The goal of this research is to rule out "natural" independence phenomena in Set Theory by maximizing your theory in terms of large cardinals and forcing axioms. Using large cardinals in ZFC, by the results of Woodin, we have a clear understanding of the 1st order theory of sets of natural numbers and what it should be. In this talk, we try to extend this understanding to the 1st order theory of subsets of \omega_1 by using large cardinals, forcing axioms, and some hypothesis from inner model theory in ZFC. This is joint work with Matteo Viale.