Obviously, I\Sigma_1 + Con(I\Sigma_1) is a sub-theory of I\Sigma_2, since I\Sigma_2 proves Con(I\Sigma_1). I am guessing (a) that it is a proper sub-theory and (b) that it does not even interpret I\Sigma_2. Yes? References? Does the same hold for I\Sigma_n + Con(I\Sigma_{n+k})? Thanks in advance, Richard Heck