It looks like you want a confirmation of your opinion that classical paradoxes have something to do with the Halting Problem. This is the case on some vague level as most well-known paradoxes rely on a form of self-reference, and so do the usual proofs of the Halting Problem. But to be more precise, we would have to understand what it means "to use a paradox in solution of the Halting Problem".

Have you looked at the article on Self-reference at the Stanford Encyclopedia of Phylosophy? In particular, section 2.4 seems to address your question quite well.

It looks like you want a confirmation of your opinion that classical paradoxes have something to do with the Halting Problem. This is the case on some vague level as most well-known paradoxes rely on a form of self-reference, and so do the usual proofs of the Halting Problem. But to be more precise, we would have to understand what it means "to use a paradox in solution of the Halting Problem".

Have you looked at the article on Self-reference at the Stanford Encyclopedia of Phylosophy? In particular, section 2.4 seems to address your question quite well. The article also describes the mathematical ideas that can be used to treat the paradoxes. I recommend it.