Following on from yesterday's post...
Suppose the hypothesis that the universe is infinite is true, and that any event with a finite probability will happen an infinite number of times. Now, given a set of observations about the universe (such as every observation you have ever made), what is the probability that those observations were made in the first stage of the universe (i.e. the one we have generally assumed that we were in) and what is the probability that those observations were made in one of the macroscopic quantum fluctuations in the time after the heat death of the universe?
The set of observations is finite, so the amount of matter required for them to be made is finite. So quantum fluctuations will cause these observations an infinite number of times for each sufficiently large volume of space. In the first stage, those observations can happen at most once for the same volume of space. So given the set of observations, it's practically certain that they are post-heat-death.
But if the universe is post-heat-death, why can we see so much? A much smaller universe would have allowed for almost as many observations (including the existance of the entire human race) but be much more likely. Perhaps the human race (and hence somebody to observe the universe) cannot exist without all those very distant galaxies.