The idea that the universe is "isotropic and homogeneous" does not require it to be infinite. For example, the universe could be shaped like a sphere's surface, which is closed and finite. The size and shape of the universe, and its ultimate fate, are answered by the question "What is the sum of the angles of a very large triangle?" (this turns out to be equivalent to measuring Omega, the density parameter of the universe).

If Omega > 1, then the universe is closed, shaped like a sphere, finite, and will collapse in a Big Crunch. If Omega = 1, then the universe is flat (but could still be finite, eg doughnut-shaped, or infinite, like the Cartesian plane), and end in heat death. If Omega < 1, then the universe is open, shaped like an infinite saddle, and will end in a Big Rip. To the best of our knowledge, Omega appears to be 1, plus or minus Omega might actually be bigger or smaller than 1.

Shape of the Universe

Fate of the Universe