Determine if for given some $L$, $S_L={L(M) : <M>\in L}$ then for any $L$, if $S_L=RE$ then $L\in R$ is True or False and explain

Question

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.

Answer 1

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.}