Could the following statement be correct?
"Every infinite recursive language has as a subset an infinite regular language."
2026-03-30 07:37:43.1774856263
Infinite recursive languages and infinite regular languages.
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No. Call a language lacunary if it is infinite and the gaps between the distinct lengths of words goes to infinity; for example, the language of words whose length is square.
By the pumping lemma, no lacunary language is regular (or context-free), and every infinite subset of a lacunary language is lacunary.