Eh, I had a lot of programming to do, and a final to study for, so I had to have some way to procrastinate =).

so I am not alone in my misunderstanding

Yeah, there are definitely plenty of misconceptions about Turing machines. One is that it's proved that a TM is equivalent to any modern computer. It's not, because you'd have to reprove it for every little architecture change, but it's generally considered a very safe assumption to make. I'm not sure whether this has been proven for quantum computers also, but, from what I know about them (admittedly not a ton on the technical/mathematical side), I imagine it's the same.

Since language, mathmatics, and logic continue to evolve, I believe they must not be fundamental properties of the universe - rather they are models we use to model the universe as we experience it.

I think we're still not sure in a lot of cases what is a model in the approximate-still-need-to-refine-the-accuracy sense, and what's a model in the if-this-is-true-then-it's-actually-analogous sense. Limits of a TM would probably fall into the second category. The idea language, encoding, and even information is somewhat hard to really define, but for the ways that those are usually thought about, the stuff we know about how we can shunt symbols around, so to speak, is very much a fundamental property of that information. Whether information (and by further extension, possibly intelligence and sentience, but that's a whole 'nother topic, and one that's even less well defined) is a fundamental emergent property of matter is very much another, very open question. Godel, Escher, Bach, is a good starting point (or, if you've already learned about the individual topics it brings together, an interesting angle on how it all ties together) for learning about this stuff. But if you're really interested, you should probably check out more rigorous and thorough treatments of the subject.

So when we use our models the examine our models, sometimes we get feedback and this causes a mess and contridictions or infinite loops.

Whether this is the cause for some of the things we see, I don't know. We can make a true statement that's subtly different though. Certain ideas or concepts (including a machine that decides the halting problem) by their hypothetical existence prove their inexistence. So if they don't exist, they do exist, but even if they do exist, we can show that the existence causes a contradiction. This is the basis for every proof by contradiction, whether in calculus, number theory, or computer science. It's why any idea that takes advantage of a computer that can solve the halting problem is not just impractical, but meaningless. There is no way, no matter how much you try, to create a machine that does that, and so the concept of it's hypothetical existence exists, but the concept itself doesn't actually exist. I'm pretty sure about those last few sentences, someone correct me if I'm wrong. That's what I've gathered so far.

Basically, there is no model that can model everything including itself.

This is true, but I think, for a different reason. To model something completely accurately, you need to hold as much information as that system. But how can a system hold all of the information of the surrounding system (even if that's less information than in the simulator) *plus* it's own information over again (you're asking it to hold all of it's information twice over, and you can't have something hold more information than it holds)? This is sort of a fuzzy way of saying it, but I haven't come up with a better way yet. I'm sure someone's formally defined this sort of thing already though, so there are probably better explanations.

A corelary is that there is no function that can predict infinite loops in all functions because the analysis of itself can cause such a loop.

Hmm... I think that's slightly off. It's not so much that the self-analysis causes an infinite loop, but that it allows a situation with a paradox to arise. Most proofs I've seen of the halting problem don't rely so much on infinite loops but more a "look, if this is possible, than it causes a paradox, the only thing we assumed was that it was possible, therefore it must be impossible." The thing is, you're not really supposed to think about the halting problem in terms of infinite loops (since there are things other than infinite loops that could cause a lack of halting). It's more like, having a decidable algorithm for that problem causes a paradox, therefore that algorithm can't exist.

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