One doesn't need the full AC in order to prove (in ZF) N X N ~ N (~ being equinumerous), whereas one does need the full AC in order to prove that for all sets A, AXA~A. So, what determines, for a particular A, whether one can prove that AXA~A? For example, does one need the full AC to prove that RXR~R? (I suspect the answer to this last question is no.) So where is the "cut-off point"?