Boris Tsirelson constructed the first Banach space that did not have a subspace isomorphic to some $\ell_p$, $1\le p < \infty$ or $c_0$.  This eventually led to the Gowers-Maurey construction of hereditarily indecomposable Banach spaces.  Tsirelson writes that his forcing construction was motivated by Cohen's forcing arguments:

http://www.tau.ac.il/~tsirel/Research/myspace/remins.html