On Friday, November 1, 2013 11:54:01 AM UTC-4, Bart Goddard wrote: > Dan Christensen <Dan_Christensen@sympatico.ca> wrote in
> > You yourself have admitted that 0^0 is ambiguous. How do YOU propose > > > to formalize this ambiguity, > > > > This is, perhaps, the heart of your illogic. "Formalize > > this ambiguity" is a nonsense phrase.
Only to slaves of convention.
> It has no more > > meaning than "skyblue pink". Second, before one > > formalizes anything, one ought to have an argument about > > why it should be formalized at all. >
> > > In the context of combinatorics, it's completely logical > > and completely formal to define 0^0 = 1. And you don't > > get it.
First of all, you don't need combinatorics (or cardinality) to construct an exponent-like function ^ on N such that 0^0=1. All you need is ordinary logic and set theory (see my Theorem 1). Again, the only problem is that, simple convenience notwithstanding, there doesn't seem to be any logically compelling reason to accept that the definition of that particular function as The Definition of exponentiation on N.