For a set S being uncountable means that for any list of elements s_1, s_2, ... of S, there will always be some element of S that's not on the list. For instan

For a set S being uncountable means that for any list of elements s_1, s_2, ... of S, there will always be some element of S that's not on the list. For instance, infinite sequences of bits are uncountable: for suppose that there's any list s_1, s_2, ... of infinite sequences of zeroes and ones. Let x be a sequence that has 0 on i-th position if i-th sequence on the list has 1 on the i-th position, and 1 otherwise. Then x is not on the list: for suppose x would be on the list as some s_n. Then by definition, x has 0 on n-th position if s_n has 1 on n-th position, and 1 otherwise, but s_n is x, a conradiction. For example, if you consider a list: 1: 01011001010100... 2: 01110001001001... 3: 11010110100110... 4: 10010110101010... and further elements, then x would start with 1010...