The Consistency of Predicative Fragments of Frege’s Grundgesetze Der Arithmetik

Reading ‘x̂(Fx)’ as ‘the value-range of the concept F ’, Basic Law V thus states that, for every F and G, the value-range of the concept F is the same as the value-range of the concept G just in case the F s are exactly the Gs. In Frege: Philosophy of Mathematics, Michael Dummett (1991, pp. 217–22) raises the question how the serpent entered Eden, that is, how we should understand the genesis of the contradiction in Frege’s system. The standard view is that the inconsistency is simply to be blamed on Basic Law V. The problem, seen this way, is that Basic Law V requires for its truth that there be as many objects in the domain of the theory as there are concepts true or false of those objects, i.e., that the firstand second-order domains should be of the same cardinality: And that, as Cantor showed us, is quite impossible. Note, however, that this reflection shows only that the theory is unsatisfiable, that is, that it does not have a standard model. And one might wonder whether this explanation really answers Dummett’s question, since it does not, or at least does not obviously, say why the theory ought to be inconsistent: why a contradiction should be derivable from Basic Law V. These two questions are not only different, but may need different answers, since the completeness theorem fails for second-order logic. Dummett’s own view is that the blame is to be ascribed, not so much to Basic Law V, as to the impredicative character of Frege’s theory, in particular, to his acceptance of an unrestricted comprehension schema. In his paper ‘Whence the Contradiction?’, George Boolos (1998) questions Dummett’s view and argues in favor of the more standard line. In the course of doing this, Boolos raises a number of technical questions