Here's something to think about, though... maybe it doesn't make a lot of sense, but perhaps you can unravel it for me.
First off, I would like to distinguish between the terms "mathematics" and "metamathematics".
By "mathematics" I will mean the (finite, or potentially infinite) set of truths derivable from logic and associated philosophies. "Mathematics" is what we write down, talk about, and use to solve problems.
By "metamathematics", I will mean the (actually infinite - see below) set of truths or principles upon which reality is based. "Metamathematics" includes those hidden processes and relationships which govern the nature of things, according to your (and others') belief in the MUH.
Therefore the problem boils down to whether or not there is really a "metamathematics" at all - you are convinced there is, and I am led to believe there is not.
In the ZFC set theory axioms, a set is not allowed to belong to itself. If the set of metamathematical truths were finite, then it would be possible (in theory) to list them all. However, such a list would contain a statement of all metamathematical truths, which would be semantically equivalent to metamathematics. So metamathematics would have to "contain itself", as it were, if there were only finitely many truths in it. Therefore, we can conclude that there are infinitely many metamathematical truths, given that the ZFC axioms, and the resolution to Russel's Paradox, are correct.
Another interesting note is that, if there is a metamathematics, I don't think our mathematics would have anything to do with it. Here's my reasoning: I'm not sure how any system can simultaneously encode its information along with a means for expressing that information, if the means for expressing it must also be included in the information. Let me try to come up with a concrete example.
Say you want to write a very simple, single program that saves a copy of itself to disk. Assume there is no OS to do it for you; the only thing you can do is write immediate data to the disk.
It's simple enough to copy the rest of the program, with the saving procedure excluded, and write that to disk. If your total program took 100 lines and the saving procedure took 20 lines, you would write 80 lines to disk and be done.
However, you want to write a program that's also capable of copying itself; one which can not only provide a representation of itself, but which can do so in such a way that the representation can also represent itself, and so on ad infinitum.
I think that with a little thought you'll agree that it's not possible to do this unless you have a third party come in and act on the whole program without the program's intervention.
So what does this all have to do with us? Well, if metamathematics really existed, it would have to encode in itself a way in which to represent it, that is, mathematics. Remember, mathematics (if it has anything to do with metamathematics) *is* the way to encode metamathematics. Anyway, I think you'll also agree it's reasonable to say that we have ways of encoding mathematics (notations, conventions, etc.) Ergo, metamathematics must be capable of representing not only a non-empty subset of itself, but also the rules for encoding this non-empty subset of rules.
And herein lies the rub: even if you assume metamathematics, there must be a higher level of abstraction - a manager - if our mathematics is to have anything to do with metamathematics. And if metamathematics is completely unrelated to mathematics, then why think of it like mathematics at all?
