From: rose@fsu1.cc.fsu.edu (Kermit Rose)
Newsgroups: sci.math
Subject: Re: Stupid question about FLT
Date: 24 Jul 92 15:28:20 GMT
Reply-To: rose@fsu1.cc.fsu.edu
Organization: Florida State University
News-Software: VAX/VMS VNEWS 1.3-4

In article <1992Jul19.195047.28807@galois.mit.edu>, jbaez@cayley.mit.edu (John C. Baez) writes...
>In article <BrJtH1.24x@cs.psu.edu> sibley@math.psu.edu (David Sibley) writes:
> 
>>"If Fermat's Last Theorem is undecidable it's true, right?"
....
>>
>>Can someone please straighten this out for me?
> 
>(A slightly dramatized (by me) account of something that always amuses
>me.)  
> 
>Forgive me if I get this messed up, since it's been a while since
>I've thought about it.  We can, for example, describe models of Peano
>arithmetic using ZFC.  One of these models would be based on the set
>{0,S0,SS0,SSS0....} where 0 is the null set and S, the successor
>operator, is given by Sx = x U {x}.  If we define addition and
>multiplication in the "obvious correct way" in ZFC, we have a formal
>description in ZFC of a model in PA.  This would be one way of thinking
>of the REAL (i.e., honest-to-god) natural numbers.  One can cook up other nasty
>"nonstandard" (i.e. fake) models of the natural numbers as well in ZFC.  I have
>always found that Boolos' & Jeffrey's book "Logic and Computability"
>gives the most elementary descriptions of some nonstandard models of the
>integers.
> 
>Of course, one can easily become suspicious of whether we really know
>what these REAL natural numbers are (i.e., how their properties differ from the
>fake ones).  First of all, what we can prove using Peano arithmetic
>about the REAL natural numbers is precisely equal to what holds in ALL models
>of the integers.  Rather curious, eh, that our axioms for the natural numbers ,
>which are supposed to capture our basic intuitions about the natural numbers,
>are only able to prove those things about the REAL natural numbers which also
>hold for all the nonstandard ones?  Even worse, there is no recursive
>enumerable set of additional axioms can be added to the Peano axioms so
>that the theorems which can then be proved are precisely those which
>hold for the REAL natural numbers.  (I.e., there will still be
>nonstandard models even if we through in more axioms in any systematic
>way.)  
> 
>......
> 
>So one should not be upset at the fact that the logician could not tell
>you exactly which were the REAL integers. :-)

So, it seems that the REAL integers are the abstract numbers underlying the 
numerals that are our model for the integer.  Therefore, I suggest that IF 
FLT is undecidable, it is only because a proof of it would require a proof 
for each of the infinite primes.  Clearly if FLT has no counterexamples, 
then it is true.  If it is undecidable, it cannot be proven false, 
therefore it is true.  And this would hold in EVERY model of the integers.
I suggest that if FLT is undecidable in one model of the integers, it is 
undecidable is ALL models. It is no paradox that if it is undecidable, then 
it is true. This is because undecidable simply means unable to prove true or 
false. If we cannot (in principle) prove FLT false, this means that it IS 
NOT FALSE.  Whereas, it might be possible that no method exist to prove FLT 
is true.  Since we say that FLT is true if it is not false, we say that if 
FLT is undecidable, then it is true.

rose@fsu1.cc.fsu.edu          To be sure I see your response, use e-mail.
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