Sure, the length of the intervals is easily compared. But saying
there are twice as many elements in the total than there are in half the range
is false. They are both aleph 1. In other words, for each unique element you can pick from [0,2], I can pick a unique element from [0,1]. I could even pick two or more. So you can’t compare the number of elements in the two in a meaningful way other than saying they both belong to the same category of infinite.
This is the whole crux of the coastline problem, isn’t it?
Sure, the length of the intervals is easily compared. But saying
is false. They are both aleph 1. In other words, for each unique element you can pick from [0,2], I can pick a unique element from [0,1]. I could even pick two or more. So you can’t compare the number of elements in the two in a meaningful way other than saying they both belong to the same category of infinite.
This is the whole crux of the coastline problem, isn’t it?