# [FOM] Question on the number line

Rob Arthan rda at lemma-one.com
Wed Nov 16 07:50:23 EST 2005

```On Tuesday 15 Nov 2005 8:34 pm, Dean Buckner wrote:
> Forgive me if this is a naïve question.  It occurs in the chapter in
> Suarez (the 16C philosopher & mathematician) that I mentioned in a
> previous posting.  Suppose one takes a line segment of length 1 and
> break it in half (assuming we can break it "in half").  Intuitively, you
> would imagine you get two identical segments of length 0.5.  But if you
> "identify" that line segment with the reals from 0.0 to 1.0, you have to
> ask what happens to the point at 0.5 -- which "half" does it belong to?
> ...

It doesn't matter, what the apparent dilemma shows is that the length of an
interval should not be affected by removing the end-points under any
reasonable formalisation of the intuitive notion of length.

Under the usual formalisation of length via Lebesgue measure you can remove
uncountably many points without affecting the length. The Cantor set
comprising all numbers with no 2s in their ternary expansion has measure 0,
so its complement in the unit interval has the same length as the whole
interval. Does that bother you too?

Regards,

Rob.

```