On Wednesday, September 18, 2013 7:11:53 PM UTC-4, Rotwang wrote: > On 18/09/2013 15:53, Dan Christensen wrote: > > > [...] > > > > > > Thus, if the Product of Powers Rule is to hold on N, 0^0 will be ambiguous -- being either 0 or 1. Unless one of these alternatives can be formally proven > > > > Obviously if you start of with a "definition" of ^ which leaves 0^0 > > undefined,
Actually, I start with a definition which leaves out any explicit mention at all of exponents of 0 or 1. I define only exponents greater than 1. Starting with this definition, I prove that x^0 = 1 and x^1=x for x=/=0. I also prove that if you want to extend the Product of Powers Rule to all of N, then we must have 0^1=0 and either 0^0=0 or 0^0=1. Until we are able to prove or disprove one of these alternatives, we should probably leave 0^0 undefined.
> then nothing can be formally proven about 0^0.
It seems you can only prove something about 0^0 if PPR is extended over all of N.
> There's > > nothing particularly special about ^ in this regard; if I defined x + y > > to be equal to its usual value when y > 0 but left x + 0 undefined then > > nothing could be proved about x + 0 either. That doesn't mean that > > partially defining addition in this way would be a sensible thing to do. > > > > > > > or shown to give rise to a contradiction, > > > > Definitions cannot give rise to contradictions. >
They most certainly can. If you define 0^0=0 AND 0^0=1, you get a contradiction because 0=/=1.
> > > > > > the prudent course > > > > You keep saying this. Why?
Because I believe it is true.
> What negative consequences do you imagine > > will follow from people defining exponentiation in the usual way?
Whatever consequences may arise from a calculation that results in a value of 1 when it should be 0. The result could be catastrophic.
The most well known theorem that makes use of 0^0 is probably the Binomial Theorem. And it can easily be restated to avoid its use, e.g. by stating at the outset that (x+0)^n = x^n, etc.