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**Die Grundlagen der Arithmetik, §§82–3.**
*(English)*
Zbl 0935.03008

Schirn, Matthias (ed.), The philosophy of mathematics today. Papers from a conference, Munich, Germany, June 28-July 4, 1993. Oxford: Clarendon Press. 407-428 (1998).

The authors discuss an important passage of Gottlob Frege’s “Grundlagen der Arithmetik” [Koebner, Breslau (1884), critical ed. by C. Thiel, Meiner, Hamburg (1986)], in particular §§82-3, containing Frege’s sketch of an existence proof of the successor that would complete his proof that there are infinitely many natural numbers. According to the authors, these paragraphs offer severe interpretative 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” (p.407). The authors give two alternative proofs, one similar to that provided by Frege himself in the “Grundgesetze der Arithmetik” [vol. 2, §§115, 117, 119, Pohle, Jena (1903; JFM 34.0071.05), reprinted Olms, Hildesheim (1962, 1998)]. Before this they sketch Frege’s development of arithmetic in the “Grundlagen” from §70 on. The paper closes with three appendices, listing the counterparts in “Grundgesetze” of some propositions in “Grundlagen”, an interpretation of Frege arithmetic in second-order arithmetic, and a translation of some paragraphs of the “Grundgesetze” in present-day notation.

For the entire collection see [Zbl 0897.00021].

For the entire collection see [Zbl 0897.00021].

Reviewer: V.Peckhaus (Erlangen)

### MSC:

03A05 | Philosophical and critical aspects of logic and foundations |

03-03 | History of mathematical logic and foundations |

01A55 | History of mathematics in the 19th century |