&quot;The distance between 0 and 2 is larger&quot; is a question about the metric of the space, not size of sets.  There are the same number of points in both sets, since f(x) = 2x is a bijection between them.<p>Try to make arguments from axioms and definitions rather than asserting things from intuition.  Intuition is often a useful tool, but (1) it&#x27;s not an argument, and (2) it&#x27;s not very helpful once you step into the infinite realm.  Incidentally, that&#x27;s why I went for programming:  it&#x27;s like math, but with no infinity (unless you&#x27;re using floats, but that&#x27;s a much easier infinity).