Re: As far as I can tell, you can prove
The surface of a ball is homogeneous, isotropic, and finite. To my knowledge, most models of the universe are that we are in an S3 manifold (three-dimensional surface of a four-dimensional ball). There are some problems with the background radiation, however, that have some suggesting that we might actually be in some toroidal structure.
But neither homogeneous nor isotropic principles are proven. Certainly, they appear to be true, but imagine an infinite flat universe where the energy was piled up in a heap somewhere. Things spread out, but for a significant region near the center of the heap, you might not observe anything indicating that.
Furthermore, if our experiences with big explosions has taught us anything, it is that they are NOT uniform. If the big bang were completely uniform, that would represent a rather unique quality. There might be a physical reason for that to be so, but I cannot think of any reason that it must be.
When you start with spherical cows, it can take a long time for them to grow legs. We might not have the toolset yet to dig into such possibilities.