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In the context of computability and turing machines

Let $L_1$ be an undecidable language and $L_2 = \{n \in \mathbb{N} | \exists x \in L_1 |x| \geq n\}$

How can I prove if $L_2$ is decidable or not?

I guess $L_2$ is not decidable in general and it depends on $L_1$ but still nothing proved. \

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Consider two cases:

  • $L_1$ is finite, and so there is a word of maximal length.
  • $L_1$ is infinite, and so there are words of arbitrarily large length.

Try to see if you can decide $L_2$ in these two cases.

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