Iterative and fixed point common belief

We define infinitary extensions to classical epistemic logic systems, and add also a common belief modality, axiomatized in a finitary, fixed-point manner. In the infinitary K system, common belief turns to be provably equivalent to the conjunction of all the finite levels of mutual belief. In contrast, in the infinitary monotonie system, common belief implies every transfinite level of mutual belief but is never implied by it. We conclude that the fixed-point notion of common belief is more powerful than the iterative notion of common belief.