FOM: Urbana Thoughts

[Harvey Friedman]

Response to Baldwin 10:25AM 6/11/00.

There is probably no serious disagreement between us.

>I think my attempts to draw careful lines in a brief
> comment at Urbana were inadequate.  Let me try again.

Agreed.

>internal/intrinsic: Those aspects of a subject which are
>developed by reflection on its own practice.

>external/extrinsic:  Those aspects of a subject which are
>developed to solve problems posed in another area.

Boolean relation theory (BRT), at least in its beginning stage (now), is
not adequately classifiable in these terms. It has aspects that are
somewhat internal/intrinsic and aspects that are somewhat
external/extrinsic.

1. BRT as internal/intrinsic. Obviously BRT is being developed primarily by
reflecting on the existing connections of large cardinals with concrete
mathematical contexts - which, in the hands of the set theorists, are at
the relatively high level of analytic sets and above. Borel diagonalization
theory of roughly 1976-1988, and now Boolean relation theory, directly
addresses this.

2. BRT as external/extrinsic.  BRT clearly is not being developed to solve
problems actually formally posed outside logic, in the strict sense of
"pose." I.e., we don't see the thinness theorem stated formally anywhere.

However, it is external/extrinsic in a weaker, related sense. Let us
concentrate for the moment on the thin set theorem. It is of course a
consequence of the classical infinite Ramsey theorem that is child's play
for the combinatorially oriented logician - although not, interestingly,
child's play for combinatorists.

But one of the interests of the Ramsey theory crowd has obviously been -
and still is - applications of Ramsey's theorem to universally attractive
situations. They clearly go to great lengths to show the power of Ramsey
theory, and are particularly interested in extremely elementary
mathematics. They are particularly interested in high school level theorems
which are trivial in structure and combinatorial in nature.

In this respect, the thin set theorem is exactly the sort of thing that
Ramsey theorists - and many other mathematicians - are looking for.
Actually, that's how I found it. First, I conceived of BRT. Then I asked

what sort of statements that mathematicians would be interested in are
actually special cases of BRT?

(Another basic example of BRT is the invariant subspace problem for Hilbert
space - which was already asked. BRT does not provide an answer, of course.)

I "knew" that the thin set theorem would be of interest to combinatorists
and a wide variety of other people. I checked this out in the real world
and found that this was true.

***So the thin set theorem WAS developed to solve a "problem" posed in
combinatorics, under an entirely appropriate sense of "problem" and "pose".
***

The further development of BRT that runs into large cardinals is supposed
to be the inevitable result of a standard process of research and
development that commonly drives mathematical research. BRT hasn't been
carried out nearly enough, but the architecture of it has been stated and
feedback has been obtained that it will be, in fact, the inevitable result
of a standard process of research and development that commonly drives
mathematical research.

3. Timing.

I want to view it as a purely historical accident that BRT doesn't directly
answer specific well defined questions posed by mathematicians. It would
have this role if we go out at the very most a handful of centuries. But
that is hard to test empirically - I just don't have that kind of long term
patience.

So one of my roles is to develop and exposit BRT in such a way that the
inevitability of BRT is made crystal clear.

>A little thought shows that it easy to get carried away by
>forcing such a classification on results.  The following
>theorem of Hrushovski is a key development
>in what I would have expected Harvey to call `pure model
>theory' -although `stability theory' was curiously missing from his list.

The list being referred to in my posting of pure model theory topics was
merely a list of topics in what I call pure model theory that Anand was
calling "not model theory but descriptive set theory" - over lunch at a
restaurant in Urbana. I know that Anand would call stability theory model
theory, so I didn't list it.

>This and a wealth of other examples show that the interaction
>between various areas of mathematics is much more complicated
>that the discussion in Urbana really touched on.

Of course.

>I did not intend to prejudge the possibility of success of
>Boolean Relation Theory (despite some cynicism) but to
>claim that if it succeeded it would be developing a new,
>interesting its own right, area -- not meeting the
>`extrinsic' demand of answering the questions of other
>mathematicians.

The sense in which it does meet an extrinsic demand - starting with the
thin set theorem - is stated above.