This is the first in a series of talks where I will be going over the history and the more recent advancements in forcing techniques used to produce models of set theory where the continuum is strictly greater than \(\aleph_1\), a projective well-order of the reals.
In the first talk we will establish preliminaries, understand the motivation for obtaining such models, and go over L. Harrington's initial 1977 construction. Subsequent talks will focus on some more recent results, including applications of the techniques to the theory of cardinal characteristics and the definability of various combinatorial sets of reals.