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Obviously it can have a countably infinite number of strings. (Take the language descibed by the regular expression 0* as an example.) But can a RL have uncountably many strings? I'm leaning toward no, but I can't really back it up.

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No, there are only countably many finite strings over any finite alphabet. Given an alphabet $\Sigma = \{s_0, \dots, s_d\}$, you can associate the string $w$ with the number $1w$ written in base-$d$, giving an injection from the natural numbers to $\Sigma^*$. (The reason we use the number $1w$ rather than $w$ is to avoid problems with, e.g., $2$, $02$, $002$, etc. all mapping to the same number.)

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  • $\begingroup$ Plus, countability survives taking subsets. $\endgroup$ – Raphael Apr 28 '16 at 11:31
  • $\begingroup$ I'm not sure if this is an exact answer. There are countably many finite strings, but infinite strings are not countable per Cantor's diagonalization argument. What makes them non-regular? In other words, a real number 0<= x <1 can be thought of as a word from the lanugage 0.[0-9]*, and that's uncountable. $\endgroup$ – MSalters Apr 28 '16 at 14:01
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    $\begingroup$ @MSalters Regular languages are, by definition, languages of finite strings. If you want to talk about $\omega$-regular languages of infinite strings, that's a whole 'nother question. Also a real number isn't uncountable: it's one thing. $\endgroup$ – David Richerby Apr 28 '16 at 14:54
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Yes,a regular language can have uncountable number of strings as a regular language can have unlimited strings by Kleene closure or sigma closure.

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  • $\begingroup$ This is wrong link. Uncountable is not the same as infinite. $\endgroup$ – jmite Apr 28 '16 at 17:07

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