
An Infinity of Exponentlike Functions on N
Posted:
Oct 4, 2013 11:27 AM


It can be formally shown that for every x0 in N, there exists a unique binary function pow on N such that: (1) pow(0,0) = x0 (0^0 = x0) (2) pow(x,0) = 1 for x =/= 0 (x^0 = 1 for x =/= 0) (3) pow(x,y+1) = pow(x,y) * x (x^(y+1) = x^y * x)
If we also have a binary function pow' on N such that:
(1) pow'(0,0) = x1 (the only difference) (2) pow'(x,0) = 1 for x =/= 0 (3) pow'(x,y+1) = pow'(x,y) * x
then it can also be shown that pow(x,y)=pow'(x,y) except for the case of x=y=0.
Notwithstanding combinatorial analogies and arguments of convenience, there does not appear to be any logically compelling reason to choose any particular value for 0^0.
Dan Download my DC Proof 2.0 software at http://www.dcproof.com

