FOM: Re: script/can't run away

[Harvey Friedman]

Fenner writes 2:11PM 9/145/98:

>Harvey,
>
>Your script sounds a lot like ones I've had a number of times with
>Goodstein sequences.  Have you ever tried *them* on other mathematicians?
>(High schoolers are facinated, too.)

I certainly don't believe you will get this kind of strong universal
reaction from professional mathematicians outside combinatorics, logic, and
f.o.m. First of all, full Goodstein sequences require iterated expansions.
The n(k) stuff appears very high on BOTH the scale of "recreational
enjoyment" AND the scale of "combinatorially fundamental." There is still
something articifical about even the watered down Goodstein sequences,
which admittedly avoid the iteration. Also you don't jump from 11 to
something incomprehensible in such a trivial way (passing from argument 2
to argument 3). Note that a great deal of the drama rests on talking about
a **specific integer** - n(3); this can be much more dramatic than growth
rates.

In my experience, the quality of reaction is very sensitive to such issues.
Since we are not talking about anything these people have worked with
before, or in most cases, anything they could imagine working on or using,
there has to be something very striking and natural about it. That's way it
is so useful to have something that is both **very recreational** AND
**combinatorially fundamental**. E.g., when I told Erdos about the
existence of n(k) for all k, and that it had a large growth rate (at that
time I didn't have a lower bound on n(3), but did have an unwritten sketch
of one for n(4)), he immediately got interested in just the theorem that
n(k) exists for all k, and demanded to know why I hadn't published it, and
made up several variants on the spot - all at the Columbus, Ohio Airport -
which he immediately wanted to consider with me. Well, he soon later got on
his airplane, and we didn't have enough time to get anywhere. I have not
yet had a chance to do research on his variants, but it does show how
intrinsically natural and compelling he found this n(k) business **just as
a piece of combinatorics**.

Having said this, I do encourage you and others on the FOM to come up with
a genuine competitor to this, backed up with a precise lower bound for the
single integer involved. I already know how to do this sort of thing for
stripped down Goodstein sequences, but I don't think it is exciting enough.
Perhaps somebody can find something else that is really striking.