Given there are a countable number of disjoint subsets, the number of sets in the sigma algebra has to be non-countable. It is a very simple one to one mapping of all the combinations from the countable number of subsets to all binary numbers between 0 and 1. Specifically, order the countable subsets. For any combination of subsets, the equivalent binary number has zeros for the sets omitted and 1 for sets included.
Example: .001110101.... is the image of the set where the 3rd, 4th, 5th, 7th, 9th, etc. sets are include while the 1st, 2nd, 6,th, 8th, etc. sets are omitted (from the union).