As usual, write [a,b] for the interval of
the real line from a to b including both a and b, (a,b) for the interval of
the real line from a to b excluding a and b, and [a, b) and (a, b] respectively for the
intervals that include a and
exclude b and vice versa.
Suppose that you want to measure the size m(I) of an interval I, but you have the conviction that
single points matter, so [a,b] is bigger than (a,b), and you want to use
infinitesimals to model that difference. Thus, m([a,b]) will be
infinitesimally bigger than m((a,b)).
Thus at least some intervals will have lengths that aren’t real
numbers: their length will be a real number plus or minus a (non-zero)
infinitesimal.
At the same time, intuitively, some intervals from a to b should have length
exactly b − a, which is a real
number (assuming a and b are real). Which ones? The choices
are [a,b], (a,b), [a, b) are (a, b].
Let α be the non-zero
infinitesimal length of a single point. Then [a,a] is a single point.
Its length thus will be α, and
not a − a = 0. So
[a,b] can’t
always have real-number length b − a. But maybe at least
it can in the case where a < b? No. For suppose
that m([a,b]) = b − a
whenever a < b.
Then m((a,b]) = b − a − α
whenever a < b,
since (a, b] is
missing exactly one point of [a,b]. But then let c = (a+b)/2 be the
midpoint of [a,b].
Then:
- m([a,b]) = m([a,c]) + m((c,b]) = (c−a) + (b−c−α) = b − a − α,
rather than m([a,b]) as was
claimed.
What about (a,b)?
Can that always have real number length b − a if a < b? No. For if we had
that, then we would absurdly have:
- m((a,b)) = m((a,c)) + α + m((c,b)) = c − a + α + b − c = b − a + α,
since (a,b) is
equal to the disjoint union of (a,c), the point c and (c,b).
That leaves [a, b)
and (a, b]. By
symmetry if one has length b − a, surely so does the
other. And in fact Milovich gave me a
proof that there is no contradiction in supposing that m([a,b)) = m((b,a]) = b − a.
