Is an infinite set of RE languages create a language that is also RE?

I think it's true, and my first intuition is to try induction to prove this statement.

Am I on the right way?



1 Answer 1


The statement is not true; any language $L$ can be written as the (countably infinite) union of singleton languages which each just contain one word of $L$.

Also, even if the statement was true, one could not prove it using (regular) induction. This is because the finite union of RE languages is indeed RE. My go-to example here is that one can show using induction that every natural number is finite, but there exists no infinite natural number.

  • $\begingroup$ I don't see why it is a counter-example. Let me write it formally: If $L_1,L_2,...$ is an infinite set of languages such that each one of them is in RE, then $\bigcup_{j=1,2,...}L_j$ is in RE. $\endgroup$
    – Geo
    Nov 15, 2022 at 12:51
  • 1
    $\begingroup$ @Geo His answer is a counter-example. Consider $L$ a non-RE language. Since any language is countable, you can count its words, $u_1, u_2, …$ Now $L = \bigcup\limits_{j\geqslant 0}\{u_j\}$, and each $\{u_j\}$ is RE (because it is finite). $\endgroup$
    – Nathaniel
    Nov 15, 2022 at 12:57
  • $\begingroup$ @Nathaniel i do not understand, L is RE, why assuming its not? $\endgroup$
    – Geo
    Nov 15, 2022 at 13:59
  • $\begingroup$ We show that a non-RE language can be written as the countably infinite union of RE languages. That shows that in general, such infinite unions of RE languages are not necessarily RE themselves. $\endgroup$ Nov 15, 2022 at 14:30
  • $\begingroup$ I might be misunderstanding something, but say you have a non-RE language $L$, how are you going to take its singleton elements and then build $L$ back if there is no TM that can tell if $w$ is in $L$. How do we even know that $w$ is even in $L$? $\endgroup$
    – Russel
    Nov 15, 2022 at 15:12

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