Is an Unhackable Kernel Really Possible?

In an effort to prevent drones from being hacked, DARPA has been developing an "unhackable system," and seems to think they're almost there. The development team "proved" mathematically that their kernel was unhackable, and they hope to use it for more than just drones (power grids, cars, phones, pacemakers, etc.).

"Known as seL4, the kernel has a few highly secure properties: it can only do what it is designed to do; its code can't be changed without permission; and its memory and data transfers can't be read without permission."

Full story can be read on New Scientist.

But nothing is really unhackable, is it?