[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 


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