On 18/09/2013 15:53, Dan Christensen wrote: > [...] > > Thus, if the Product of Powers Rule is to hold on N, 0^0 will be ambiguous -- being either 0 or 1. Unless one of these alternatives can be formally proven
Obviously if you start of with a "definition" of ^ which leaves 0^0 undefined, then nothing can be formally proven about 0^0. There's nothing particularly special about ^ in this regard; if I defined x + y to be equal to its usual value when y > 0 but left x + 0 undefined then nothing could be proved about x + 0 either. That doesn't mean that partially defining addition in this way would be a sensible thing to do.
> or shown to give rise to a contradiction,
Definitions cannot give rise to contradictions.
> the prudent course