I’m not sure if this is a genuine paradox but it does seem very unintuitive and I’d appreciate if someone can explain it so that it seems reasonable and/or explain the flaw in the reasoning.
Consider the following countable collection of open intervals: I1 is ( 0, 1/2 ), I2 is ( 1/2, 1/4 ), I3 is ( 1/4, 1/8 ), etc. The sum of the lengths of the intervals is 1.
Now cover each rational number with one of the intervals.
There can be at most one uncovered irrational between any two successive open intervals.
There are a countable number of open intervals so there can be at most a countable number of irrationals left uncovered.
So we’ve covered the entire real line, except for a countable number of points, with a countable number of open intervals the sum of whose lengths adds up to 1.
Or have we?
P.S. I asked a similar question a year ago but didn't write it up clearly. I hope this is better.