Dear Dirk,
I think that Dedekind was aware of the problem. He successfully dealt
with the secon-order cathegoricity of arithmetic in 1888 and he knew that
Euclid's axiomatization is satisfied by denumerable models.
I think you are right guessing that "he just did not have the
conceptual and mathematical tools for giving a satisfactory proof", if the
proof, to be satisfactory, must be a modern model-theoretic one.
For a more 'naive' proof he had just to consider the 'completenss'
of his definitions of the real numbers (sections on R do not produce new
numbers) and "prove" or "postulate" the coincidence between geometric and
arithmetic continuum. I think this last step was the most troublesome,
because it can not be proved at all, and at the end of the XIX century
even Cantor was not comfortable with such postulating.
It is not casual that few years after Hilbert to this aim had to
propose his strange "completeness axiom". (Incidentally in my e-mail I
wrongly wrote that in this axiom the required model is the 'least': it
must obviously be the 'greatest' because the denumerable model must be
ruled out).
Yours sincerely
Luigi Borzacchini