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[HM] completeness, Hilbert vs. Weierstrass-Cantor
Posted:
Nov 24, 2005 3:45 AM
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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 Bolzano-Weierstrass axiom, since it was crucial in both Bolzano's
classic paper and Weierstrass's lectures. (Perhaps one might also use
"Weierstrass-Cantor".) It was from here that Cantor took it, and used it for
his first proof of the non-denumerability 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 Bolzano-Weierstrass
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 Bolzano-weierstrass principle to the cut
axiom.
Another interesting point is that Dedekind's cut axiom is not independent
from the Archimedean axiom, but the Bolzano-Weierstrass 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. 197-198.
There Bernays refers mainly to a paper by P. Hertz in 1934, but also to
several publications of Baldus, where the equivalence of the
Bolzano-Weierstrass 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. E--41018 Sevilla, España
Tel: +34.954.557825
Fax: +34.954.551668
http://www.pdipas.us.es/j/josef/
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