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Determine if for given some $L$, $S_L=\{\ L(M) | <M>\in L \}$ then for any $L$, if $S_L=RE$ then $L\in R$. Correct or Incorrect and explain why.

I think the claim is incorrect, and I'm trying to explain it with cardinals. I know that for each language we have $\aleph0$ turing machines and $|RE|$ is also $\aleph0$ but I know that is countable, actually, I don't know how to make the connection if $S_L=RE$ then is somehow is not counable.

Edit: I am looking to solve it with cardinals.

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The statement is incorrect*. Consider that

$$ RE = \{L(M) \mid M \text{ is a turing machine} \} $$

Thus

$$\{ \langle M \rangle \mid M \text{ is a turing machine}\} \subseteq L \implies S_L =\{\ L(M) \mid \langle M \rangle \in L \} = RE$$

Therefore if I set

$$L = \{ \langle M \rangle \mid M \text{ is a turing machine}\} \cup \{ \#\langle M \rangle \mid M \text{ is a TM that always halts}\}$$

where # is some character that does not occur in turing machine encodings, we have $S_L = RE$ but $L \not\in R$ as the halting problem could be reduced to it.

* when assuming that $M$ in $S_L$ ranges over all turing machines. Otherwise the statement is ambiguous.

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  • $\begingroup$ This is correct except for the incorrect statement that $S_L=RE\iff \{\langle M\rangle \mid M\text{ is a turing machine}\}\subseteq L$. Only one direction is correct, which is $\{\langle M\rangle \mid M\text{ is a turing machine}\}\subseteq L \implies S_L=RE$, which is the direction you actually use $\endgroup$
    – nir shahar
    Jul 28 at 11:30
  • $\begingroup$ @nirshahar Thanks for the hint. But can you actually come up with any $L$ for which the other direction actually doesn't hold? $\endgroup$
    – idmean
    Jul 28 at 11:36
  • $\begingroup$ Yes, for example $L_{M_0}:=\{\langle M\rangle \mid M \text{ is a TM}\} \setminus \{\langle M_0\rangle \}$ for any TM $M_0$ of your choice. Note that $\exists M: \langle M\rangle \in L \land L(M)=L(M_0)$ $\endgroup$
    – nir shahar
    Jul 28 at 11:39
  • $\begingroup$ @nirshahar Oh, of course, yes. Thanks, I wasn't thinking straight. $\endgroup$
    – idmean
    Jul 28 at 11:43
  • $\begingroup$ Thanks for this solution, But actually, I am trying to prove it with cardinal numbers reason, as I started to prove it to show that is not countable. $\endgroup$
    – John D
    Jul 28 at 11:57

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