Must the universe recur, be infinite, or be infinitely divisible?

Question

When I was an undergraduate, I had this thought: Suppose that everything is made of atoms (I mean, pieces which cannot be separated further) and the universe has a finite amount of space.

Let $N$ be the total number of atoms in the universe, $V$ the total space of the universe and $v$ the volume of every atom. For every atom there are at most $\frac{V}{v}$ possible placements inside the universe, so all the possible ways to place all atoms inside the universe is at most $\bigl(\frac{V}{v}\bigr)^N$.

This number is clearly $<\infty$ which means that an infinite number of times everything in the universe is placed exactly in the same place.
(Which means that if we wait for many, many years, we could probably see an infinite number of times Bolt winning at the Olympic games, setting a "new" world record at $9.58$ seconds)

My question is:

Is it true that (based on the arguments above) one of the following propositions must hold true?

1. Not everything is made of atoms.

2. The universe's space is infinite.

3. An infinite number of times, everything in the universe is placed exactly in the same place.

   • Note : I consider quantum matter (non - continious situation - positions)

Answer 1

Not everything is made of atoms.

This is true, although I am not sure what you mean by "everything". Photons are not made of atoms, but as far as we know, all "normal" matter is based around atoms, although you can have loose electrons, in a plasma say. During chemical reactions, or in particle accelerator collisions, in neutron stars and in "ordinary" stars, not everything is made of atoms.

The universe's space is infinite.

That we don't know......We can only see what could be a small part of a much bigger universe.

An infinite number of times, everything in the universe is placed exactly in the same place.

Nobody can tell you if that is true or not, with 100 percent certainty, but I have never heard of a theory that predicts this.

I am not being facetious with you, but the soft question tag is gone because you are asking specific physics based questions, (and they are hard, not soft questions to answer, such as "is the universe infinite"). Lots of people are trying to work that out.

Answer 2

Maybe more of a comment, but as Myridium comments, the answer to question 1 is yes. For instance, you can have many photons in exactly the same place. But even particles that are not able to overlap have infinitely many possible positions in the Universe. Just like if you move a football, you're not restricted to move it an integer times its diameter.