I have been very interested in the countable ordinals for awhile now, but one thing has eluded me despite my research into the subject. What is a well-ordering of the natural numbers corresponding to $\epsilon_1$?
By $\epsilon_{1}$ I mean the second solution to the equation $\omega^a = a$. I have a basic understanding of the ordinals up to $\epsilon_0$, and I have a (unproven) basic method for constructing a well-ordering of the natural numbers corresponding to those ordinals. The ordinal $\epsilon_0$ is massively hard to comprehend but I have constructed an ordering on finite trees that has order type $\epsilon_0$, and after a long search found a way to convert finite trees to natural numbers. However, I have yet to find anything that gives a well-ordering of the natural numbers or trees or pumpkins or any countable set with order type $\epsilon_1$. Even a hint of an idea of the possibility of constructing an ordering on the natural numbers corresponding to $\epsilon_1$ would be helpful. Also, if someone knows a process for going even farther, that would also be appreciated.