On Thursday, September 19, 2013 1:02:30 PM UTC-4, Peter Percival wrote: > Dan Christensen wrote: > > > > > > > > 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.
That is not part of the definition. That is a derived result assuming PPR on N.
> 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. >
In group theory, we define a set of binary functions with certain properties on an arbitrary set. I guess you could say that, here, I am defining a set of binary functions with certain properties on N -- the set of exponential functions. It turns out to have only 2 elements that satisfy PPR.
> > > > Apart from the base of 0, > > > > Why apart from? Why are you leaving it out?
We can't divide by 0. Unless you want to assign a value to 0/0 as well.
> 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? >
Usually real numbers, but real or natural numbers -- it shouldn't matter.