On Wednesday, October 30, 2013 1:24:56 PM UTC-4, Bart Goddard wrote: > Dan Christensen <Dan_Christensen@sympatico.ca> wrote in > > news:firstname.lastname@example.org: > > > > > On Wednesday, October 30, 2013 11:51:13 AM UTC-4, Bart Goddard wrote: > > > > > >> statements. You're proposing to extend exponentiation from > > >> > > >> N to N_0, but somehow exponentiation isn't extended to 0. > > >> > > > > > > On the contrary, I have defined exponentiation on all of N_0. > > > No, you haven't. That's the problem. If you "define" > > exponentiation for 0 in such a way that you have to > > exclude 0 from all your formulae, then, in fact, you > > have not defined exponentiation on N_0.
If, for some reason, you don't want to call it a definition, call it a theorem. In practice, it makes no difference. We know that such functions can be shown to exist -- an infinite number of them, in fact. And, apart from the value assigned to 0^0, they are all identical.
Again, since you snipped it, here is my definition of ^:
1. ^: NxN --> N 2. x^0 = 1 for x=/=0 3. x^(y+1) = x^y * x
These are properties of all exponent-like functions. And from this definition (or "theorem," if you insist), we can derive the so-called Laws of Exponents for non-zero bases, thus formalizing what has been the practice of mathematicians for nearly two centuries.