 However, second-order logic does not have a model theory in the same sense as first order logic does. There is no Compactness Theorem to produce non-standard models. One reason why first order logic has such a rich model theory is that first order logic is relatively weak. Countable first order theories that have infinite models have models of all infinite cardinalities. This is a consequence of a weakness of first order logic: its sentences cannot limit the size of the model unless the size is finite. In model theory this weakness is turned to a strength: the structure of models of first order theories can be analyzed in interesting ways. See the entry on  model theory  for examples of this.