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

[John Baldwin]

Note that the completeness is not of the axioms but of the model. This
is a formulation of what we now call the completeness of the real numbers.
It is equivalent to the least upper bound axiom.  Such a second order 
axiom is necessary to characterize the reals as an ordered set.

I hope someone can respond to Rodin's historical questions.





On Sat, 27 Jan 2007 Andre.Rodin at ens.fr wrote:

> 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
> more?)
> I will be grateful for any hint or a reference.
>
> Thanks in advance
> Andrei Rodin
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John T. Baldwin
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Department of Mathematics, Statistics, 
and Computer Science  M/C 249
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