:No one disputes that 0^0 is 1 when interpreted as the cardinality of maps from the empty set to the empty set.  But that is not what the real-to-real-power exponential function is.  When we say , we certainly don't mean that the cardinality of the set of maps from a set of cardinality  to one of cardinality  is !  That would be simply gibberish.  And the meaning of the exponential function is not based on case logic; there is no special case for zero. 
 :So the conclusion must be that the real-to-real-power exponential function is simply a different function from the natural-to-natural-power one.  They're notated the same, and called by the same name, but that does not make them the same. 
 :They form a nice commutative diagram with large ranges of the inclusion map, and so you could if you wanted to make a grand unified exponential function that incorporates them both.  That's what Bo wants to do.  But in practice no one does that.  For the very good reason that the grand unified exponential function isn't good for anything.