[FOM] Is anyone working on CH?

[Arnon Avron]

On Sun, Oct 21, 2007 at 01:12:02PM -0700, Martin Davis wrote:
> Arnon Avron wrote:
>  > One remark: one way in which it has already affect
>  >"normal" mathematics is by throwing from it propositions
>  > that have been proved to be undecidable. Are there now
>  > (what you call) "normal" mathematicians that are trying
>  > to solve the problem of CH? Will anybody put it now
>  > anywhere on the list of the most important open problems
>  > of mathematics? Yet CH certainly was a problem of
>  > "normal" mathematics at the time Hilbert formulated his
>  > famous list of problems!
> 
> My answer to the two questions you raise is: Yes, yes!
> 
> Hugh Woodin has been carrying out a fascinating 
> attack on CH. You can read about it in two recent 
> numbers of the Notices of the AMS, hardly an obscure publication.

I am aware of course of Woodin's work (how can someone who 
regularly reads FOM for many years be unaware of it?). This is
the reason I included in the formulation of my 
question the three word in brackets "(what you call)". 
I am afraid  that *Harvey* does not include in "normal
mathematicians" mathematical logicians and set theorists
like Woodin (following in this the people he does take as "normal
mathematicians"). Needless to say, if we do include them
then the answer to Harvey's question about the
relevance of the incompleteness phenomena to "normal
mathematics" is immediately "yes".

 Personally, I take the work in mathematical logic 
as no less "normal" (and frequently more important)
than the work done in what Harvey calls "normal mathematics".
Still, it should be admitted that VERY few people would
include nowadays CH in the list of important problems
in mathematics, and even fewer actually work on it. 

> G?del certainly regarded CH as one "of the most 
> important open problems
> of mathematics" and, diffidently mentioning 
> myself in the same sentence, I do too.

Again, I doubt that you too would be recognized as doing
"normal mathematics" by the "normal mathematicians" -
and it seems to me that this would now apply even
to Goedel himself. In any case, Goedel belonged to
a different generation, and his views were formed long
before the results of Cohen and himself. So he hardly 
can serve as a counterexample to my thesis that 
because of its undecidability in ZF (and beyond), 
CH is not considered NOW by most "normal mathematicians"
as an important problem of "normal mathematics". 

> 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.
 
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.

> People who have thought about the matter suggest 
> that using various work-arounds  the proof could 
> be brought down to something like 3rd order 
> number theory. Is it really so improbable 
> that FLT is unprovable in PA?

This you should ask Harvey, not me. Recall that for me it is 
rather probable that Goldbach conjecture is undecidable in PA, so
certainly the same is probable for me concerning FLT. 

Arnon Avron