
Re: The ambiguity of 0^0 on N
Posted:
Sep 20, 2013 11:40 AM


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

