(noob rant warning) What bugs me about the decidability thing, is Turing proved 2-undecidability in the action space {halt, loop}, but I still feel like we miss an opportunity to try 3-decidability {halt, loop, paradox} and throw out the functions upon which his proof hinged, namely, the ones that invoke the is_halting function and do the opposite.
Also, the whole, "we make this other program that does the opposite" argument implies the test of `is_halting` is actually a test of some other function "do_opposite" that wraps "is_halting" and does the opposite. That's not exactly fair, that's a test of the opposite function, not the "is_halting" function. Furthermore, the inner "do_opposite" evaluated by is_halting, is a different invocation of the do_opposite source. (I.E. Fregeian Sense and Reference, a different referent for the same sense).
Just because somebody outside the is_halting function can do something counterproductive, doesn't necessarily mean the specific invocation of do_opposite within the closure of is_halting is impossible to classify. Furthermore, is_halting could theoretically refuse to play ball and crash the program at runtime, or induce a compile error before the game can even begin, if we try to create a paradox. Has anyone actually even witnessed a real paradox in real life? Maybe the universe is deeply anti-paradox and the whole argument is a bunch of humbug.
Can someone please tell me how I'm fooling myself? Cuz I'm neck deep in coding this thing and building all the helper functions and it would save a bunch of time to know why it's wrong. Every proof seems to boil down to "muh contradiction" which feels like, ok, so what? If the caller wants to do_opposite, that's their problem, not on is_opposite. A function which can decide halting for all the non_opposite functions (which is, every one that matters in practice, right?)
Plus, who knows what an anti-paradox machine can accomplish? Maybe there's one weird trick quantum scientists or thermodynamic reversible computer engineers hate, and you can solve hard problems by finding the ones which both halt and don't.
Just feels weird how we all accept this quasi-religious belief that we shouldn't even try to decide if programs halt simply because you can't categorize every single program into 2 buckets. What about 3 buckets i.e. 3-decidability of the ternary halting problem?
sorry for long comment, fuck it, ill post code, it sucks and doesn't run, just tinkering, but https://github.com/bionicles/halts published for your pleasure. No promise it will ever actually work.