All we know is that if a language is countable than it must be recognizable. However, a recognizable language may or may not be decidable.

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    $\begingroup$ $\Sigma^*$, the set of all words, is countable. Therefore, every language is countable. It is certainly not the case that every countable language is recognizable. $\endgroup$ – Shaull Nov 29 '16 at 20:14
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    $\begingroup$ Every language whose alphabet is finite is countable. Check also this answer: cs.stackexchange.com/a/12665/30527. $\endgroup$ – André Souza Lemos Nov 29 '16 at 21:32
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    $\begingroup$ @AndréSouzaLemos ​ : ​ ​ ​ Even languages with countably-infinite alphabets are countable. ​ ​ ​ ​ ​ ​ ​ ​ $\endgroup$ – user12859 Nov 29 '16 at 21:52
  • $\begingroup$ @Shaull Make an answer? $\endgroup$ – Yuval Filmus Dec 1 '16 at 16:15

Your premise is incorrect. The set of all words, $\Sigma^*$, is countable (as long as the alphabet is countable, and we usually refer to finite alphabets anyway).

Every language is a subset of $\Sigma^*$, and thus every language is countable. In particular, there are undecidable and unrecognizable languages.


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