 But the problem is that second-order logic with Basic Law V as a whole requires that there be more concepts than extensions. The requirement that there be more concepts than extensions is imposed jointly by the Comprehension Principle for Concepts and the new significance this principle takes on in the presence of Basic Law V. The Comprehension Principle for Concepts asserts the existence of a concept for every condition on objects expressible in the language. Now although it may seem that this principle, in and of itself, forces the domain of concepts to be larger than the domain of objects, it is a model-theoretic fact that there are models of second-order logic with the Comprehension Principle for Concepts (but without Basic Law V) in which the domain of concepts is not larger than the domain of  objects.[3]  However, the addition of Basic Law V to Frege’s system forces the domain of concepts to be larger than the domain of objects (and so larger than the domain of extensions), due to the endless cycle of new concepts that arise in connection with the new extensions contributed by Basic Law V. However, as we saw in the last paragraph, Vb requires that there be at least as many extensions as there are concepts.