[FOM] inconsistency of P

[Arnon Avron]

Timothy Y. Chow wrote:

> On Mon, 3 Oct 2011, aa at post.tau.ac.il wrote:
> > Can Nelson understand what is Pi^0_0 statement? How?
> > Can he  understand what is a formula? How?
> > Can he  understand what is a proof? How?
> > 
> > I cannot understand how he can use all these
> > concept and yet claim to doubt the consistency of P
> > (how does he understand the notion of consistency of P
> > at the first place?)
> 
> Obviously, the best person to answer these questions is Nelson himself.  
> However, I'll suggest how someone, not necessarily Nelson, could address 
> these questions.  For ease of explanation, I'll pretend that that 
> "someone" is myself.
> 
> I can "understand" all these concepts in the sense that I have assimilated 
> myself into mathematical society by learning the rules for manipulating 
> sentences involving these terms, well enough to publish papers that obey 
> these rules and garner me some measure of social recognition.  But 
> privately, I regard almost all mathematical statements as merely "true 
> according to a certain story" (as Hartry Field might say).  Is it true 
> that Oliver Twist was younger than Fagin?  Yes, according to a certain 
> story.  Do I understand what that question means?  Yes, as well as anyone 
> else.  Do I believe that Oliver Twist *really* exists (or existed)?  No.
> 
> I said "almost all" mathematical statements, not "all."  The exceptions 
> are purely finite statements that I can directly verify, or that a finite 
> machine whose operation I trust can directly verify.  "Tic-tac-toe is a 
> draw" is O.K.  "PA proves that sqrt(2) is irrational" is O.K.  "2^31 - 1 
> is prime" is O.K.  What about "2^127 - 1 is prime"?  That's starting to 
> get a little uncomfortable.  "PA proves `2^127 - 1 is prime'" is O.K., but 
> "2^127 - 1 is prime" is less clear.
> 
> By this point it should be clear that I have no problem with "there is a 
> proof of `0=1' in PA using at most one million symbols."  As for "PA is 
> consistent," I know what it means according to a certain story, and I know 
> that those who believe that the story is really true would then conclude 
> that, in particular, there is no proof of `0=1' in PA using at most one 
> million symbols.  But I myself don't believe that the story is really 
> true, and the most concise way to state that in a way that will be widely 
> undertood is to say that I don't believe that anyone knows that PA is 
> consistent.

This is indeed a very nice story that you are telling above, but it is indeed
nothing but a story -  a story I believe  that you do not really believe , and 
in fact neither does even Nelson. 

First, you refer above to  "finite statements" that are "exceptions".
But this takes us back to square zero: How can "you" understand
the meaning of "statement"? And what for god sake do you mean by
a "finite stastement"?? Since there is no infinity, every statement
(whatever this term means for "you") is necessarily finite, isn't it?

I should say that I cannot really understand this one big game of 
trying desparately to prove something "you" think is meaningless,
and pretending not to understand the concepts that "you" yourself use.

But let's forget about "your" views, and return to Nelson.
He was trying to prove the inconsistency of P, and later claims
that the problem of the consistency of P is still open. Now
do you say that according to the story he tells himself
by this he means something completely different than what I (for example)
mean? P is after all an infinite theory. So when one talk about about its
(in)consistency one talks aboput a property of an infinite object. And
if he claims that he refers only to its finite representation
(actually, one of its infinitely many representations, depending
e.g. on your choice of "symbols")
using schemas, then still he should be able to understand the notions
of strings, formulas, formal proofs, substitutions of strings for 
strings, etc. These are all recursively defined. So how can he
understand it, no matter what story he tells himself? Or is he meaning
something else? What? And if he is only playing according to
other people's story, how does he know that he is doing so correctly
if he does not understand the bic concepts and rules of the "story"?

 Note, by the way, that Nelson was *not* claiming that he has shown
anything of the form "there is a proof of `0=1' in PA using at most 
one million symbols." At least in his message to FOM he gave no bound
on the alleged proof of `0=1'. Note also that what he did was to
give an *outline* of a complicated proof of `0=1'. Obviously, like
anyone of us he was confident that if the outline is correct then an 
actual proof can be constructed out of it. It is beyond me
on what this confidence was based, given that he pretends not to understand 
the collection of natural numbers, or the collection of formulas of P,
or the collection of formal proofs in P.

Richard Heck wrote:

> As I understand him, Nelson is prepared to accept (i) Q and (ii) as
> much induction as is interpretable in Q. If so, however, then he has
> no problem understanding such notions. In particular, the theory
> I\Delta_0 + \omega_1 is interpretable in Q, and all of these notions
> can be defined there and their basic properties proven.
>
> Indeed, Visser once remarked that I\Delta_0 + \omega_1 is "just
> right" for syntax. That is why it has played such a significant role in
> the study of interpretability.

Again: what does it means that Nelson "accept Q"?. Does it mean
that he accepts that the collections of formulas, proofs and theorems
of Q are well-defdined? And in what basis does he "accept" it?
Because he reconizes the theorems of Q to be true? True about what?

Similarly, anyone who talks about I\Delta_0 + \omega_1 already 
understand entities that are at least complicated and "doubtful"
as the natural numbers. o

To sum up: I am conviced that a lot practice in doublethought
is needed for someone like Nelson in order to really believe
that he believes his official views.

Arnon Avron