As was said above, the details of Frege's proof of Theorem 288 are not relevant to the central concerns of the present paper. But, as the proof is somewhat difficult to follow, and as certain features of it are interesting for other reasons, it is worth explaining here. I shall discuss only those theorems which Frege proves in the relevant sections of Grundgesetze, omitting the proofs of any and all theorems proven earlier in the book, i.e. all theorems earlier than Theorem 264.55
Before we begin, we need one additional definition:
Thus, F^(QπR)(〈a,x〉)(ξ,η) is a relation, defined by induction, according to Frege's method: it is the relation which holds between ξ and η just in case 〈ξ,η〉 belongs to the (QπR)-series beginning with 〈a,x〉.
Recall that Theorem 288 is:
Func(Q) & ¬∃z.F(Q)(z,z) & F=(Q π Pred)(〈x,l〉,〈y,n〉)→ Nz:Btw(Q;x,y)(z) = Nz:Btw(Pred)(l,n)(z).
That is: If the Q-series is simple and if 〈y,n〉 is a member of the (Q π Pred)-series beginning with 〈x,l〉, then the number of objects Q-between x and y is the same as the number of objects Pred-between 1 and n.
The proof of Theorem 288 requires two lemmas. The first of these is Theorem 287, whose proof is given in the notes:56
¬∃z[F=(Pred)(l,z) & F(Pred)(z,z)].
The second lemma is Theorem 284:
Func(R) & ¬∃z[F=(R)(m,z) & F(R)(z,z)] & Func(Q) & ¬∃z[F=(Q)(x,z) & F(Q)(z,z)] & F=(QπR)(〈x,m〉,〈y,n〉)→ Nz:Btw(Q;x,y)(z) = Nz:Btw(R;m,n)(z)____________________
This paper is dedicated to the participants in a seminar, informally known as 'Frege's Three Books', which George Boolos and I taught together in the Spring of 1993. Regular members, other than myself and George, were Emily Carson, Janet Folina, Michael Glanzberg, Delia Graff, David Hunter, Darryl Jung, Josep Macia-Fabrega, Ofra Rechter, Lisa Sereno, and Jason Stanley. I shall be a happy man if, one more time in my life, I have the opportunity to teach a seminar which is half as much fun as that one was.