 These results indicate a weakness in the expressive resources of first-order languages like \(\LKe\). No satisfiable set of sentences can guarantee that its models are all denumerably infinite, nor can any satisfiable set of sentences guarantee that its models are uncountable. So in a sense, first-order languages cannot express the notion of “denumerably infinite”, at least not in the model theory. (See the entry on  second-order and higher-order logic.)