Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



[HM] completeness, Hilbert vs. WeierstrassCantor
Posted:
Nov 24, 2005 3:45 AM


Dear friends,
yesterday I was writing to a colleague on a topic that may be interesting to more people. The question was about Hilbert's completeness axiom, Cantor's version stating that every infinite sequence of nested intervals has a common point, and something "everybody knows", that both are equivalent.
As you know, Hilbert's "completeness" axiom (Vollständigkeitsaxiom) first appeared in his 1899 axiomatization of the real numbers, and then was incorporated into the second edition of GdG. Alternative statements are Dedekind's cut axiom, first published in 1872, and the principle that Cantor published in 1874 concerning nested intervals.
Regarding the name of this principle, I think it would be desirable to call it the BolzanoWeierstrass axiom, since it was crucial in both Bolzano's classic paper and Weierstrass's lectures. (Perhaps one might also use "WeierstrassCantor".) It was from here that Cantor took it, and used it for his first proof of the nondenumerability of the set of reals. So although Cantor was the first to use it in a publication, the origins was well known (and Cantor himself acknowledges it in a paper of 1884 (see his Abhandlungen, p. 212, where you find interesting comments).
As a matter of fact, Cantor does not present the BolzanoWeierstrass principle as an axiom in his paper of 1874. Rather he offers a proof based on the theorem that an infinite sequence of reals that is monotone and bounded has a limit. The explanation for this is that Cantor's redaction was based on some of Dedekind's letters. Dedekind had proven the theorem about infinite sequences on the basis of his "cut axiom" of continuity. Thus, in his letter he was reducing the Bolzanoweierstrass principle to the cut axiom.
Another interesting point is that Dedekind's cut axiom is not independent from the Archimedean axiom, but the BolzanoWeierstrass principle is. So the "Cantor axiom" is a natural candidate for replacing Hilbert's completeness axiom, as Paul Bernays remarks in vol. III of Hilbert's works, p. 197198. There Bernays refers mainly to a paper by P. Hertz in 1934, but also to several publications of Baldus, where the equivalence of the BolzanoWeierstrass axiom with Hilbert's, and their independence from the Archimedean axiom, is analyzed.
Best wishes,
Jose Ferreiros
 Jose Ferreiros Departamento de Filosofia y Logica, Universidad de Sevilla Camilo Jose Cela, s/n. E41018 Sevilla, España Tel: +34.954.557825 Fax: +34.954.551668 http://www.pdipas.us.es/j/josef/



