Is this language countable : $L= \{ w : w \in (1 + 0)^{*} \}$ This is my take :

Epsilon ---> 1

0 -->  2

01 ---> 3

10 ---> 4

11 ---> 5

001 ---> 6

010 ---> 7

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So therefore we can count them.

But  based on this video : https://www.youtube.com/watch?v=oe-ZAJQz9Cc&amp;index=5&amp;list=PLsFENPUZBqiqbnD-WatYxUhRWLMNDoMun

They should not be countable, but i did not understand that video, the guy says all possible languages over $\{0,1\}^{*}$ are uncountable!
The language $$L= (0+1)^*$$ (the set of all strings over $0$ and $1$) is countable. Furthermore, any subset of $L$ is also countable. However, the set of all sublanguages of $L$
$$S = P(L) = \{M \mid M \subseteq L\}$$ (a set of sets in fact) is not countable. This video proves this using the diagonalization argument. 

$S$ and $L$ are two different sets. In fact, $L$ is a member of $S$.