User blog comment:Wythagoras/All my stuff/@comment-10429372-20130715145943/@comment-5150073-20130716171030

Suppose that the function f(n) has order type alpha. Then fn(n) would have order type alpha+1, and we can have the function f(n) = In(n). If In n (n) won't grow faster, it means that ordinal alpha is the fixed point for alpha = alpha+1, which is by definition, a non-existent ordinal, and the function isn't consistent.