On Friday, November 1, 2013 6:05:18 AM UTC-4, Bart Goddard wrote: > Dan Christensen <Dan_Christensen@sympatico.ca> wrote in > > news:firstname.lastname@example.org: > > > > > Here, I prove 0^x = 0 for x=/=0 > > > > The proof is fatally flawed. If y = 1, then z = 0, > > in which case the "proof" is threaded with the > > "uspecified" phrase "0^0."
On the contrary, this is not a problem at all. Again, 0^0 is a natural number, but no value is assigned to it in my definition of ^ on N. This formalizes the longstanding practice (since Cauchy in the early 19th century) of leaving 0^0 undefined. Again, I define ^ as follows:
1. ALL(a):ALL(b):[a e n & b e n => a^b e n] (i.e. ^ is a binary function on N)
2. ALL(a):[a e n => [~a=0 => a^0=1]]
3. ALL(a):ALL(b):[a e n & b e n => a^(b+1)=a^b*a]
These 3 statements characterize ALL exponent-like functions on N (Theorem 1). And ALL exponent-like functions on N are identical except for the value assigned to (0,0). (Theorem 2).
No value is assigned to 0^0, but as readers can see for themselves from the theorems posting here, it is really quite easy to work around this minor inconvenience.