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Paul wrote:
> Jan Hidders wrote:
> >> Doesn't this only apply if you are considering the set of all relations
> >> over all domains? What if you restrict yourself to a finite set of
> >> domains? I can't see how Cantor's Paradox would apply in this case.
> >
> > It wouldn't. But the question at hand was whether the collection of all
> > relations can be a domain. If you are going to postualte the set of
> > domains a priori, then the set of relations over those domains will of
> > course not be one of those postulated domains.
>
> OK. but domains can be thought of as simply sets. So then the
> "collection of all domains" is like the "set of all sets" which, by
> Cantor's Paradox, isn't actually a well-defined set.
In "naive" set theory, yes, in ZF, no. There is no problem with defining a "set of all domains" provided that the set satisfies ZF axioms.
>
> So we can't even get to the stage of considering the set of all
> relations over all domains, because "all domains" is meaningless in set
> theory! No wonder Cantor went insane... :)
>
> Paul.
Received on Mon Jul 11 2005 - 11:01:20 CDT