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Do any decision problems exist outside NP and NP-Hard?

Why do there exist some problems that are NP-hard but not in NP?

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    Simply put, there are problems that are harder than all problems in NP, and if they could be solved quickly we could use them to solve any problem in NP. One simple problem with this property is the halting problem. It's trivially not in NP because is not computable. To see that it can solve any problem in NP, consider the following. Suppose we have an algorithm H that solves the halting problem. Let A be any algorithm solving some problem in NP (A does not need to be efficient, it just needs to work), and let x be some input to A. Let C be the following program define C() { if (A(x)) return else loop forever } By passing C to our halting algorithm H, we learn whether C halts or loops forever. Equivalently we learn whether A(x) returns true or not. Since A and x were arbitrary, we can solve any problem in NP with H.

Tim Wilson at Quora Visit the source

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There exist complexity classes that are above NP. A NEXPTIME-hard problem is also NP-hard, but is not NP.   A problem is NP-hard if it is at least as hard as the hardest problems in NP. But it can be very harder.

Luc Gillibert

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