[FOM] Paradise vs. Hell

[Dana Scott]

On Dec 18, 2005, at 4:52 PM, Doron Zeilberger wrote:

> Dear Dana,
>
> Thanks for the very interesting feedback and reminding me
> that I had to be shocked for the last 60 years. Somehow the
> Fox et. al. statement is even more concrete and counterintuitive.
> Thanks also for alerting me about Harvey Friedman's even more
> shocking discoveries, but I believe that if you exclude even
> the potential infinity, then I can sleep in peace.
>
>    Best wishes
>             Doron
>
> P.S. May I post your interesting response as feedback to my
> Opinion?

Hey Doron!

Thanks for the reply.  We all could use some peaceful sleep
these days, but I really wish you would take a peek at Harvey's
brief paper on his new incompleteness results.  (It is a six-page
abstract without proofs but with much attention given to putting
the statement into context.)  He says:

> 'Beautiful' is a word used by mathematicians with a semi
> rigorous meaning.  We give an 'arguably beautiful' explicitly
> Pi01-sentence independent of ZFC. See Proposition A from section 1.

The PDF-file can be downloaded from:

   http://www.math.ohio-state.edu/%7Efriedman/pdf/Pi01120905.pdf

He makes a clear distinctions between things that are provable
and analogous statements which are independent.  And I don't feel
that excluding potential infinity is the right move to happiness here.

As regards Fox, his statements are very, very nice (and he ties up
many loose ends), but I don't agree that he is "more concrete"
than the old Erdős-Kakutani result.  (BTW, here is the reference:

	"On non-denumerable graphs", Bull. AMS, vol. 49 (1943),
	PP. 457-489. )

But we don't have to argue about this point considering all that
Fox has done.

I did some more searching and found a copy of his excellent
paper (at a non-MIT site!) as follows:

    http://www.math.ucsd.edu/~sbutler/seminar/FoxRado1.pdf

I notice he credits you (and others) "for helpful advice and
comments concerning this subject matter" -- though he does not admit
to doing the Devil's work!

As regards measurability, I find it especially interesting that Fox
and a collaborator have shown in ZF set theory that the equation

	x1 + 2 x2 + 4 x3 = x4

is NOT 4-regular if you assume the Axiom of Choice, but IT IS
if you assume Dependent Choice and Every Set of Reals Lebesgue
Measurable.  (The latter assumptions were proved consistent with
ZF -- assuming an inaccessible cardinal -- by Solovay back in
1970.)  There is no question that measurable sets are better than
non-measurable ones!  (And Fox indicates other reasons to think
so.)

Indeed, the well known proof from Choice for finding a non-measurable
set by picking one representative out of each coset of the quotient

	(R/Z)/(Q/Z)

(where R/Z is the additive group of reals mod 1, and Q/Z is the subgroup
of rationals mod 1), shows that choosing arbitrarily from a suite of
uncountably many sets is highly non-constructive.

So maybe the right advice here is simply: AVOID BAD SETS!

All the best,

        -- DANA

P.S.  And, yes, you can post my comments.  I would like to hear other
comments for other people as well.