TITLE: How do I resolve this paradox of counting line segments?
QUESTION [6 upvotes]: If I have some uncountable proper subset $T$ of transcendental numbers on the real line, by definition I can't count them.  Now suppose I take the complement of this set.  I now have a set of line segments defined in-between each pair of points, one fewer segments than there were transcendental numbers the original set.  But every line segment contains a rational number and therefore the number of segments is countable.
This appears to contradict the independence of the continuum hypothesis from some reasonable maths theory in which the above description is expressed.
Where am I going wrong?

Update: Is the problem that I am defining line segments between distinct rational numbers with potentially no lower limit on length?

REPLY [2 votes]: While almost every idea necessary to understand this "paradox" is in an answer or comment somewhere, I haven't yet seen anyone concisely point out the actual error in the OP. So here goes:
"Now suppose I take the complement of this set. I now have a set of line segments defined in-between each pair of points, one fewer segments than there were transcendental numbers the original set."
This assertion is false.
The intuition is presumably that if we have a set of $100$ real numbers, and take the complement of that set in the real numbers, then the result is in fact $99$ open line segments (together with two infinite rays that are irrelevant).
But for infinite sets, it's quite unlikely that the complement of the set in the real numbers consists of open line segments. There's just no reason that it should, despite our intuition about finite sets. (And indeed, if we examine the proof for complements of finite sets, we see that it depends upon each element of our set having a "next element to its right", and now we've merged back to other answers.)