Common belief in monotonic epistemic logic

We study the extent to which the notion of common belief may be expressed by a finitary logic. We devise a set of axioms for common belief in a system where beliefs are only required to be monotonic. These axioms are generally less restrictive than those in the existing literature. We prove completeness with respect to monotonic neighborhood models, in which the iterative definition for common belief may involve transfinite levels of mutual belief. We show that this definition is equivalent to the fixed-point type definition that Monderer and Samet elaborated in a probabilistic framework. We show further, that in systems as least as strong as the K-system, our axiomatization for common belief coincides with other existing axiomatizations. In such systems, however, there are consistent sets of formulas that have no model. We conclude that the full contents of common belief cannot be expressed by a logic that admits only finite conjunctions.