Shortly after Paul Cohen developed forcing to demonstrate that the Continuum Hypothesis is independent of ZFC, Robert Solovay adapted the technique to prove that the axiom of choice is essential (sort of) in the construction of a nonmeasurable subset of the real line. Solovay's model of ZF has many nice features and many classical mathematical problems are actually questions about what is true in Solovay's model. Decades later, work of Shelah and Woodin showed that Solovay's model is largely immune to independence phenomena - offering an explanation for why forcing isn't so useful in subjects like number theory, low dimensional topology, and partial differential equations. I'll explain all of this for the nonlogician - and explain what I mean by the oxymoron "essential (sort of)".