My treatment of dimensionality has been stalled for a while. It's partly because I've been busy, it's also partly because when I encounter a conceptual issue, I have a hard time writing anything else. The conceptual issue that stalled me out this time is still unresolved, so I thought I'd share it and look for input. The formal statement of the problem is as follows:
Imagine a line segment whose length, L0, is one meter. This segment is, of course, composed of infinitely many points. Said another way, the cardinality of the set of all points in the segment is |S0| = ∞. Imagine now that I bisect the segment. After one bisection, the length of the right half of the segment is L1 = 50cm. And the cardinality of the right half of the segment is still infinity. |S1| = ∞.
Now imagine that I repeat this operation several times. After n bisections, Ln = (½)n meters. And |Sn| = ∞. Consider the limit n → ∞. If my expression for Ln is correct then, lim(Ln) = 0 which is to say that the segment length approaches zero. But if my expression for |Sn| is correct, lim(|Sn|) = ∞ which is to say that even after infinitely many bisections, the segment will still contain infinitely many points.
The problem I'm having is that a segment of length 0m is a point right? But a set of infinitely many adjacent collinear points is a segment. Aren't these two conclusions contradictory? Did I make a mistake?
Please leave any thoughts or ideas at all in the comments. I'm really curious about this one.
