Well, but thinking of mathematics, why not develop the mathematics of a limitless space?
Put the philosophical thoughts of an infinite space to an exact, sophisticated, modern, high level mathematical model that you love so much. A space where there is no edge and all is there, no outside, no time, just the space where all is.
Then from the astronomy, from the physics, put the galaxies to move and develop, in their space- time, there in the everlasting universe of this space. It is possible when you estimate a radius for it.
The movements of the galaxies have no effect to this space. They can move as they move, and for their movements we find from astronomy the empirical explanations that we don't yet exactly know.
So you see the axioms have meaning for the interpretations of the evidence. The mathematics is not enough, and it, in high level, is only necessarily needed for practicing astronomers.