First, we need to establish that two infinities are equal in cardinality (aka size) if all their elements can be 1:1 mapped to each other.
So, to go from the reals within [0, 1] and [0, 2], we can multiply by 2. This maps every value within [0, 1] to every value within [0, 2], so these are of the same cardinality.
Where things get interesting is the proof that the reals within [0, 1] are of greater cardinality than every integer.
Say we have an arbitrary mapping from every integer to a real within [0, 1]:
This list contains every integer, but it does not contain every real number because we can always come up with a new one by ensuring at least one digit is different in each existing real:
0-> …8… ≠ 91-> …7… ≠ 82-> …7… ≠ 83-> …9… ≠ 04-> …2… ≠ 3
⋮ ⋱
0.98803… is not within the list
Therefore, no 1:1 mapping between the integers and reals exists. Because the limiting factor is the amount of integers, the cardinality of the reals is greater than that of the integers.
May as well go through the proofs:
First, we need to establish that two infinities are equal in cardinality (aka size) if all their elements can be 1:1 mapped to each other.
So, to go from the reals within [0, 1] and [0, 2], we can multiply by 2. This maps every value within [0, 1] to every value within [0, 2], so these are of the same cardinality.
Where things get interesting is the proof that the reals within [0, 1] are of greater cardinality than every integer.
Say we have an arbitrary mapping from every integer to a real within [0, 1]:
0 -> 0.89236… 1 -> 0.47389… 2 -> 0.84776… 3 -> 0.18790… 4 -> 0.90542… ⋮ ⋱This list contains every integer, but it does not contain every real number because we can always come up with a new one by ensuring at least one digit is different in each existing real:
0 -> …8… ≠ 9 1 -> …7… ≠ 8 2 -> …7… ≠ 8 3 -> …9… ≠ 0 4 -> …2… ≠ 3 ⋮ ⋱ 0.98803… is not within the listTherefore, no 1:1 mapping between the integers and reals exists. Because the limiting factor is the amount of integers, the cardinality of the reals is greater than that of the integers.
Edit: https://en.wikipedia.org/wiki/Cantor’s_diagonal_argument