On Wednesday, October 30, 2013 4:42:11 PM UTC-4, Bart Goddard wrote: > Dan Christensen <Dan_Christensen@sympatico.ca> wrote in > > news:firstname.lastname@example.org: > > > > > > > If, for some reason, you don't want to call it a definition, call it a > > > theorem. > > > > No. It is neither. > > You're claiming to extend exponentiation to N_0, but > > all of your rules say "nonzero base."
Nothing wrong with that. Mathematicians have being doing this implicitly for nearly two centuries. Now, it has a rigorous foundation.
> seen. If you keep saying "nonzero base", then you > > haven't done anything about 0, so there is no extention. > > > > >In practice, it makes no difference. > > > > Of course it makes a difference. If you extend > > exponentiation to N_0, then 0^0 has to mean something, >
0^0 is an unspecified natural number.
> if you make it mean anything but 0 or 1, then you > > get contradiction. Of 0^0 doesn't mean anything, > > then you haven't done any extending.
Again, my definition refers specifically to 0-exponents, and I have referred here to a theorem concerning 0-bases that can be derived from that definition. Sure sounds like "extending" to me.