[FOM] Collecting independence results

[Harvey Friedman]

On 10/5/04 9:47 PM, "Daniel Ian Rosenbloom" <dirosenb at fas.harvard.edu>
wrote:

> For my undergraduate thesis, I'm considering doing a project on
> independence results in set theory that have applications to other areas
> of mathematics.  To that end, I am searching for any such results that
> exist scattered across articles and would benefit from a more
> consolidated/general/modern/detailed exposition and/or results that are of
> great general interest to mathematicians.  So far, I'm looking at
> Whitehead's conjecture, Kaplansky's conjecture, and Shelah and Soifer's
> work on AC and chromatic numbers of the plane, but I want to find more.
> 

These seems like a very good project. You may or may not want to get into
the more futuristic aspects of independence results. Let me explain.

My efforts in independence results are focused almost entirely on examples
that are far more concrete than the ostensibly set theoretic examples you
mention. 

The vast majority of mathematicians think that one can avoid independence
from ZFC if one stays within the simultaneously concrete and mathematically
interesting - at least mathematicians think this at the subconscious level.
In this way, mathematicians feel "protected from logic".

Refuting this ideology has been a major ongoing challenge, and the surface
has barely been scratched. This ideology will be profoundly and
spectacularly and productively demolished during this century.

You may want to look at the independence results discussed in my Rademacher
lecture series, especially towards the end, where independence from all of
ZFC is discussed. Also, look at "Selection for Borel relations", which will
appear in Logic Colloq '01, ASL.

These independence results concerning Borel selection involve problems
worked on by some functional analysts, and immediately accessible and
natural to anyone taking a course in real variables. The novelty here is
that they involve only Borel measurable functions.

The independence results concerning Boolean relations among sets of integers
do not involve problems worked on by any mathematicians. The novelty here is
twofold:

1. They involve only sets of integers, and in some cases, only finite sets
of integers.
2. Boolean relation theory promises to be a new well studied subject of the
future, of intrinsic mathematical interest.

UNFORTUNATELY, these two manuscripts are now on a defunct preprint server,
and I have not quite finished refurbishing my website, which will support
easy access to all of my unpublished work of significance, including, of
course these two. I will be maintaining this website with frequent updates.

I will send you pdf files of both of these documents in a separate email, as
document transfers are not permissible on the FOM.

Harvey Friedman