I am supposed to find a language $$L\subseteq \Sigma ^*, \Sigma \subseteq \mathbb{N}$$ that fullfills the pumping lemma and is not in RE and therefore not in coRE. I've never constructed a language with a given constraint, I've only shown that a certain language satisfies a given one. Not wanting to waste a bunch of time guessing some languages and prooving that they meet my criteria, I am looking for a hint on how to start. I was thinking maybe I could find a non-trivial index set that fulfills the pumping lemma, thus by Rice's Theorem: $$L \notin REC \implies L \notin RE => L \notin coRE$$

I am supposed to find a language $$L\subseteq \Sigma ^*, \Sigma \subseteq \mathbb{N}$$ that fullfills the pumping lemma and is not in RE and not in coRE. I've never constructed a language with a given constraint, I've only shown that a certain language satisfies a given one. Not wanting to waste a bunch of time guessing some languages and prooving that they meet my criteria, I am looking for a hint on how to start.