The argument begins by giving a necessary condition for a theory to be complete, that every element of reality features in the theory.
It goes on to note that for noncommuting operators/quantum observables, the wave function cannot simultaneously be an eigenstate for both. Hence, it is not the case that both quantities are represented in the theory. From this they conclude that either quantum mechanics is incomplete or else not both of the quantities are real.
The second part of the argument sees the famous example of an entangled pair of particles. The argument here now relies on what they take to be a sufficient condition for reality, that a quantity be predictable without disturbing the system. However, if I allow the two particles to travel far from each other, it appears that by making measurements on one particle I can predict both of two noncommuting quantities of the other system (admittedly, not simultaneously) without disturbing it. Both quantities then are real.
Thus they conclude that quantum mechanics is not complete.
… Need to talk about separability and locality. Then link to Bell's inequalities.
The original article is
A thorough treatment is in
Arthur Fine, The Einstein-Podolsky-Rosen Argument in Quantum Theory, Stanford Encyclopedia of Philosophy (web)
Anthony Sudbery, Quantum mechanics and the particles of nature: An outline for mathematicians (in chapter 5)
A survey is in