Date: Oct 30, 2013 1:47 PM
Author: Dan Christensen
Subject: Re: Formal proof of the ambiguity of 0^0

On Wednesday, October 30, 2013 1:24:56 PM UTC-4, Bart Goddard wrote:
> Dan Christensen <Dan_Christensen@sympatico.ca> wrote in
>
> news:04511d59-4dcc-4e23-8e6f-8172781a8fe5@googlegroups.com:
>
>
>

> > 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.

Dan
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