On Friday, September 20, 2013 8:54:45 AM UTC-4, 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 extra-logical symbols do you have and what extra-logical > > axioms govern them? If ^ isn't among the primitive extra-logical > > symbols how is it defined in terms of them? >
In lines 1-14, 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 15-17, I give the axioms/definitions for ^.
In lines 18-41, I prove that x^1=x for x=/=0.
In lines 42-83, 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 number-theoretic results (lines 1-14) 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 1-17), let's discuss it.