Comment by mistercow

Yeah, apparently I didn’t communicate that well enough. But I think this is a subtle and common point of confusion.

There are machines which are very, very hard to determine whether they halt or not, and so people end up thinking that there must be specific machines for which no Turing machine can decide halting correctly. But that’s just not true. Every “attempted halting decider” has its own infinite set of failure cases, which are specific to that machine, and not fundamental to the input machines.

This feels really strange, and trying to turn that other intuitive sense into something meaningful and reasonably formal is an interesting exercise, but it’s tricky.