That's just it, in that case we are part of the process, because we affect the operation / output. Any halting detector looking the whole thing, which must include us, would show that we always negate what comes out of what we're doing, and would say so.
Just like you did. You've accounted for all output, and demonstrated that it always negates us.
By that, if a halting detector cannot exist, then how do we exist? We can tell if a hypothetically-impossible device (Q calling P and reversing the output) will loop forever or not.
By changing the detector,
you're surely a defector
from the point of the proof.
If what comes out is made moot,
give the whole thing the boot:
look at it from the roof.
edit: previous comment requested my argument in rhyme :)
edit2: second verse!
If the halt-checker is wrapped, isn't it not?
Since we've just solved it, why can't a 'bot?
Just what's going on here?
So if it's internal...
... this damned, infernal...
Lets just go get a beer.
Just like you did. You've accounted for all output, and demonstrated that it always negates us.
By that, if a halting detector cannot exist, then how do we exist? We can tell if a hypothetically-impossible device (Q calling P and reversing the output) will loop forever or not.