Einstein, Podolsky and Rosen’s argument (1935, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?) is another interpretation of the famous Gödel incompleteness argument (1931, Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I) in terms of quantum mechanics. The close friendship between the Princeton refugees Einstein and Gödel might address that fact. However the outlines of a common set-theory structure interpretable in both ways are much more essential concerning the incompleteness (or openness) of infinity: 

An arbitrary infinite countable set “A” and another set “B” so that their intersection is empty are given. One constitutes their union “C”, which will be an infinite set whatever B is. Utilizing the axiom of choice, a one-to-one mapping “f ” exists. One designates the image of B into A through f by “B(f)” so that B(f) is a true subset of A. If the axiom of choice holds, there is always an internal and equivalent image as B(f) for any external set as B. Thus, if one accepts that B(f) coincides with B, whether an element b of B belongs or not to A is an undecidable problem as far as b(f) coincides with b. However if the axiom of choice is not valid, one cannot guarantee that f exists and should display how a constructive analog of “f” can be built. If one shows how f to be constructed at least in one case, this will be a constructive proof of undecidablity as what Gödel’s is. 

The EPR argument interprets the same structure.