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

Giovannini, Eduardo N.

Theoria, Segunda Epoca 28, No. 76, 139-163 (2013).

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:

(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’s remarks: In footnote 8, where the author mentions the linear form of the completeness axiom V.2, to be found in the seventh edition of Hilbert’s GdG, it would have been helpful to mention [W. Weber, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1938, 376–382 (1938; Zbl 0021.24305; JFM 65.0635.01); F. Bachmann, Jahresber. Dtsch. Math.-Ver. 53, 49–56 (1943; Zbl 0060.32506)], where flaws in that axiom were mentioned and fixed. In footnote 21, where the works mentioned above by Baldus, Schmidt, and Hertz are cited, and two versions of the Cantor axiom mentioned, one strong enough to imply the Archimedean axiom, while the other consistent with a non-Archimedean hypothesis as well, the author could have mentioned the much earlier work of K. Th. Vahlen [Jahresber. Dtsch. Math.-Ver. 16, 409–421 (1907; JFM 38.0116.01)], where an axiom Ver, attributed to Veronese, is mentioned, equivalent to the weak version of the Cantor axiom, and the fact that the usual Cantor-Dedekind axiom can be split into the Archimedean axiom and Ver proved (see the review of [P. Cantù, Giuseppe Veronese e i fondamenti della geometria. Milano: Edizioni Unicopli (1999; Zbl 1158.51300)] for details). In footnote 29, the author could have mentioned [J. Panvini, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 7, 153–159 (1953; Zbl 0053.41806)].

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)

Keywords:

Hilbert; Euclidean geometry; axiom of completeness; continuity axioms; axiomatic method; philosophy of geometry

Citations:

Zbl 0002.32302; Zbl 0011.03404; Zbl 0021.24305; Zbl 0060.32506; Zbl 1158.51300; Zbl 0053.41806; JFM 38.0501.01; JFM 56.0488.05; JFM 38.0116.01; JFM 57.0702.01; JFM 60.1215.02; JFM 65.0635.01