  Bell's theorem cannot be proved if complementary measurements have to be
represented by random variables which cannot be added or multiplied. One such
case occurs if their domains are not identical. The case more directly related
to the Einstein-Rosen-Podolsky argument occurs if there exists an `element of
reality' but nevertheless addition of complementary results is impossible
because they are represented by elements from different arithmetics. A naive
mixing of arithmetics leads to contradictions at a much more elementary level
than the Clauser-Horne-Shimony-Holt inequality.
