"Are there more real numbers between 0 and 1 or between 0 and 2?" If you ask this question to a present day mathematician, he/she would answer that they have the same amount of numbers. Why? Because for every x in the set of numbers between 0 and 2 (call this set A), there is a corresponding number x/2 in the set of numbers between 0 and 1 (call this set B). Thus both set A and B have the same number of elements. But this type of reasoning seems very subjective to me. If instead of mapping from x -> x/2, you map from x -> x/3, then you conclude that there are more elements in set B! Furthermore, if you map from x -> x, then you conclude that set A is bigger! Thus, by changing your mapping you can just about say any thing: |A| > |B|, |A| < |B|, or |A| = |B| ! I don't get it. I thought math is suppose to be objective, not subjective?