FOM: Grothendieck universes

[Colin McLarty]

>you seem to be implicitly assuming that all
>existing discussions of derived functor cohomology in the literature
>rely on Grothendieck universes.  Is this really the case?

        Derived functors between Abelian categories are themselves universe
sized sets. If you quantify over derived functors you are quantifying over
universe-sized sets. Further, every treatment I have seen of derived functor
cohomology proves the "universal property of derived functors" and uses it
freely. This property is itself a relation of the derived functor to "all
delta functors...." where delta-functors are also universe sized sets.

        If anyone knows of a textbook on cohomological number theory that
does not use this apparatus, please let me know. 
  
>Regarding Friedman's work on the incompleteness phenomenon, you claim
>to understand and appreciate the familiar and valid f.o.m. point that
>ConZFC lacks ``genuine mathematical meaning''.  However, are you using
>this phrase in the same sense in both the Friedman and Grothendieck
>contexts?

        Certainly Friedman and Grothendieck would not agree on all instances
of "genuine mathematical meaning", and would agree only in part on the
definition. Both agree it is opposed to excessive abstraction, and technique
for its own sake.

>Perhaps you are playing on some extremely subtle distinction between
>``the proofs'' and ``the methods''?  If so, what is the distinction?

        I say the method has vital uses. But to use only that method, would
render the field meaningless. This is a very common thing. Many methods are
quite fine, while it would be a poor idea to use them all the time.