[FOM] Re: Atlanta Meeting
Timothy Y. Chow
tchow at alum.mit.edu
Wed Jan 19 13:33:01 EST 2005
Thanks for the detailed synopsis of those events at the AMS meeting!
Harvey Friedman <friedman at math.ohio-state.edu> wrote:
> Simpson's 1999 SOSOA book (Subsystems of second order arithmetic) had
> served as such a text until it went out of print shortly after it
> appeared. However, there is no definite timetable for its republication.
Is there anything we can do to help this process along? Or does the
publisher hold all the cards?
> Furthermore, both Martin and Woodin emphatically stated that we know the
> correct theory of second order arithmetic. Neither Martin nor Woodin
> explained what "correct theory" means here, and why these truth value
> determinations are so compelling. [They are referring to "all projective
> sets of real numbers are determined in the sense of infinite game
> theory".]
As usual I need a remedial lesson---what exactly is the theorem in
question here?
> Show that over ZFC, any equivalence of the continuum hypothesis requires
> a lot more quantifiers. Show that over ZFC, any statement consistent
> with ZFC that settles the continuum hypothesis, requires a lot more
> quantifiers.
This is interesting. How many quantifiers does (a straightforward
formalization of) projective determinacy use? What about V = L?
> Kleiner indicated, to my surprise, that acceptance by the mathematics
> community in this case is many years into the future (I don't remember
> the figure but I thought it might have been at least ten years!). He
> mentioned one reason: Perelman leaves many claims as exercises for the
> reader. These exercises are nontrivial.
I asked Kleiner about this not long ago and he said (I believe he would
not mind my quoting his email), `It is unclear if anyone will ever
publicly announce that the proof is "correct". A more likely scenario is
that various readers will make nuanced statements describing the status of
the verification process.'
It seems that this kind of situation is getting increasingly common: A
major open problem is reportedly solved, but the writeup is not cleaned up
to everyone's satisfaction. The system of rewards is such that people
can't get much credit for doing the cleanup, unless they can arrange to
have the cleanup be an incidental byproduct of some other work that is
judged to be meritorious for some other reason. This doesn't seem to be
a very satisfactory situation (unfortunately, I don't see how to solve
it). In the case of the Poincare conjecture, there is actual cash riding
on the solution. How much of the cash will go to solvers of the
"nontrivial exercises"? My guess is $0. (Unless, of course, one of
the exercises turns out to be wrong in a really big way. But even then
...imagine if there had been a cash award for the classification of finite
simple groups. All the cash might have been awarded long before
Aschbacher and Smith's recent books were even announced, since lots of
people considered it proved a long time ago.)
Tim
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