[FOM] Foundational Challenge

[Harvey Friedman]

On Sun, Jan 12, 2020 at 6:18 PM Timothy Y. Chow
<tchow at math.princeton.edu> wrote:
>
> On Sat, 11 Jan 2020, Harvey Friedman wrote:
> > We go merrily along in this style. What is the simplest case where a
> > nasty time consuming situation arises? There may be some need to know
> > something about the isomorphisms themselves. Of course, here we might as
> > well prove that for two relevant gadgets, there is a unique isomorphism.
>
Tim Chow:
> I understand why you repeatedly ask for the "simplest nontrivial case,"
> but I suspect that this may be akin to asking for the smallest number of
> grains of sand in a heap.

"What is the simplest nontrivial case" is a figure of speech, commonly
used, and not to be taken literally the way you sometimes do. It means
"give me the simplest nontrivial case that comes into your mind".
Nothing more, nothing less.
>
> That is, there is a good chance that any specific example that one might
> try to exhibit ends up being trivial to handle.  This doesn't necessarily
> mean that problems can't pile up to the point where they become a
> significant nuisance to handle.
>
Chow goes on at some serious length about issues regarding exact
rigorous formulations surrounding elliptic cubes.

Looking at the issues Chow is referring to, I feel like these are
exactly the kind of issues I deal with everyday when I have to decide
how to set up a new field for Tangible Incompleteness. I have major
issues of choice of generality, and major issues of exactly what the
shape of definitions should be, and so forth. Now granted it is not
because I need to figure out how to do rigorous exposition of existing
material. It's more like I need to come up with the right frameworks
and right details in order to maximize naturalness and simplicity and
interest.

But handling this kind of thing efficiently and simply in the
classical set theoretic foundation is second nature to me. So what
Chow is writing there doesn't even come close to making me lose any
faith in the set theoretic approach.

Of course we must not lose sight when we think there is a serious
awkwardness in the set theoretic approach, what we lose when we bring
in non set theoretic stuff for the cure.

One thing I would recommend against is to replace something
philosophically coherent like set theory with something that is, at
least not right now, philosophically coherent. Unless one highlights
that no one has made it philosophically coherent. Together with a plan
for making it philosophically coherent.

Harvey Friedman