Die Grundlagen der Arithmetik, §§82-3
GEORGE BOOLOS AND RICHARD G. JNR. HECK
Reductions of arithmetic, whether to set theory or to a theory formulated in a higher-order logic, must prove the infinity of the sequence of natural numbers. In his Was sind und was sollen die Zahlen?, Dedekind attempted, in the notorious proof of theorem 66 of that work, to demonstrate the existence of infinite systems by examining the contents of his own mind. The axioms of General Set Theory, a simple set theory to which arithmetic can be reduced, are those of Extensionality, Separation ('Aussonderung'), and Adjunction: ∀w∀z∃y∀x(x∈y ↔ x∈z ∨ x=w). It is Adjunction that guarantees that there are at least two, and indeed infinitely many, natural numbers. The authors of Principia Mathematica, after defining zero, the successor function, and the natural numbers in a way that made it easy to show that the successor of any natural number exists and is unique, were obliged to assume an axiom of infinity on those occasions on which they needed the proposition that different natural numbers have different successors.
In §§70-83 of Die Grundlagen der Arithmetik, Frege outlines derivations of some familiar laws of the arithmetic of the natural numbers from principles he takes to be 'primitive' truths of a general logical nature. In §§70-81, he explains how to define zero, the natural numbers, and the successor relation; in §78 he states that it is to be proved that this relation is one--one and adds that it does not follow that every natural number has a successor; thus by the end of §78, the existence, but not the uniqueness, of the successor remains to be shown. Frege sketches, or attempts to sketch, such an existence proof in §§82-3, which would complete his proof that there are infinitely many natural numbers.
§§82-3 offer severe interpretive difficulties. Reluctantly and hesitantly, we have come to the conclusion that Frege was at least somewhat confused in these two sections and that he cannot be said to have outlined, or even to have intended, any correct proof there. We will discuss two (correct) proofs of the statement that every natural number has a successor which might be extracted from §§82-3. The first is quite similar to a proof of this proposition that Frege provides in Grundgesetze der Arithmetik, differing____________________