[FOM] inconsistency of P

[Timothy Y. Chow]

Arnon Avron wrote:

> 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 agree that one has to practice a lot of "doublethink" to keep everything 
straight, but I'm not convinced that you have articulated a serious 
objection here.  Suppose someone says they don't believe in inaccessible 
cardinals.  I don't think it would be a very convincing attack to accuse 
them of saying that they must actually believe in inaccessible cardinals 
because if they didn't, wouldn't every cardinal be "accessible" and hence 
wouldn't their statement be vacuous?  Knowing what someone "means" isn't 
the same as believing that what they're talking about *exists*.

> 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?

This is a somewhat more cogent objection.  I think the response is this: 
he understands the recursion as long as the recursion is feasible.  
Beyond that, his understanding shades off into uncertainty.  It's not so 
different from saying that I understand what a computer with a large 
amount of memory is.  Given some knowledge of computer engineering, I can 
comprehend what it means to build a large array of hard drives.  I know 
what it means to add another hard drive to the array---up to a point.  
Eventually, the array will get too large for the Earth.  What will we do 
at that point?  The operation of the drive might depend on the Earth's 
gravity.  Even if you have no trouble imagining that, what if the memory 
is so large that there is no way to build a storage system that won't 
decay in the billion or so years that it takes a signal from the CPU to 
reach the storage unit?  If Nelson has an analogous kind of "engineering" 
understanding of the natural numbers, then you won't be able to discover 
the discrepancy between his understanding and your understanding in 
practice.

> 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"?

He may not, in fact, know that he is doing so correctly.  But if you take 
this kind of skepticism seriously, then I think you will find yourself 
taking the Kripe/Wittgenstein skepticism about rules and private language 
seriously.  Forget Nelson; how do you and I, who profess to know the 
natural numbers, know that we have the same understanding and are 
correctly following the same rules?  You and I have only had a finite 
amount of interaction, so maybe we *don't* have the same understanding but 
have radically different understandings that just so happen to coincide on 
the finite segment that we have "accessed" and confirmed against each 
other using our finite human bodies and minds.

> 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.

This is the easiest objection of all to meet.  The confidence of an 
engineer is all that is needed here.  It is not "absolute" confidence, of 
course.  After all, he retracted it when a design flaw was discovered.  
More to the point, once you have a specific, concrete proof, there's no 
need whatsover to believe in the set of natural numbers as a completed 
totality for this purpose.  No infinite induction schema is needed for any 
fixed, finite proof.

Tim