We know there are difficult computational problems. P vs NP is more narrow than that; it’s sometimes phrased as “are there problems that are not difficult to verify but difficult to solve?”, where “difficult” means that it cannot be done in asymptotically polynomial time.
The point is that you can’t use the result that there exists a hard function, since all you know is that the function is hard, not whether it’s in NP, which is a basic problem for your scheme.
Your counting argument for Turing Machines also likely have this problem, and even if not, I see no reason why I couldn’t relativize the results, which is a no-go for P vs NP proof attempts.
We know there are difficult computational problems. P vs NP is more narrow than that; it’s sometimes phrased as “are there problems that are not difficult to verify but difficult to solve?”, where “difficult” means that it cannot be done in asymptotically polynomial time.
Yes, I am familiar with the definition of PvsNP. That’s not what I am asking.
The point is that you can’t use the result that there exists a hard function, since all you know is that the function is hard, not whether it’s in NP, which is a basic problem for your scheme.
Your counting argument for Turing Machines also likely have this problem, and even if not, I see no reason why I couldn’t relativize the results, which is a no-go for P vs NP proof attempts.