[FOM] Foreman's preface to HST

[Marc Alcobé]

Ok. So we could find a distinction between "mathematical problems
normally considered important" and "maybe not so important, but still
foundational problems". I do not exclude that the investigation of the
latter could be developed within ZFC (or whatever fragment or
extension of it).

Naturally, mathematically motivated foundational problems (in the
sense of being necessary for the solution of those "mathematical
problems normally considered important") are more appealing for
mathematicians, specially for the ones who are working on areas
particularly near to those mathematical problems, than those
foundational problems that smell like being "only" (or simply "too
much") philosophically motivated.

However, no matter where the motivation comes from, when a certain
level of abstraction over a given problem is attained, the problem
becomes a genuinely mathematical one, and it might pose questions
that, I believe, are of genuine mathematical importance.

2010/4/27  <joeshipman at aol.com>:
> If you only care about solving the mathematical problems normally
> considered important, and not about philosophy, "first order logic +
> ZFC" appears to provide a good enough foundation that it is sufficient
> to search for additional set-theoretical axioms and not to go outside
> that framework. For example, the important results that were obtained
> using category theory can be easily obtained within ZFC if you add the
> "Grothendieck Universes" axiom (inaccessibles are unbounded among the
> cardinals).
>
> This is merely a practical observation; if you know of any serious
> mathematical investigation of questions which can be stated in the
> language of set theory that cannot easily be conducted in the framework
> of first-order-logic + ZFC, I'd like to hear about it.
>
> -- JS