[Solved]: Can a recursive language be uncountable?

Problem Detail: 

Does there exist a recursive language $L$ whose cardinality is uncountable?

I would like to have an explanation whether Turing Machine can encode uncountable languages and whether we can use this to reject the initial question.

Asked By : revisingcomplexity

Answered By : Yuval Filmus

Languages are collections of words. Words are finite strings.

As Shaull stated in his comment, every language over a finite alphabet is countable. (In fact, every language over a countable alphabet is also countable.)

Languages of infinite words, sometimes called $\omega$-languages, are considered in computer science. For example, they are the subject of $\omega$-automata theory. But the Turing machine formalism is about the usual notion of language.

Best Answer from StackOverflow

Question Source : http://cs.stackexchange.com/questions/41963