FOM: RE: Re:The missing 1%

[Matt Insall]

Hasan Keler writes:


> I asked:
>
> >   "ZFC is a foundation of the 99% of mathematics."
> >But what is the remaining 1% ?
> >And why is ZFC not a foundation of this 1% ?
>
> Edwin Mares wrote:
>
> >I should think that category theory is the remainder.
>
> To be honest, I don't know much category theory. Are categories proper
> classes in general rather than sets? Is this your reason why you don't
> consider category theory as something not in the reach of ZFC? What if we
> use NBG?

Actually, some categories are sets.  These are, as I recall, called ``small
categories''.  In a model-theoretic sense, however, I expect that even the
theory of categories can be realized, if it is consistent, by a model within
some model of ZFC.  One can certainly do this with certain large categories,
such as the category of topological spaces.  Basically, if V is a given
model of ZFC, then there should be a model W of ZFC for which V is a member
of W.  Then the proper classes of V are sets in the model W, so in
particular, the category, in V, of all topological spaces, is a model, in W,
of the category of all topological spaces in V.  I see a never-ending
sequence of interpretations of each theory in the other, and each should
have similar, aye, even related, foundational questions.  For instance, I am
certain there is an appropriate formulation of AC in terms of categories,
which, for some theories of categories holds, and for other such theories,
does not hold.  I am not a category-theorist, however, so I have not seen
enough category-theoretic literature to know whether any of it discusses
these questions, in particular.  I have, however, wondered at times, how
category-theory supposes to get around a paradox related to that of Russell,
for I have seen references (in Herrlick and Strecker, for example) to a
``category of all categories''.  Having been well-indoctrinated that one
should not refer to an kind of ``blurb'' of all ``blurbs'', I shuddered (I
think visibly), when I read such a thing.  Perhaps a lurking
category-theorist would care to comment, and help assuage my concerns?



 Name: Matt Insall
 Position: Associate Professor of Mathematics
 Institution: University of Missouri - Rolla
 Research interest: Foundations of Mathematics
 More information: http://www.umr.edu/~insall