Dan Christensen <Dan_Christensen@sympatico.ca> wrote in news:firstname.lastname@example.org:
>> You're claiming to extend exponentiation to N_0, but >> >> all of your rules say "nonzero base." > > [snip] > > Nothing wrong with that.
In mathematics, contradiction is the ONLY sort of wrong there is. And you're definitely contradicting yourself. You're extending without extending.
> Mathematicians have being doing this > implicitly for nearly two centuries. Now, it has a rigorous > foundation.
No, they have not. I'm trying to imagine a mathematician "extending" real number multiplication to the complex numbers, but declaring, in all parts of his definition, that it doesn't apply to non-real numbers. And, as if that wasn't bizarrely stupid enough, to follow up with "therefore it doesn't matter how you define complex multiplication...."
> 0^0 is an unspecified natural number.
If it's not specified, then it's not defined. If it's not defined, there is no extention.
> 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.
But you've shown yourself to be a particularly bad judge of what an extention is, so that fact that it sounds like it "to you" means nothing. (And, in fact, actually argues against.) You can't claim an extention to N_0, unless 0^0 is defined.