Find a set A of rational numbers such that (A, $\le_Q$) is isomorphic to sup{$\omega$, $\omega^\omega$, $\omega^{\omega^\omega}$, $\cdots$}.
This is a question that is listed in "Introduction to Set Theory" by Hrbacek and Jech on page 123. Here, $\omega$ represents for the ordinal corresponding to the set of natural numbers. The prior question was to find such a set that is isomorphic to $\omega^\omega$, and I've constructed A as {$\prod {{1} \over {p_i^{a_i}}} $} there, where $p_i$ represents for the $i$th prime number and $a_i$ being its exponents.
Honestly, I've tried my best but couldn't construct such a set corresponding to sup{$\omega$, $\omega^\omega$, $\omega^{\omega^\omega}$, $\cdots$}. And now I have another question: what is the biggest ordinal number that is isomorphic to (A, $\le_Q$) where $\le_Q$ denotes the usual ordering in rational number? According to Well Ordering Principle, we know that Q can be well-ordered, then (Q, $<_W$) is isomorphic to a unique ordinal, but clearly $\le_Q$ isn't a well ordering. Is there no limit in this procedure?
