> > I have shown that we need only explicitly define ^ in N for exponents > of 2 or more.
Actually you haven't, you have shown that in that case 0^0 might be either of 0 and 1. So your "definition" leaves x^y undefined at one point; therefore it is no definition. The purpose of a definition is to give the thing defined a precise meaning, you haven't given 0^0 a precise meaning.
> Apart from the base of 0,
Why apart from? Why are you leaving it out? What is even odder, having left it out you seem to think that 0^0 being undefined is a good thing. No it isn't, it's a job half done.
> we can derive values for > exponents 0 and 1. In that sense, the usual definition has redundant > elements.
>> > > Many standard textbooks leave 0^0 undefined. This is not the realm of > cranks or radicals. On the contrary, it is the most conservative > option.
Are they talking about x^y on the reals?
-- Sorrow in all lands, and grievous omens. Great anger in the dragon of the hills, And silent now the earth's green oracles That will not speak again of innocence. David Sutton -- Geomancies