Re: categorical foundations

[Andre.Rodin@ens.fr]

>
> The basic notions are in fact not very articulate in themselves, and
> throughout the history of mathematics it has taken further ideas to
> articulate them.  Bill saw how to articulate these and many more,
> quite directly, in categorical terms not assuming any prior set
> theory.  That articulation works even if you do not take it as
> foundational.  But it gets a natural foundational character in the
> framework of the category of categories -- thus CCAF, the axiomatic
> theory of the category of categories as a foundation.
>

I agree with you about generalities concerning pre-formal and formal
concepts. A reason why I say CCAF is not a satisfactory categorical foundation
is different. ETC is the formal basis of CCAF and ETC relies on a pre-formal
notion of set or collection just like ZF or any other axiomatic theory built
with Hilbert-Tarski axiomatic method. Elements of a new properly categorical
method of theory-building are present in the "basic theory" (BC) that follows
ETC. (I mean, in particular, the "redefinition" of functor in BC as 2-->A, etc.
The standard definition of functor given earlier in ETC never reappears in BC.)
However in CCAF these new features are not yet developed into an autonomous
axiomatic method - or into a new way of formalisation of pre-formal concepts,
if you like. In my understanding, such a method should meake part of
categorical foundations deserving the name.  CCAF remains in this sense
eclectic, it is a half-way to categorical foundations.

best,
andrei






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