On Thursday, September 19, 2013 11:51:30 AM UTC-4, Peter Percival wrote: > Dan Christensen wrote: > > > > > > > > Let's see your proof that 0^0=1. Sorry, simply defining it as such > > > won't do in this context. > > > > You don't like "my" proof because its conclusion follows so readily from > > the definition?
Your "proof" consists of:
(1) 0 in N (Definition) (2) x^0=1 for all x in N (Definition) (3) 0^0=1 (Conclusion)
Sorry, not good enough in this context.
All proofs follow from definitions. Do you think a > > proof of one step is less good than one of many steps? Why? > > > > > > > > Technically, + on N is not simply defined. It is a construction based > > > Peano's Axioms (or their modern equivalent) and set theory. > > > > Here's the definition: > > > > x + 0 = x > > x + y' = (x + y)' > > > > ' being successor. No set theory required. >
It is a good working definition, but you really need to first prove the existence of such a function. For that you need set theory.
> > > > You could > > > probably construct an exponent function on N with 0^0=1, but you > > > could also construct one with 0^0=0. And they would agree on every > > > value but that assigned to 0^0. > > > > So what? I might have 0^0 = 5.214. >