countability

Question

if a language is not recursively enumerable, then is it uncountable language? I believe every language over ∑ is subset of ∑* which is a countable set and as subset of countable set is countable therefore every language itself is countable whether it is recursively enumerable or not.

Comments

Lang is uncountable .Does this even mean anything!!

Answer 1

It’s a general statement, so I disagree with your reasoning, as the alphabet may not be finite. In that case, languages that are uncountable exist. 

The general case explanation I could find best explained here: https://www.youtube.com/watch?v=x0LMIZTIohg

In conclusion, both countably and uncountably infinite languages exist which don’t fall under RE.
This question is analogous to asking are all countably infinite languages RE? Whose answer is NO for the exact reason that non-RE languages can also be countably infinite.