explained what makes me absolutely certain that understanding 
what is meant by "1, 2, 3, 4, and so on indefinitely"
forces one to see with 100% certainty that N is a model of PA,
and so that PA is consistent. If you claim that there
is some circularity in my thinking I'll accept, and note
that we cannot even start to talk about such issues
(what is a formula of PA? What is a proof in PA? What does it
mean to say that PA is inconsistent) without being involved in
such circularity, i.e. understanding the things we pretend
not to understand.

> The induction that you
> describe for first-order properties is conceptually more complicated
> than the induction needed to grasp the successor operation.  And of
> course, we have technically precise ways of demonstrating that the
> greater conceptual complexity is real and not illusory.  Given this,
> I don't find it convincing to argue that induction for first-order
> formulas is part of the *fundamental meaning* of the natural
> numbers, any more than I find it convincing to argue that the power
> set axiom is part of the *fundamental meaning* of the set-theoretic
> universe.

 Really, Tim? (or are you still playing the devil's advocate?)

Arnon