I tried setting 9=x^x^x^x^..., and by similar logic, we can take x^[both sides] and get x^9 = x^x^... = 9. Which means x is a ninth root of 9; I think we may a

I tried setting 9=x^x^x^x^..., and by similar logic, we can take x^[both sides] and get x^9 = x^x^... = 9. Which means x is a ninth root of 9; I think we may as well pick the real root. However... arc> (expt 9 1/9) 1.2765180070092417 arc> (expt (expt 9 1/9) that) 1.3656685405664577 arc> (expt (expt 9 1/9) that) 1.3957179500635313 arc> (expt (expt 9 1/9) that) 1.405994788813235 arc> (expt (expt 9 1/9) that) 1.409526784112037 arc> (expt (expt 9 1/9) that) 1.4107427255115605 arc> (expt (expt 9 1/9) that) 1.4111615739170247 This doesn't look like it's converging to 9. This tells me that you can't just say "N = x^x^x^..." and expect there to be an x satisfying that condition, any more than there's an x satisfying "3 = 0*x". Actually, that one's arguably solvable with x = ∞. A better example is "|x| = -2", or perhaps "3 = f(x) where f(x) = 1 if x>0, 0 otherwise". So I guess the takeaway is that the function "x^x^x^..." is a kind of "decision" function, if you know what I mean--one that requires making some sharp distinction somewhere, like the absolute value and "if x>0 then 1 else 0" functions. The x^x^... function is actually a limit, and evaluating it requires deciding that it does approach a limit in the first place. You can't expect these "decision" functions, as I call them, to always be as robust and invertible as usual mathematical functions. This result is probably surprising only because the limit is implicit.