[FOM] inconsistency of P
Timothy Y. Chow
tchow at alum.mit.edu
Mon Oct 3 11:10:04 EDT 2011
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.
Tim
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