Are there intractable problems?

Question

Obviously, there is no problem in NP which can be shown to be intractable (by which I mean: not in P).

Is there a problem (outside of NP) which can be shown to be intractable (not lying in P)?

Answer 1

What you want is the Time Hierarchy Theorem, which broadly speaking states that for any "reasonable" function $f$ there are problems that take about $f(n)$ time to decide.

Concretely, each of the known EXPTIME-complete problems would be an example of something that has been proved not to lie in P.

Here is a related question on compsci.

Answer 2

NP-hard / NP. A problem is in NP-hard when every problem L in NP can be reduced in polynomial time to H. Under the assumption of $P \neq NP $ ,NP-hard is intractable. Therefore NP-hard / NP is intractable.

https://en.wikipedia.org/wiki/NP-hardness