Using Theorem 1 (specifying x0=1) , a binary function ^ on N can be constructed such that:
1. x^0=1 2. x^(y+1) = x^y * x
Using this a definition, the usual Laws of Exponents can be derived:
1. The Product of Powers Rule: x^y * x^z = x^(y+z) 2. The Power of a Power Rule: (x^y)^z = x^(y*z) 3. The Power of a Product Rule: (x*y)^z = x^z * y^z
However, using Theorem 1 (specifying x0=0 instead) , another binary function ^ on N can also be constructed such that:
1. 0^0=0 2. x^0=1 for x=/=0 3. x^(y+1) = x^y * x
The two functions are almost identical, differing only on the value of 0^0 itself (see Theorem 2).
How to choose? For the obvious reasons, 0^0=1 is much more convenient. But the fact is, there does not seem to be any logically compelling reason to choose either alternative. The only real alternative seems to be to leave 0^0 undefined.