Assume that the universe is infinite.

Because of quantum effects, for any given volume of space there is a finite (albeit very large) number of ways that matter and energy can be arranged within it.

Therefore, using the pigeonhole principle we can prove that for any particular volume of space there must be SOME configurations that repeat.

(For example, it’s not too surprising to think that if we look at spheres of space one centimeter in diameter, there are probably a large number of them that contain only one hydrogen atom in their exact center. That’s probably a pretty common configuration.)

My question is: In an infinite universe must ALL possible configurations of matter occur? We know that some must repeat, but is it possible that some configurations don’t exist anywhere, even in an infinite universe?

Is there a version of Earth where everyone looks like Spongebob Squarepants and flies to the moon on weekends in boats pulled by zombie swans to eat moon cheese?

What I just described is a VERY UNLIKELY configuraton of matter. It couldn’t exist as the result of natural processes. It could only occur through countless random quantum effects all accidentally combining to create a very, very unstable combination of matter and energy.

How about an Earth were everyone flickers back and forth between a male and female version of themselves 10 times a second?

Are there, in fact, some things that are so unlikely that they can’t exist even in an infinite universe?