No, that's a misunderstanding of what hypercomputation is. If you can write down or talk yourself through a proof of why a program will halt, you have solved a

No, that's a misunderstanding of what hypercomputation is. If you can write down or talk yourself through a proof of why a program will halt, you have solved a computable problem. By definition. When you say "we can decide a whole lot of programs halt", you're talking about plain Jane computation. If you have knowledge that any program that I give you will or will not halt, then you're a hypercomputing oracle for the halting problem, and you can do magic, basically. There's a world of difference between the two situations. And I'm fairly sure (but not positive) that there's no way for you to prove in finite space and time that you are an oracle for the halting problem. Edit: for more on that last bit, see https://pdfs.semanticscholar.org/19f9/0c34cdda43efdcf0831b2f9a037b57ddac21.pdf https://pdfs.semanticscholar.org/19f9/0c34cdda43efdcf0831b2f...