Re: [Axiom-developer] Complex exponentiation and 0

[William Sit]

"Page, Bill" wrote:
> 
> On Monday, June 21, 2004 4:01 PM William Sit
> address@hidden wrote:
> 
> > ...
> > Martin Rubey wrote:
> > >  > ...
> > >  > card.spad: error "0**0 not defined for cardinal numbers."
> > ...
> > In the case of card.spad, the function x^y, where x and y are
> > cardinal numbers of sets X and Y respectively, x^y is the
> > cardinal number of the set of all maps from Y to X. For example,
> > when x = 2, 2^y is the cardinality of the power set of Y. Now
> > when x = 0 and/or y = 0, one (or both) of X and Y is the empty
> > set and since one cannot define what a "map" is to or from an
> > empty set, it may seem justified to leave 0^y, x^0, and 0^0
> > undefined.
> >
> 
> I am inclined to reject this argument on the general grounds of
> the current treatment of categories with initial objects.
> 
> In the category of cardinal numbers a "map" is a morphism and 0
> is initial. I think card.spad should be understood as implementing
> such a category, although strictly speaking of course Axiom does
> not (yet?) fully conform to category theory in this respect.
> 
> Regards,
> Bill Page.

I agree, in other words, you would recommend that x^0 =1 (empty product,
including x = 0, and there is a unique morphism from 0 to any x) and 0^y = 0 for
any y \ne 0. But this is still just a convention which agrees with the general
algebra cases. Note I used "may seem justified", to give a plausible reason why
Axiom is using the "undefined" convention.