I have a problem with this part of the explanation: >However, this definition extends quite naturally from the positive integers to the non-negative integers,

I have a problem with this part of the explanation: >However, this definition extends quite naturally from the positive integers to the non-negative integers, so that when x is zero, y is repeated zero times, giving y^{0} = 1, which holds for any y. Hence, when y is zero, we have 0^0 = 1. When y is zero, don't we have 0^0 = 1 x [y zero times]? Maybe I'm conceptualizing it incorrectly, but I'm envisioning an empty space where y would be, akin to an empty set. 1 times an 'empty set' is _not_ 1, it's one empty set, which rubs me as another way of saying nothing/zero, not 1. I know my language is imprecise and I'm probably describing empty sets and the definition of zero incorrectly. The point is that the last step of his explanation doesn't sit right with me. Just because Y exists zero times does not mean you can just throw it out of the multiplication.