[FOM] inconsistency of P
Staffan Angere
Staffan.Angere at fil.lu.se
Sat Oct 8 17:49:21 EDT 2011
Arnon,
do you take "all men are mortal" to be understandable by someone who does not know about every so-called "African chief" Frege hasn't heard of? I think most of us would, because understanding is not really extensional. To understand what a "statement" is, you do not have to assume that there is a class of all statements, and that the understanding somehow involves this class. It is enough to know the rules for forming wff's. Likewise, once could understand P by just knowing the axioms, and being able to (ideally) produce proofs of arbitrary provable formulas, and possibly counterexamples to non-valid ones. Indeed, what else could anyone require? We are, as has been stated here before, finite beings. We simply cannot grasp infinities extensionally.
To take an analogy: understanding chess does not require one to believe that there exists a class of possible games that satisfy the rules. One could even believe that there could appear situations where the rules of chess at the same time prohibit and allow a certain move, but that none of these situations have been actualised yet. Such a belief might, of course, be held to be inductively dubious (people have, after all, played a lot of chess), but it would not disqualify anyone from understanding the game.
Best wishes,
Staffan
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
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