Subject: Expressiveness of category theory and set-theory.

From: categories <[email-masked]>
Date: Thu, 14 Mar 1996 17:11:56 -0400 (AST)

Date: Thu, 14 Mar 1996 20:59:55 GMT
From: Ralph Loader <[email-masked]>

[email-masked] writes:
>Date: Wed, 13 Mar 1996 13:40:25 -0800
>From: james dolan <[email-masked]>
>
>-This is an interesting example.  You state a question in mathematical
>-English, and then criticise ZF for being able to express this question,
>-while category theory cannot.
>
>
>i at least am curious just what is this mathematical question you
>claim to have in mind which zf can express but which category theory
>can not.  obviously it wasn't "Does any simple group appear as a zero
>of the Riemann Zeta function?" since that is mere gibberish and not a

This question clearly is mathematical English, in the sense that it is in
the English language, and uses mathematical jargon.

It doesn't make sense.  I wasn't arguing with that.  Ruling out statements
from being "mathematical" *merely* on the grounds that they offend your
mathematical taste, quickly becomes disasterous.  Especially when we're
(implicitly) talking about a formalisation in a formal language such as
that of first order set theory.

Anyway, as a good example, take a set-theoretic proof of Borel
determinancy.  Can anyone give a reasonable category theoretic proof of
Borel determinacy?  [This is a result that Friedman showed is a theorem of
ZF but not of Z, if my memory serves me correctly.]  This could well
suffice to answer my concerns w.r.t. the adequacy of category language, by
showing how even arguments needing replacement can have category theoretic
analogues.

Ralph.

P.S.  My personal opinion on foundations is fairly agnostic, in case anyone
mistakes my concerns with rejection of category theoretic foundations.