
Re: The ambiguity of 0^0 on N
Posted:
Sep 19, 2013 10:41 PM


On Thursday, September 19, 2013 7:11:39 PM UTC4, fom wrote: > On 9/19/2013 12:16 PM, Peter Percival wrote: > > > Dan Christensen wrote: > > > > > >>> There is a simple definition of exponentiation on N, based on > > >>> > > >>> cardinal arithmetic. It doesn't have any special cases, but > > >>> > > >>> 0^0 = 1 falls out of it quite naturally: > > >>> > > >>> > > >>> > > >>> A^B = {f  f:B>A} > > >>> > > >>> > > >>> > > >>> In English, the cardinal number of set A raised to the power > > >>> > > >>> of the cardinal number of set B is the cardinal number of the > > >>> > > >>> set of all functions from B to A. No exceptions, no special > > >>> > > >>> cases. > > >>> > > >> > > >> Thanks, but as with any analogy, it may not be perfect. > > > > > > It's not an analogy, it's a fact. I've just taken off my bookshelves > > > the first set theory book that came to hand. It's Kunen's Set Theory, > > > an Introduction to Independence Proof. On page 31 he defines cardinal > > > exponentiation just as Michael F. Stemper does! You know, don't you, > > > that the finite cardinals are the same as the natural numbers? > > > > > > > > > > And, as noted elsewhere, whereas Peano and Dedekind began with > > 1 as their base element, Frege's logicism involves 0 as a base > > element because of classbased construction. >
You are not really going to start arguing whether or not 0 is a natural number, are you? Really now!
Dan Download my DC Proof 2.0 software at http://www.dcproof.com

