User blog:ArtismScrub/Proof that negative tetration is impossible

At first, you'd think that: a↑↑(-b) = 1/(a↑↑b) analogous to exponentiation. However, this does not work with how tetration is defined: a↑↑(b+1) = a(a↑↑b)

Instead, this definition can be generalized backwards towards:

a↑↑(b-1) = loga(a↑↑b)

So, this means:

a↑↑1 = a by definition

a↑↑0 = loga(a) = 1 (this is true for all a)

a↑↑(-1) = loga(1) = 0 (this is true for all a)

a↑↑(-2) = loga(0) = nope (as loga(n) approaches 0, the result approaches negative infinity)