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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.

dar_bur
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1 Answers1

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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.

Shaull
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