New axiomatizations for logics with generalized quantifiers

We give a complete axiomatization for admissible fragments of L {ie257-1}(Q). This axiomatization implies syntactically Gregory's characterization of L {ie257-2} sentences with no uncountable models ([5]). This is then extended to stationary logic. To obtain these results, we employ Ressayre's methods ([16], [17]) augmented with an application of game sentences. In section 4 we prove a result emphasizing the naturalness of Gregory's result.