RE: [Axiom-developer] Complex exponentiation and 0

[Page, Bill]

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.