fom
Posts:
1,968
Registered:
12/4/12


Re: The ambiguity of 0^0 on N
Posted:
Sep 19, 2013 7:11 PM


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.
If Dan is going to argue outside of established mathematics, then he must justify his pretheoretic assumptions. He cannot simply ignore "settheoretic" views when, in fact, that is one principal reason for including 0 as a base element in the modern theory.

