0^0 still wishes it were equal to 1 (but it still isn’t)
Here’s a little supplement to the last post. I mentioned before that there are several reasons somebody might try to define 0^0 := 1. The big reason before (flawed though it turned out to be) involved calculus and limits. But there’s an entirely different way to interpret the expression y^x which puts 0^0 in an intriguing different light.
If X and Y are sets, with |X|= x and |Y| = y (X has x many members and Y has y many), then the number of distinct functions f:X -> Y is given by y^x. Think about it in the case X just has 2 elements: the first element has y different places to which it could be mapped, and likewise for the second element, leading to y^2 many functions.
So then 0^0 should count the number of functions from the empty set to itself.
There are two possible replies to this neat observation:
(1) Of course there is exactly one such function - it takes in nothing and sends nothing nowhere. A useless, trivial, but perfectly understandable function. Hence 0^0 = 1.
(2) The very idea of a function rests on sending elements to elements. This idea is meaningless when it comes to the empty set - there are no elements! So 0^0 is nonsensical.
Now here’s a reply back to (2): Mathematicians actually do allow functions to be defined on the empty set - they’re called empty functions and they don’t do anything interesting, but they’re required for making certain definitions in category theory work nicely. Any two empty functions f: { } -> Y and g: { } -> Y (with Y nonempty) are identical, and this reflects the fact that y^0 = 1.
(See http://en.wikipedia.org/wiki/Empty_function)
I can’t find any math literature that says there’s no way of making sense of a function from the empty set to itself. It seems to me such a function wouldn’t break anything. In other words, I dig response (1) above. But unfortunately, this still isn’t any kind of hard proof that 0^0 = 1.
