Disabeling interrupts to solve critical section issue on multiprocessor system Assume I have a multicore system and a critical section.

I understand how disableling interrupts on a single core systems solves the critical section problem.

So assume I could disable interrupts on all cores atomically (Why is that not possible?). 
Why would that solve the problem?

As I understand, there would still be n processes running at the same time that may interfere with each other. They just wont be stopped until interrupts are enabled again.

Can somebody explain why that would (according to my professor) work?
We are generally concerned about controlling concurrent access to kernel data structures.  User-level processes don't have access to kernel data structures, so they can't do any harm from user space.  The only risks are (1) if a user process does a syscall or causes a context switch to kernel mode, (2) if an interrupt arrives.  (1) could be guarded by explicitly checking for a lock, or by other methods.  (2) could be guarded by disabling interrupts, if you had a way to disable them on all cores.