 One way to think about Henkin models is that they fill the gaps left by full models, just as irrational numbers fill the gaps left by rational numbers. They are needed in order to obtain a smooth theory of second-order logic. They manifest “paradoxical” phenomena that we do not see among full models. For example we obtain non-standard models of number theory, countable models of the axioms of real numbers, etc. The categorical sentences characterizing mathematical structures among full models now suddenly have also other “models”, namely Henkin models, and they come in all infinite cardinalities.