 We’ve now reconstructed the inconsistency in Frege’s system by representing his logic and Basic Law V in a modern system of second-order logic. Philosophers have diagnosed the inconsistency in various ways, and it is safe to say that the matter is still somewhat controversial. In this subsection, we discuss only the basic elements of the problem. Most philosophers and logicians agree that the reason second-order logic can’t be extended by Basic Law V is that the resulting system requires the impossible situation in which the domain of concepts has to be strictly larger than the domain of extensions while at the same time the domain of extensions has to be as large as the domain of concepts.