I am reading through the EPR paper and follow most of it. The authors argue that either QM must be incomplete (let's call this statement A), or incompatible observables can not have simultaneous physical reality (statement B). The authors define what they mean by complete and physically real. They go on to show that rejecting A forces us to also reject B. Since one of these statements must be true, it follows that must accept A, i.e. QM must be incomplete.
I am having trouble seeing exactly where the authors invoke the !A to demonstrate !B. They construct a joint quantum state and show that by measuring the position or momentum of one particle the position or momentum (respectively) of the other can be known perfectly. It is not clear to me that the assumed completeness of QM is exploited anywhere in this argument.
update: In Arthur Fine's analysis of the EPR argument he writes,
Indeed what EPR proceed to do is odd. Instead of assuming completeness and on that basis deriving that incompatible quantities can have real values simultaneously, they simply set out to derive the latter assertion without any completeness assumption at all. This “derivation” turns out to be the heart of the paper and its most controversial part.
This articulates my confusion about/objection to the EPR argument well.