Whether we use Dedekind cuts or Cauchy sequences to construct
the ordered field of real numbers (and it doesn’t matter which),
we are taking the ordered field of rational numbers and
enlarging it to be “complete” in some sense or another. But we
also aren’t making it too much bigger. The universality property
we got from completing the uniform structure already gives
evidence of that, but there’s another property which we can show
is true of , and which shows that the real numbers aren’t too
unwieldy. In The Sand Reckoner, the ancient Greek mathematician
Archimedes once set about the problem of should the number of
grains of sand in existence to be finite. He does this by
determining a (very weak) upper bound: the number of grains of
sand it would take to fill up the entire universe, as he
understood the latter term. He writes: There are some … who
think that the number of the sand is infinite in multitude; and
I mean by the sand not only that which exists about Syracuse and
the rest of Sicily but also that w