It seems to me "Is there a proton in this volume of space" does not obviously capture all possible options for matter within a universe. Without getting to more esoteric descriptions of what matter is composed of, it seems that other factors, such as velocity/energy of said particles, may turn it into a "bigger" infinity than the infinity that there are universes. But of course I know nothing about these things, and this is a real physicist so I am obviously wrong. Can some here explain it?
OK, first to cover what I can understand of his basic principles. It appears that his "level 1" interpretation of a multiverse treats the
observable universe (a.k.a. the Hubble volume around us) as strictly equivalent to a universe. Due to the limitations of light speed, everything outside said observable universe cannot effect us, so may be regarded as a completely separate and isolated place - i.e. a different universe. I can see a basic problem with this idea, which I will get to a bit later, but let's call this his first assumption.
His second assumption is that this larger
non-observable universe extends forever, in all directions, with no larger scale patterns in matter distribution. This seems iffy with most descriptions of the Big Bang theory that I've seen, which assign large but finite numbers to parameters such as the rate of expansion of space. But I'm no expert on those.
Let's take both of these assumptions as true. The multiverse can now be thought of as a grid of spheres (yes, I know spheres don't tesselate, this is touching on the problem with his first assumption that I'll get onto later), each sphere being a universe comparable to our observable universe. Strictly the universes are only separated along the three conventional space dimensions, and anyone with an FTL drive can therefore travel between them. There would be no fundamental difference in a trip to Mars and a trip to "parallel Mars" other than the latter being further away. He is arguing that every possible universe would be found in these spheres, because there are only a finite (but very large) number of ways of arranging matter in a finite sphere.
It doesn't really matter how many parameters we take into account related to particles, velocity etc. It's still an absurdly big, but
finite, number. His second assumption however is that the greater universe is literally
infinite. Therefore you will be able to fit any finite number of these finite observable universe bubbles into it. Indeed, if we're getting into different kinds of infinity, you will be able to fit an
infinite number of each finite universe into the
infinite multiverse. The second infinite in that sentence being infinitely larger than the first infinite! Also, if the probability of matter arrangement A is more likely that matter arrangement B, while you will have an infinite number of copies of each, you will still have a larger number of copies of A.
"Infinity" tends to cause some issues when it shows up in any mathematical model. All of this relies on his second assumption that matter really does go on forever in all directions. Once you've stuck an "infinity" into your basic model, more infinities will necessarily follow.
Now to get back to his first assumption, and the basic problem with it. He assumes that each observable universe, i.e. Hubble volume, is effectively an isolated universe within a larger multiverse. This doesn't appear to work as a concept, because said "observable universes" necessarily overlap, and hence can influence each other. Here on Earth, our observable universe extends something like 47 billion light years in each direction. Anything more than 47 billion light years away cannot effect us, and we cannot effect it due to light speed. He therefore argues that it can be treated as a separate and isolated universe.
But consider some alien world 40 billion light years from us. They can in principle see and be affected by us, but they can also see, and be affected by things far beyond our observable universe. Anything that is more than 47 billion light years from earth, but less than 47 billion light years from them is still part of their observable universe. Are they in the same "universe" as us in his model, or not? Their "universe" contains part of ours, but also part of another from our point of view - the "universes" in his model cannot be isolated and independent. (The tessellating spheres issue above is also a giveaway that universes in this model overlap). This also screws around with the second assumption, which requires that what happens in one universe bubble does not influence the probability of what happens in the next. If you have a big chunk of matter on the edge of our observable universe, its gravity is very definitely interfering with things beyond our observable universe in this model.
I was actually reminded a lot of the ways we think about crystals in chemistry. You can extend a crystal lattice indefinitely in any direction, but that doesn't mean you'll encounter every possible arrangement of atoms. As the crystal grows, the position and bonding is dependent on the atoms already in place. You can end up with a totally different endlessly repeating lattice depending on the conditions and the arrangement of the initial "seed" crystal at the beginning. Grab an (idealized) diamond and no matter how big it gets, you'll only see one type of carbon bonding, despite the fact carbon atoms are capable of taking other arrangements. The probability of graphite is definitely non-zero, but it's still totally absent from the multiverse-analog lattice.
Since the above multiverse model requires overlap and influence from one universe to the next, it would be valid to end up with a similar restriction on the possible arrangements of matter in each universe. I don't think his assertion that you'll see every possible arrangement of matter within each bubble works given the assumptions he's made. Nor am I completely convinced his two assumptions are even compatible with each other.