[FOM] Is anyone working on CH?

[joeshipman@aol.com]

Davis:

>> Arnon, you mention elsewhere Fermat's last
>> theorem. As I'm sure you know, the proof of FLT
>> that was found is not even formalizable in ZFC,
>> going far beyond the bounds you suggest as
>> already causing trouble. (ZFC + an inaccessible
>> will do.)  I haven't heard any mathematicians
>> question whether FLT has indeed been proved because of this.

Avron:

>I do question it.
>Moreover: I think that most mathematicians simply
>do not know that the current alleged proof goes beyond ZFC,
>and if the meaning of this is explained to them then at
>least some of them might have some reservations whether it was
>really proved.
>
>BTW: I have already said on FOM that for me
>FLT will be considered proved only when it reaches the
>point when the proof is presented  in a textbook style
>so that any mathematician who wish to devote a reasonable
>time for it can follow and check the proof for himself. I
>take it as unbearable and against the whole spirit of mathematics
>that someone like John McCarthy would say that he knows
>he will never be able to understand the proof (as he said
>here on FOM not long ago). A proof that is accepted
>only because of the authority of some experts
>and is not really accessible to the majority of mathematicians
>is not a proof yet (and most probably, may I add, contains
>some very-difficult-to-spot subtle mistakes). As an
>example of a completely different situation I can give
>Paul Cohen results, which Paul Cohen himself made accessible
>for any interested mathematician by writing his famous
>manuscript on CH.

There are two separate phenomena occurring here which it is important 
to distinguish.

The first phenomenon is that although proofs can be extremely long, 
intricate, and difficult (so that only the specialist in the area is 
capable of following them, while still being fully rigorous and 
reliably verified by the mathematical community), over time proofs of 
important theorems tend to get "boiled down" in a way that makes them 
more accessible (or else completely different and simpler proofs are 
found). If John McCarthy was merely saying that he did not intend to 
devote the necessary years to learning as much number theory as is 
currently needed to truly understand Wiles's proof, but hoped that a 
simpler proof will be found that he could understand, this is not an 
"unbearable situation".

In fact, I am not sure yet whether I agree or disagree with McCarthy's 
attitude toward this particular proof in my on case; for reasons 
related to my own research I have recently begun to study number theory 
again, starting with Lang's text, but I expect it to take the 
equivalent of two semesters of study to get fully through Lang's text, 
and another two, should I choose to continue, to shore up my background 
in algebra to the point where I could tackle with full confidence the 
book "Modular Forms and Fermat's Last Theorem" (edited by Cornell, 
Silverman, and Stevens), which builds up to a proof of Wiles's theorem 
over several hundred difficult pages.

In other words, although it is unfortunate that the proof has not yet 
been boiled down or simplified to the point of being accessible to an 
ordinary mathematician with a more reasonable amount of effort, it is 
not "unbearable", because the breadth of understanding an acceptance of 
Wiles's result in the mathematical community (as evidenced by the 
aforementioned book, which has more than two dozen contributors) 
persuades me that the theorem really has been proven, with a very small 
possibility (less than 0.1%) that there is an error that everyone has 
overlooked.

The second phenomenon is that some advanced results in mainstream 
mathematics, including Wiles's, involve extremely abstract work of 
Grothendieck and others in algebra and algebraic geometry that makes 
frequent use of the "Grothendieck Universes" axiom (equivalent to a 
proper class of inaccessible cardinals). I am informed that it is 
usually easy to eliminate this axiom in favor of the axiom that there 
are infinitely many inaccessibles, more difficult but straightforward 
to eliminate the axiom in favor of the axiom that there exists ONE 
inaccessible cardinal, and very difficult to eliminate all use of 
inaccessibles in any generality (though apparently specialists feel 
that they can eliminate all use of inaccessibles in the proof of any 
specific result by being more explicit and constructive at various 
points in the proof).

This is unsatisfactory for three minor reasons (the referenced work was 
never published in refereed journals, it was never translated into 
English, and no one has ever produced a proper textbook giving all the 
necessary details in a manageable amount of space), and one major 
reason: THERE IS NO METATHEOREM which would allow anyone to conclude 
that Wiles's result is provable in ZFC.

Several years ago I queried this discussion forum rather intensively, 
and no one was able to report EITHER that there was such a metatheorem, 
OR even that some specialist in number theory had asserted on his own 
authority that the use of inaccessibles in the proof of Wiles's 
particular result could be eliminated by, say, the application of 
technique X at points Y and Z.

I regard THIS situation as scandalous and close to "unbearable", not 
because the axioms of inaccessibles [or their corresponding 
arithmetical consistency statements] are so dubious, but because the 
community of "mainstream mathematicians" who have understood and 
validated Wiles's proof are so insouciant about foundations that they 
don't even consider it necessary to REMARK that the use of 
inaccessibles can be eliminated in this case, let alone actually do it 
(and let even further alone formulating and proving a metatheorem so 
that it doesn't have to be done separately for each result that depends 
somewhere on Grothendieck Universes).

-- JS
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