Comment by alexarnesen

I like the approach, but infinities can be counter intuitive. See Cantor, Dedekind. The problem is your experiment. We don't say "there are N reals between 0 and 1", we talk about cardinality. The sets (0,1] and (1,3] in R1 don't have "the same number of items" in them; they have the same cardinality.