 Frege’s derivations of Hume’s Principle were invalidated by the fact that it appeals to the inconsistent Basic Law V. Neverthelss, we briefly describe in this subsection, for interested readers, Frege’s derivations. In Gl, §73, Frege sketches an informal proof of the right-to-left direction of Hume’s Principle using the above informal definition of the number of \(F\)s. The derivation appeals to the fact that a concept \(G\) is a member of the extension of the second-level concept concept equinumerous to F if and only if \(G\) is equinumerous to \(F\). In other words, the proof relies on a kind of higher-order version of the Law of Extensions (described above), the ordinary version of which we know to be a consequence of Basic Law  V.[9]  Here is a reconstruction of Frege’s proof in Gl, §73, extended so as to cover both directions of Hume’s Principle.