However, the existence of a halting state does not imply that all execution paths eventually reach it. Certainly I have never seen a definition (of an algorith

However, the existence of a halting state does not imply that all execution paths eventually reach it. Certainly I have never seen a definition (of an algorithm, rather than, say, a total language) that builds in that requirement; otherwise, what would be the objects considered in the halting problem? By this definition, they couldn't just be algorithms, or else "RETURN TRUE" would be a correct decision procedure.