In article <email@example.com>, Rotwang <firstname.lastname@example.org> wrote:
> 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.
If one has two or more definitions which are to all compatible with each other, they CAN cause contradictions. > --