Mathematics is full of cases where the same notation means different things. You’ve kind of dodged the question of what 0^0 is by talking about set theory, because it is (in some sense) arbitrary that X^Y means Y -> X or its size (cardinal exponentiation), and in analysis may mean exp(y log x). We accept that for natural X,Y except (0,0) these must agree but that does not imply that they should agree at (0,0), because the definitions are simply not the same.

If you are serious about math, which it seems you are, I would be a bit more careful about “transporting” notation from one field from another and arguing about correctness.

The usual way to transport a definition from a discrete domain to a continuous one is a technique called analytic continuation. I am curious if there is an analytic continuation of the discrete X^Y which contains 0^0=1, and what that would look like, but at that point you’re definitely not talking about exponentiation any more.

See: https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero

See: https://math.stackexchange.com/questions/11150/zero-to-the-z...