I gave that a quick glance, but will read in detail later.
From what I gathered, he is referring to the “finiteness” of the envelope of matter that we call the “universe”.
If this envelope is growing, and filling up a void, it still means that the classical definition of the universe, i.e. the envelope of matter plus the space that it is advancing into, is infinite.
Thought about your “beyond the material universe” question.
We were discussing the probabilities of earthlike planets and techno societies. The “infiniteness” of the “beyond” is irrelevant if it contains no matter, as it adds nothing to the probability of life.