Dear Benjamin
Thanks for your comments. For sure mathematics is our upper bound. We can go just as far as math allow us, and there will be times we'll need to first extend math; but that's not new in physics.
Well, you talked about the incompleteness theorem. Some days ago I found something very interesting about it, and I was willing to discuss about it. A theory is complete in the sense of this theorem when
"if it is consistent, and none of its proper extensions is consistent." [Wikipedia on Complete Theory]
A closed theory must be consistent, but ideally, a closed theory should allow extensions! Every time you create a new theory that extends another one or uses the framework of another theory you are creating a proper extension! Therefore, a closed theory that is complete in this sense is useless! You cannot extend it, you cannot apply it! Every model of a theory is a proper extension because a model adds new assumptions and so allows new results! If a theory is complete, then you cannot add a new assumption to it! Because either the assumption or its negation is already part of the theory!!! Complete theories are useless for physics!!!
About item 1
You have shown a great point: if a physical theory is based on mathematical elements that are not clear and "closed" enough then all the theory runs in trouble and cannot be really closed. Set theory is really problematic when you analyze it in depth; however, there are parts of physics that only requires results of set theory which are really established. This parts I would say they are safe...
About item 2
Well, for me when interpretation mathematically works in on situation while another on doesn't work, then they are not just different interpretations. In this case the formalism or something else is different, and we need other criterions to choose which one is the best.
About item 3
Well, it is not impossible for a mathematical system to be contradictory, but it is very hard, or at least much harder than if the system as not described mathematically. But, when you system is very simple and clear, it is very different for it to have a contradiction, and if you have applied it successfully, than is even harder. But, even so you may never prove mathematically that it is consistent. But math evolved too much after Russell. Many of his viewpoints are not followed anymore, and, his approach is not good enough for physics. In fundamental physics everything is becoming more simple, unified and elegant, but Russell did the opposite with math! he took thousand pages to define the simplest mathematical notions! If physics started using his theory, one would have to understand his thousands pages before understanding the simplest closed theory!
For sure there is a gap to be filled between our current theories and the best we can do. And I've been working at this program. In my arXiv paper I've tried to provide a closed formulation of part of a large part of quantum theory. It is a new interpretation and formulation, but also a closed theory. That's what I think. I hope others would point me what are the missing points for it being a completely closed theory, it is not perfect yet. However, the purpose of finding a closed theory is exactly what gives value to my work, if not, it is just another interpretation of QM. Well, I saw that you are interested in modern algebra, so you will like my new formulation: numbers, bras, kets and operators are all included in the same mathematical structure. I'll take the time to read and rate you essay, please rate mine too!
Best Regards
Frederico
P.S.: what do you think about this view on godel's theorem?