Re: How analogous are categorial and material set theories?

[Neil Barton <bartonna@gmail.com>]

Thanks to everyone for their replies. I provide some quick
clarifications to some of Patrik's comments below:

``You now introduce the word "core", and I would need to understand
what that mathematically and logically really means, at least
intuitively.''

I mean ``core'' in the trivial sense. The axioms defining what it is
to be a category (of some kind) are just first-order. The same is
true, of course, for ZFC. However, in the set theory context it seems
at least sensible to try and provide a second-order (quasi)
categoricity proof, despite the diversity of models for first-order
ZFC (and indeed we have one by the work of Zermelo, Shepherdson,
etc.). To attempt the same for category theory would seem to be an
absurd strategy---the point of categories is that they can be
instantiated in many non-isomorphic models.

``Yes, indeed, across diverse contexts, with "context" somehow related
to "system" in Hidekazu Iwaki's reply involving "mathematical
system".''

I cannot find Hidekazu Iwaki's reply (this is rather strange, I do not
know why). However it seems like similar usage to me: I don't have a
formal definition of `mathematical context', but the point is just
that there's diverse non-isomorphic models where the same categorial
properties pop up. e.g. (somewhat obviously for this list, but I found
it striking learning some category theory) the notion of topos appears
all over the place in diverse structures, and allows us to have a
reasonable notion of internal logic.

Your comments on the different notions of hierarchy are interesting,
but I will have to think about this some more before I have anything
reasonable to add.

Best Wishes,

Neil

-- 
Dr. Neil Barton
Postdoctoral Research Fellow
Kurt Gödel Research Center for Mathematical Logic
University of Vienna


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