I know it exists but not sure how to prove it. I assume it needs something to make sure the elements are Halting problems, but they are undecidable so I am not sure where to start.
2026-03-26 13:45:20.1774532720
Prove there exists a Turing machine $M$ that decides an infinite subset of $HALT_{TM}$.
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Well, if I understood your question correctly, finding an infinite decidable subset of $HP$ isn't that hard, since it contains many simple machines. For example, let's denote $M_0$ a TM that instantly rejects, then the language $$L=\left \{ \left \langle M_0 \right \rangle ,x |\ x\in \Sigma^{\star}\right \} \subseteq HP$$ is decidable by checking $M$ is encoding $M_0$ and accepting/rejecting by that.