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"Points on the line" is fine for
the first, second, ..., tenth cut at a
definition. Sure, completeness
is the biggie for the reals compared
with the rationals, algebraics, etc. Still, as in the OP, mentioning Dedekind cuts
is okay since it is one way to establish
completeness, but there is much more,
e.g., as in John C. Oxtoby,
Measure and Category. and even that doesn't fathom all that
is special about the reals. E.g.,
for just a little more, there is
the continuum hypothesis, that little
thing! The OP wants to say that by mentioning
Dedekind and completeness he is getting
at what the reals really are; no,
instead he is just cutting one layer
deeper of something that has
likely some infinitely many layers
available. Yes, yes, yes, I know; I know;
the reals are the only complete,
Archimedean ordered field, okay, after
we have defined completeness,
Archimedean ordered, and field and
explained why these are important. So, back to "points on the line" --
it's actually pretty good for a first
cut. |