What is your take on Gödel's rotating universe, and does it support the cosmological principle?

Gödel's rotating universe is not the one you see in geocentrism. It was saying that the universe can move, as opposed to Newton's absolute universe that cannot move, which is the natural course of general relativity: you get away from the Newtonian system that says only the earth can move while the universe stays put. But Gödel's universe rotates very slowly; it might take a million years to make one complete rotation, whereas in the geocentric system the universe rotates once every twenty-three hours, fifty-six minutes, and about four seconds every day. Gödel may have known about a geocentric universe that Ernst Mach talked about, but he was not talking about the same thing as geocentrism, and other men after him who talk about a rotating universe mean something moving only a few millimeters a year. Neither of those has anything to do with the cosmological principle, which basically says there is no distinguishing characteristic from one end of the universe to the other, that everything is homogeneous, like a glass of homogenized milk with no globules to mark up from down or left from right.