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.
> 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.
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?
> 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 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.
>> 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. Why do you imagine that the consequences of 0^0 returning 1 when it "should" be 0 would be any more catastrophic, or more likely, than the consequences of 0^0 not returning 1 when it "should" return 1?
Why yes, rewriting every textbook that states the binomial theorem, any theorem about continuity of polynomials, power series, cardinal exponentiation or any number of other mathematical results, as well as changing the behaviour of all those programming languages I mentioned, would almost certainly be easier than getting you to acknowledge that your quarter-baked ideas about how things should be might carry less weight than those of the people who write mathematics textbooks and design programming languages. But that doesn't answer my question. Why do you imagine the negative consequences of continuing to define 0^0 = 1 are more serious, or more likely, than those of changing the definition on the whims of Dan Christensen? In pondering this question, it's worth bearing in mind that time we had every textbook and web page about category theory rewritten after you sort-of read the introduction to a wiki page on the subject and decided that the conventional definition of morphism was wrong. And then we had them all changed back again a couple of weeks later, when you learnt a bit more about the subject and decided that the definition was right after all. Remember that, Dan? I mean, as far as we can tell it only cost the world a few thousand man-hours and several hundred acres of rainforest that had to be cut down to provide the paper for all those new editions, but the full repercussions may not be known for some time.