The axioms demand that either one function is eventually dominated by the other, or both functions are of the same order. But which of these is the case will strongly depend on which subsequence you look at.
You may have missed the same subtlety that I did. Because pi is irrational, the functions are different at all integers. Therefore, in the total order, these two functions cannot have the same order.
That still doesn't resolve which one is larger though.
Well, as presented in Tao's post, the set Ω can be either the natural numbers or the real numbers. So I'm assuming the "subsequence" is a (perhaps uncountable?) set of real parameters, in the latter case.
