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

[W.Taylor at math.canterbury.ac.nz]

Quoting "Timothy Y. Chow" <tchow at alum.mit.edu>:

> Arnon Avron wrote:
>
>> I would add that I *do not believe* people who pretend to doubt the  
>>  consistency of PA.

I would have said the same, up to the point where I got a private
communication from Edward Nelson about this point, perhaps 10 years ago or so.
I had politely suggested (something like) that his stance was a posture
of principle, and that he didn't really expect to find PA inconsistent.
His reply was that no, he fully expected to find such an inconsistency,
and that he was working continuously to dig one up.

> 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.

Excellent point.    I like to say that N is a crystal-clear model;
but that all of its properties are not; the latter is just set theory,
whose intended model is clear to many, and unclear to many.

> 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?

I agree with the conclusion, but not necessarily the antecedent.
I recall raising a storm on a math forum by suggesting that
(for math reasons) we *knew* TPC was true, but we didn't have a proof.
Someone else made a supporting statement to counter the howls of derision,
that surely we *know* that chess with black starting with a queen off
was a forced win for white, but we have no proof of it?

> 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.

Exactly so.  As in the chess case, a proof could be so horrible,
that we can never "behold" it all, even with computer help.
Such things are well known in math logic OC - the existence of
infeasibly long irreducible proofs.

For the record, I, like Arnon, feel that N is an obviously correct model
of PA, and that serious ontological doubt is almost ludicrous;
(while realizing that similar things have been said in the past
about cases that seemed clear-cut at the time).

Bill Taylor


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