Begin with the empty set then construct the set of the empty set, then construct the set of all subsets of the latter set, then at each level of construction construct the next level as the set all subsets of that level that are definable using formulas restricted to that level, i.e. follow Godel's construction of stages, then continue that iterative construction, lets have a first limit level among levels of that construction, lets continue this construction transfinitely as long as at each level $L_{\alpha}$ there exists a set $x$ constructed at a prior level such that there is a well ordering on the class of all levels from the empty set till $L_{\alpha}$, that is isomorphic to $x$, i.e. the height of each level corresponds to the well ordering of some set already formed at a prior level. 

> Question: what is the limit to this construction? 

>Is it $L_{\omega_1}$? or is it some fixed countable point, like $L_{\omega^{\omega^{\omega^{.^{.^{.}}}}}}$?

I tend to think it is $L_{\omega^{\omega^{\omega^{.^{.^{.}}}}}}$, and that this (I suppose) would make it of the strength of $PA$?