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nova

I like my high school math consistent with set theory.

ø : empty set, 1 : {ø}, A : nonempty set, ~= : isomorph to.

A^ø ~= 1, because there is only one function ø->A, the empty function.

ø^A ~= ø, because there is no function with empty codomain and nonempty domain.

ø^ø ~= 1, because there is again one function ø->ø, the empty one.

So yes, 0^0 = 1.

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This! The empty function is the function whose graph is the empty set. When you stop thinking of functions as symbolic expressions and make friends with the empty set, it's all clear as crystal. The empty set has perplexed a lot of people over the ages, so it's unsurprising that 0^0 prompts puzzlement. But with the benefit of modern hindsight, there's really no reason to stay confused. Set theory in the large is still a great mystery but we got the empty set well figured out by now.

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