[FOM] Hilbert's Vollstandigkeitsaxiom and Hilbert's Hotel

Andre.Rodin@ens.fr Andre.Rodin at ens.fr
Sat Jan 27 08:14:03 EST 2007

Dear FOMers,
I have few questions concernig Completeness axiom (Vollstandigkeitsaxiom) of
Hilbert's "Grundlagen der Geometrie":

"To a system of points, straight
lines, and planes, it is impossible to add other elements in such a manner that
the system thus generalized shall form a new geometry obeying all of the five
groups of axioms."

This axiom absent in the first edition of Grundlagen of 1899 appears in the
second edition of this work. Earlier Hilbert used a similar axiom in his
lecture "Ueber den Zahlbegriff" delivered in 1899 and published in 1900.

My first group of question is historical: Are there historical evidences
explaining Hilbert's motivation behind the introduction of this axiom? How
exactly Hilbert formulated the problem the Vollstandigkeitsaxiom was supposed to
treat? How he discovered this problem?

The second group of questions concerns the axiom itself. Obviously Hilbert had
in mind an infinite model of his geometry: points, straight lines and planes
are infinitely many. Whatever is the cardinality of this model M, it is always
possible to intruduce into M new elements - up to a countable number of new
elements - and so obtain another set M' of the same cardinality. This property
of infinite sets is often associated with Hilbert's name through the popular
story of "Hilbert's Hotel". Hilbert's Vollstandigkeitsaxiom implies that
although M' is isomorphic to M as a set M' unlike M is NOT a model of the
theory in question. Hilbert provided no justification of why M with the
required maximal property should exist (in any appropriate sense). Perhaps he
didn't really think about models in set-theoretic terms. I wonder how the
problem looks like from the today's viewpoint. Can Hilbert's intuition
concerning the existence of maximal models of mathematical theories be
justified? Was it plainly wrong? How "completeness" in Hilbert's sense relates
to semantic completeness? How it relates to categoricity? (Hilbert's
completeness implies categoricity but not the other way round. Can one say
I will be grateful for any hint or a reference.

Thanks in advance
Andrei Rodin

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