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**Completeness and continuity in Hilbert’s Foundations of geometry: on the Vollständigkeitsaxiom.**
*(Spanish.
English summary)*
Zbl 1269.03002

This is a detailed analysis of the reasons why Hilbert chose to use the Vollständigkeitsaxiom V.2 rather than a more traditional axiom of completeness such as a version of the Cantor-Dedekind completeness axiom. Given that, in 1894, in his first lecture on the foundations of geometry, given while still in Königsberg, Hilbert had used a variant of the Bolzano-Weierstrass theorem as a continuity axiom (as well as in a postcard to Felix Klein of August 14, 1894, published in 1895 in the Mathematische Annalen and added in later editions to the appendices to the Grundlagen der Geometrie (GdG)), it was not for lack of alternatives that Hilbert felt the need to introduce V.2 as an axiom stating a maximality condition.

There are two reasons that are likely to have influenced Hilbert’s choice:

There are two reasons that are likely to have influenced Hilbert’s choice:

- (i)
- the fact that a traditional completeness axiom of the Cantor-Dedekind variety would have implied the Archimedean axiom (which a properly reformulated V.2 does not, as shown by H. Hahn [Sitzungsber.Kaiserl.Akad.Wiss., Math.-Naturw.Kl.116, 601–655 (1907; JFM 38.0501.01)]), and Hilbert wanted to build up geometry in small steps (a Stufenaufbau), enabling an analysis of the effect each particular axiom has in the axiomatic edifice;
- (ii)
- the fact that, according to Hilbert, neither Cantor’s axiom on nested sequences of closed intervals (which, in its version involving lengths of intervals converging to zero, does not imply the Archimedean axiom, a fact for which the author cites R. Baldus [in: Atti del congresso internazionale dei matematici, Bologna 3–10 settembre 1928, 4, 271–275 (1931; JFM 57.0702.01); Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturw. Kl. 1930, No. 5, 12 S. (1930; JFM 56.0488.05)], A. Schmidt [Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturw. Kl. 1931, No. 5, 1–8 (1931; Zbl 0002.32302)], P. Hertz [C. R. Soc. Physique Genève (Suppl. aux Arch. Sci. Physiques etc. 16) 51, 179–181 (1934; Zbl 0011.03404; JFM 60.1215.02)]), nor Dedekind’s axiom on cuts can be considered ‘purely geometric’ (p.153).

Reviewer: Victor V. Pambuccian (Phoenix)

### MSC:

03-03 | History of mathematical logic and foundations |

51-03 | History of geometry |

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

01A60 | History of mathematics in the 20th century |

03B30 | Foundations of classical theories (including reverse mathematics) |