FOM: Russell paradox for naive category theory

[Till Mossakowski]

Stephen G. Simpson wrote ( Tue May 11 22:53:01 1999 ):
> I still say that the small/large distinction is *not* essential for
> category theory, because you could simply work within the usual
> set-theoretic foundational framework (formalized as ZFC) and restrict
> your attention to set-size categories.  The small/large distinction
> would then be replaced by set-theoretic cardinality considerations.
> In this way you would lose little or nothing, at least insofar as
> applications of category theory are concerned.
> 
> Why don't category theorists fully accept this?  My tentative answer
> is that many category theorists harbor some sort of smoldering
> resentment against the set-theoretic foundational framework.

Please have a look at Adamek, Herrlich, Strecker: Abstract and concrete
categories, p. 385. On this page, both small/large distinctions
and cardinality considerations (involving regular infinite cardinals)
appear. Appearantly, the authors are very aware of the set-theoretic
foundations. The page is about the deep characterization theorems
I mentioned, and it would just make no sense without the small/large
distinction.

Till Mossakowski