**APPENDIX 1. FREGE'S PROOF OF THEOREM 288, AND REFLECTIONS ON THE LEAST NUMBER PRINCIPLE**

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.

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Before we begin, we need one additional definition:

*F*^(T;〈a,x〉)(b,y) ≡df*F*=(T)(〈a,x〉,〈b,y〉)

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)

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*'The Development of Arithmetic'*and

*'Definition by Induction'*; nevertheless, the discussion which followed--indeed, all our discussions at the conference--had great effect on my thought about these and other issues. Many thanks are due to Matthiass for organizing the conference.

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.

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*'Definition by Induction'*.

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*F*=(Pred)(l,n), then, since Pred(0,1), we have

*F*=(Pred)(0,n), by Theorem 285. But then ¬

*F*(Pred)(n,n), by Theorem 145.

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