In the same way I can show you a mapping that associates every single real number with a unique clock face (all the clock faces a properly working clock can produce in this case) but I can also show you clock faces (an infinite number of them in fact) that are not involved in this mapping; I can show you clock faces not associated with a real number, thus the number of all possible clock faces must have a larger cardinality than the real numbers.
This is not sufficient to show that you have a larger cardinality. This is essentially claiming that if I have sets A and B, and a bijection between A and a proper subset of B, then A and B must have different cardinality. This is wrong. To see a counterexample, take say the map from the positive integer to the positive integers from n → n+1. In this example, A and B are both the positive integers. Since A=B they must be the same cardinality. But we have a 1-1, onto map from A into a proper subset of B since the map only hits 2,3.4… and doesn’t hit 1.
What Cantor did is different. Cantor’s proof that the reals have a larger cardinality than the natural numbers works by showing that for any map between the positive integers and the reals, there will be some reals left over. This is a different claim than exhibiting a single map where this occurs.
This is not sufficient to show that you have a larger cardinality. This is essentially claiming that if I have sets A and B, and a bijection between A and a proper subset of B, then A and B must have different cardinality. This is wrong. To see a counterexample, take say the map from the positive integer to the positive integers from n → n+1. In this example, A and B are both the positive integers. Since A=B they must be the same cardinality. But we have a 1-1, onto map from A into a proper subset of B since the map only hits 2,3.4… and doesn’t hit 1.
What Cantor did is different. Cantor’s proof that the reals have a larger cardinality than the natural numbers works by showing that for any map between the positive integers and the reals, there will be some reals left over. This is a different claim than exhibiting a single map where this occurs.