Exactly!
I'll hunt out my standard reference on Godel's incompleteness theorem, because Roger Penrose describes this very well and should allow me to express this better. This is the very heart of the matter. Basically, there is a logical contradiction to the concept of a set that contains all other sets, itself included, a contradiction that leads directly to Godel's incompleteness theorem. I'll try to explain later.
This isn't quite right. It's nearly right, but not quite.
It's basically a variant on the Russell Paradox, I think:
The set of all sets that do not have themselves as a member cannot be complete. There cannot be a compete set of sets without contradiction.
Where I think Godel is useful is in the fact that although you cannot prove the 'Godel statement' of a particular set of axioms, you can
know that it is true! Godel statements take a self-referential form: to simplify, X says that 'it is not possible to prove X', and Godel's theorem proves that every system has to contain a statement of this nature. But X is a well formed expression, whatever it is, so it has to be true if the system is consistent. So you do indeed find that you cannot prove X - which itself shows that X is true. This is true of all systems of logical axioms. It is true of mathematics itself, which must always contain a Godel statement.
The leap you now have to make is something of a leap of faith, and it is this statement:
Mathematics, a complete system of all possible logical axioms, is formally equivalent to the universe, the set of all things including itself.
Santino has already mentioned a reason for this to be true: every event has a cause. If this is true, then it must also be true that mathematics provides a complete description of that system of cause and effect.
Now, this is a hunch on my part - an intuition perhaps - but I strongly suspect that many of the apparent artefacts of the universe - the relative strengths of the four fundamental forces, for instance - which need to be
exactly what they are in order for any universe to exist at all, are necessarily as they are in just the same way that π is necessarily exactly what it is and not another number.
This does away with the necessity for a multiverse in order for it to be true that 'everything that can happen happens'. That is already true - that's what the universe already is.