Is the the set of real numbers in the interval $[0,1]$ whose decimal containing only $0$ and $1$ digits countable?

Question

I have tried to enumerate this into a list but it does not seem possible.

e.g.: you could have [0,1000 ; 0,1100 ; 0,11100... ; ...] but then it would be impossible to have [0,0100... ; 0,01100... ; 0,0011100...] etc.

There doesn't seem to be an enumeration that contains all possible elements of the set, therefore it isn't countable.

Is this correct?

Thank you.

Answer 1

You can define a bijection from these numbers to the set of binary reals between $0$ and $1$ in an extremely obvious manner.

Answer 2

If $A$ is a subset of $\Bbb N$ define the number $f(A)$ in your set setting the $n$-th digit in $f(A)$ to $1$ iff $n \in A$. Or in a short formula $$f(A)_n = \chi_A(n)$$

where $\chi_A$ is the characteristic function of $A$. Then $A \to f(A)$ is a bijection between your set and the powerset of $\Bbb N$, which has the same size as $\Bbb R$ and which is uncountable by Cantor's theorem ($|X| < |\mathscr{P}(X)|$ for all $X$).