A simple example of blocking the Craig trick

William Craig observed that, under quite general conditions, a theory with a recursively enumerable axiomatization is also recursively axiomatizable. The paper discusses these conditions and builds a simple example of a theory for which the conditions of the Craig’s theorem are not satisfied; in particular, the constructed theory has a recursively enumerable axiomatization, but does not have a recursive axiomatization with the same set of inference rules. At the same time, extending the set of inference rules with some rules adm