Let's define x^x^(x...) as a closed form equation: y=x^y By simple substitution, y=x^(x^y), y=x^(x^(x^y)), and so on. But the closed form is easier to reason

Let's define x^x^(x...) as a closed form equation: y=x^y By simple substitution, y=x^(x^y), y=x^(x^(x^y)), and so on. But the closed form is easier to reason about. The equations given are equivalent to nonlinear systems of the form: y=x^y A=x^y where a=2 or 4. A little manipulation (logs are base x): y = x^y Assumption: x>1 (log is not defined for all values if base is <1) log y = log x^y log y = y Which is only true if y=1, * provided that x>1. But y is taken to be not 1 as a premise, and then x is derived to be greater than 1. The proof thus assumes a contradiction; naturally, it's not difficult to prove a contradiction if you assume one. The trick here was to hide the assumption in a strange equation. * - Actually, on further review, this turns out to be false. Everything else was correct, though, and as it happens log(sqrt(2)) 4 != 4, so this still shows the flaw.