The Cantor Set is amazing!

Let S₀ = [0, 1].
Let S₁ = [0, 1/3] ∪ [2/3, 1].
Let S₂ = [0, 1/9] ∪ [2/9, 1/3] ∪ [2/3, 7/9] ∪ [8/9, 1].
In general, for non-negative n, let S_n+1 = (S_n ∪ S_n + 2) / 3.
Let S = the intersection of all the S_i.
S is the Cantor Set.

How much was removed from [0, 1] to obtain S?
1/3 + 2/9 + 4/27 + ... = 1.
What is the size of S?
Consider the base three representations of the members of S and the base two representations of the members of [0, 1].

Thank you, Georg Ferdinand Ludwig Philipp Cantor (1845-1918).

I thought of the Cantor Set when I heard of a problem in 2016.
The open interval (0,1) includes all the rational numbers between 0 and 1 and the length of the interval is 1.
The collection of open intervals (0,1/2),(1/2,1) includes all the rational numbers between 0 and 1 except 1/2 and the sum of the lengths of the intervals is again 1.
The collection of open intervals (0,.5),(.4,1) includes all the rational numbers between 0 and 1 but the sum of the lengths of the intervals is 1.1.
The problem is to find a collection of open intervals that includes all the rational numbers between 0 and 1 and the sum of the lengths of the intervals is less than 1.