On Thursday, September 19, 2013 9:31:44 AM UTC-4, Rotwang wrote: > On 19/09/2013 06:23, Dan Christensen wrote: > > > 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. > > > > But your definition leaves 0^0 undefined. So nothing can be proved about > > it without an additional assumption. >
Yes. That additional assumption is the extension of the Product of Power Rule to all of N. But even then, you are left with two possibilities: 0^0=1 or 0^0=0.
> > > > > > I also proven 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. > > > > Yes, I know. "If you want to extend the Product of Powers Rule to all of > > N" is an additional assumption. But what I'd like to know is, what's so > > special about the Product of Powers Rule? I mean, your "definition" > > implies that x^0 = 1 for x != 0 - let's call this the Power of Zero > > Rule. Why do you want to extend the Product of Powers Rule to all of N, > > but not the Power of Zero rule? >
You may be onto something here. If you don't want to make this extension -- the most conservative option, I suppose -- you also could not assign a value to 0^1.
> > > > > > Until we are able to prove or disprove one of these alternatives, we should probably leave 0^0 undefined. > > > > Why? Note that I'm not asking you to repeat your argument that if one > > defines ^ a certain way one finds that there are two different functions > > that satisfy that definition; everybody already knows that. I'm asking > > why anyone should define ^ the way you do in your OP, rather than the > > much simpler and more common way that it's usually defined, which has > > the additional benefit that there is a unique function that satisfies > > it, and that PPR can be derived from it, rather than just assumed. >
It was much simpler to allow unrestricted comprehension for sets. But it eventually led to the now well-known contradictions (e.g Russell's Paradox). So, simpler is not always better.
> > > > > > It seems you can only prove something about 0^0 if PPR is extended over all of N. > > > > Or if you use a sensible definition of ^. > > > > > > >>> the prudent course > > >> > > >> You keep saying this. Why? > > > > > > Because I believe it is true. > > > > Why? >
> > > > > >> 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. > > > > But the consequences from a calculation that results in "undefined" when > > it should be 1 could be similarly catastrophic.
No more catastrophic than division by zero which is similarly undefined, and is treated by computers as an error condition.
Again, the notion of 0^0 being undefined is not some radical notion. Many standard textbooks make this assumption. It is probably more mainstream than assuming 0^0=1. I'm sure that a poll of all math instructors would confirm this.