I think the problem comes from the mapping you are choosing. You can construct a map that between (0, 2) and (0, 1) that is onto but not one-to-one. But to sa

I think the problem comes from the mapping you are choosing. You can construct a map that between (0, 2) and (0, 1) that is onto but not one-to-one. But to say that (0, 2) and (0, 1) have the same size as sets (cardinality) means that there exists a bijective map between the two sets. You just haven’t found out what that mapping is. The mapping you are thinking of is not a bijection so from your perspective you are not seeing the sets as being the same size. Consider this. The set of non-negative integers is {0, 1, 2, 3, 4,....} and the set of even non-negative integers is {0, 2, 4, 6, ...}. One might think the first set is twice as large as the second set. And under a natural way of looking at it this appears to be true. But consider this mapping: 0 -> 0 1 -> 2(1) = 2 2 -> 2(2) = 4 3 -> 2(3) = 6 . . . Under this map I’ve bijectively mapped the set of all non-negative integers to the set of non-negative even integers. This is how intuition can fail when dealing with infinite sets. Under one way of looking at things one set appears to be larger than the other. But the definition of cardinality (size of the set) deals with whether or not a bijection exists and does not care that other mappings might give the appearance of one set being bigger than another set. All infinite sets have this property: They can be put in bijecitve correspondance with a proper subset. This can not happen with finite sets. Here’s a bijection from [0, 1] to [0, 2]. f(x) = 2x.