So you start with a 1-dimensional stick, remove the middle third of it, leaving 2 pieces. From each of these 2 pieces, remove the middle third. Etc. Whatever is left at the end of infinitely many stages, call "the Cantor Dust".
The short version of my problem: Here are two things I read about the Cantor Dust: a. It does not consist of isolated points. b. It also does not contain any segments of nonzero length. (a) and (b) seem to me incompatible with each other. I don’t see what the third alternative could be.
Longer version: Here is what seems to me the case:
- At the end of the process, there are aleph-null pieces of the stick left. This must be so, since the cuts only occurred at locations with rational coordinates (calling one endpoint of the stick “0” and the other end “1”). So there are aleph-null cuts, so there must be only aleph-null pieces.
- Each piece has a size of zero. This must be so, since the measure of the stuff removed is 1, so the total measure of the Dust is 0.
- If a piece has a size of zero, it’s a point. Imagine overlaying one of the pieces of Cantor Dust on top of a geometric point. How far would it stick out? Zero distance. Therefore, the piece would coincide with the point, and therefore the piece itself is just a point.
- Therefore, the Cantor set consists of aleph-null points. (From 1-3.)
- But I also read that the Cantor set contains uncountably many points.
What has gone wrong here? In the sources I looked at, no one addresses the "long version” argument above. None even address the "short version", which surprises me.
Amendment: I intend "pieces" to be the largest connected parts that are left (so a piece might be a single point). I intend the term to capture the sense in which, after the first stage, we say "there are two pieces"; after the second stage "there are four pieces", etc.