Scientific American recently published an article by Max Tegmark, which provides calculations for a particular kind of infinite universe.
http://space.mit.edu/home/tegmark/PDF/multiverse_sciam.pdf

Although I think his use of the word “infinite” is a bit loose, he is careful to define exactly what he means, and his logic is scientifically valid. His argument is as follows:

Assume the following:

  1. Space is flat (not curved in upon itself)
  2. The universe has no “edge” (“infinite” in the sense that it continues without limit)
  3. The laws of physics are invariant (the laws that we know apply everywhere)
  4. The universe is filled with matter in about the same density everywhere (and he gives values)
  5. There are limits to how closely fundamental particles can be packed together (and he gives a value)
  6. Nowhere in the universe is hotter than a certain highest temperature (and he gives a value)

Then it follows that the number of fundamental particles that can fit in to a specific volume can only be arranged in a certain number of ways before necessarily repeating. For any given volume this means there should be an identical copy on average no more than a certain distance away, and that distance can be calculated by physics and probability.

He gives 3 calculations:

  1. For a volume similar to a human, the distance is 10^10^28 m. [This is the maximum distance to the nearest identical copy of you or me]
  2. For a 100 light year sphere, the distance is 10^10^92 m. [Ditto, includes all events for the next 100 years]
  3. For a Hubble volume, the distance is 10^10^118 m. [Ditto, for the size of our observable universe]

That last number is a one followed by 10^118 zeroes. The number of zeroes is a one followed by 118 zeroes, which is itself larger than the number of fundamental particles in the observable universe.

You will have to make your own decision as to whether you are happy with the 6 assumptions, and whether the result is satisfying. I do not find them so. He asks us to accept that observations made over a distance of 10^24 m (our observable universe) are applicable over distances of 10^10^118 m, a proposition which can never be tested. I find it strains credulity, and it is certainly beyond science.

He says: “In infinite space, even the most unlikely events must take place somewhere.”

This is false. There is nothing in his logic that goes to prove that every possible arrangement of fundamental particles must be found somewhere, and every reason to doubt it. As a simple example, this statement would demand that a region the size of our Hubble volume devoid of all matter and another absolutely full of particles must exist somewhere, but that would violate assumption (4).

Also he asks: “How could space not be infinite? ”

Although there are bits of science that are relevant, fundamentally we cannot answer this question with science. If it is not, we shall not know this or the reason why. indeed, the whole proposition lies outside science.