[d_tr]

It is a superset of (0,1), which is uncountable because it is an open interval. All open intervals except the empty one are uncountable. What kind of proof would you like?

There is also no such thing as an uncountable list. Lists can be indexed by integers, which makes them all countable.

[raoof]

the proofs that I've read shows that the list of infinite binary strings between [0,1] are uncountable because the invert of the diagonal of the list is not on the list, do you know any proof that does not refer to the invert of the diagonal?