So to recap: 1) If we define f(x)=x^x^x^... as the limit of the sequence x, x^x, x^x^x, ... then f(sqrt(2))=2, not 4, and in fact there's no such x that f(x)=4

So to recap: 1) If we define f(x)=x^x^x^... as the limit of the sequence x, x^x, x^x^x, ... then f(sqrt(2))=2, not 4, and in fact there's no such x that f(x)=4. 2) If we instead "solve" the equation y=x^x^x... by dubious manipulations with infinite expressions, we get x=y^(1/y), so y=2 and y=4 both lead to x=sqrt(2). When playing with infinite expressions, you must always keep in mind how they're defined in terms of limits. If you learn to do that, many puzzles with infinity will stop bothering you forever, like Zeno's paradox, 0.999...=1, etc.