Virgil
Posts:
8,833
Registered:
1/6/11


Re: The ambiguity of 0^0 on N
Posted:
Sep 20, 2013 3:42 PM


In article <f2365b80cd8a4eca8d994a4be4625161@googlegroups.com>, Dan Christensen <Dan_Christensen@sympatico.ca> wrote:
> On Friday, September 20, 2013 8:54:45 AM UTC4, Peter Percival wrote: > > Dan Christensen wrote: > > > > > fom, how about some honest toil to prove 0^0=1? Choose any axioms or > > > definitions you like. Just not 0^0=1 itself or anything like x^0=1 for > > > all x in N. That would be just too easy! ;^) > > > > > > > > > > > > I think the time has come for you to say what axioms and definition > > > > _you_ are using. Let's take first order logic with = for granted, what > > > > primitive extralogical symbols do you have and what extralogical > > > > axioms govern them? If ^ isn't among the primitive extralogical > > > > symbols how is it defined in terms of them? > > > > See my proof. The link again: http://dcproof.com/Ambiguityof0tothe0.htm > > In lines 114, I list the definitions and properties of the natural numbers > required for the proof. They are treated as axioms for the purposes of this > proof, but you can think of them as theorems or definitions. > > In lines 1517, I give the axioms/definitions for ^. > > In lines 1841, I prove that x^1=x for x=/=0. > > In lines 4283, I prove that x^0=1 for x=/=0. > > In the remainder of the proof, I attempt to fill in the gaps  assigning > values to 0^0 and 0^1. I show that if we assume the Power of Products Rule > (pretty standard stuff), then 0^1 must be 0 and 0^0 must be either 0 or 1. > > > > > > > > You have been shown proofs that 0^0 = 1 and you reject them, > > See my reply to fom. > > > so what > > > > theory (= logical system) of natural numbers do you wish your > > > > interlocutors to work in? Spell it out in detail. > > > > I haven't developed the entire system of natural numbers here. I have just > listed some numbertheoretic results (lines 114) that are required for my > proof. They are stated in a formalized version of the language of everyday > mathematics. If you take exception to any of them, let's discuss it. > > But you need not use my system, although that would be nice. Use whatever > system you like. Improvise one on the spot if you must. Just don't assume > what you are trying to prove as you have repeatedly tried to do here. > > > > > > > That a theory of numbers can be cooked up in which 0^0 = 1 is a wff that > > > > can't be proved is clear. But so what? > > > > There is nothing weird about my number system. Again, the notion that 0^0 is > undefined is not some new, radical idea. It has been around for centuries and > is really quite mainstream. (See that Wiki article I cite above.) If you > polled all the world's math instructors, I am sure that the vast majority > would agree that 0^0 should be left undefined. > > Again, if you disagree with any of my assumptions or definitions (lines > 117), let's discuss it. > > Dan > Download my DC Proof 2.0 software at http://www.dcproof.com
I can see that f(x,y) = x^y is continuous on domain {(x,y): x > 0 and y > 0} and can even be extended continuously to both {(0,y): y > 0} and {(x,0): x > 0} but there is no way to extend that function CONTINUOUSLY to {(x,y) x > 0 and y > 0} to {(x,y) x >= 0 and y >= 0}.
This suggest that for real functions like f(x,y) = x^y on {(x,y): x > 0 and y > 0} there is no universally acceptable limit value at f(0,0). 

