mathematical physicsmathematics
Gödel's incompleteness theorems basically sets the fact that there are limitations to certain areas of mathematics on how complete they can be.
Are there similar theorems in physics that draw the line as to how far one can get in physics as far as completeness?
No, physics is not rigorous in the sense of mathematics. There are standards of rigor for experiments, but that is a different kind of thing entirely. That is not to say that physicists just wave their hands in their arguments [only sometimes ;) ], but rather that it does not come even close to a formal axiomatized foundation like in mathematics.
Here's an excerpt from R.Feynman's lecture The Relation of Mathematics and Physics, available on youtube, which is also present in his book, Character of Physical Law (Ch. 2):
There are two kinds of ways of looking at mathematics, which for the purposes of this lecture, I will call the the Babylonian tradition and the Greek tradition. In Babylonian schools in mathematics, the student would learn something by doing a large number of examples until he caught on to the general rule. Also, a large amount of geometry was known... and some degree of argument was available to go from one thing to another. ... But Euclid discovered that there was a way in which all the theorems of geometry could be ordered from a set of axioms that were particularly simple... The Babylonian attitude... is that you have to know all the various theorems and many of the connections in between, but you never really realized that it could all come up from a bunch of axioms... [E]ven in mathematics, you can start in different places. ... The mathematical tradition of today is to start with some particular ones which are chosen by some kind of convention to be axioms and then to build up the structure from there. ... The method of starting from axioms is not efficient in obtaining the theorems. ... In physics we need the Babylonian methods, and not the Euclidean or Greek method.
The rest of the lecture is also interesting and I recommend it. He goes on (with an example of deriving conservation of angular momentum from Newton's law of gravitation and having it generalized):
We can deduce (often) from one part of physics, like the law of gravitation, a principle which turns out to be much more valid than the derivation. This doesn't happen in mathematics, that the theorems come out in places where they're not supposed to be.
This is an interesting questionning, rather than question, as I am not sure what is the question stated, because too many issues are questionned or questionnable
I think it is different, but very much related to a previous question regarding the physical provability of the Church-Turing thesis stating that any computing device that can be built will compute no more than what is computable by a Turing machine.
One issue with the Church-Turing thesis is that the concept of proof in an axiomatic theory is fundamentally the same as the concept of a computer program, i.e., a Turing Machine. This is not in the sense that a program can churn out theorems and proofs, as in Ron Maimon's presentation of Gödel's proof, but because a program can be "read" as a proof of its specification ("Given a value x such that P(x) holds, there is a result y such that Q(x,y) holds.") and conversely a proof may be read as a program that actually computes whatever is stated by the theorem. This presentation is of course a very simplified version of a result by Curry and Howard (1980) which is still researched.
Hence one major issue with the possible limitations of calculability, whether there are such physical limitations or not, and whether we can prove or not the existence of such limitations, is that mathematical proofs are directly concerned by the same limitations. One crucial aspect of such limitations, which I will address below is the denumerable nature of intellectual processes.
We can assume, we must assume, that our way of doing mathematics, including physical theories, is consistant. This is really (to me at least) a physical view of it: we do find inconsistencies, usually called paradoxes, but there are resolved (have been so far) like experimental inconsistencies in physics, by refinements of the théories and evolution of concepts to overcome the difficulty and state the problems appropriately.
Assuming mathematics is essentially consistent is essential, because whatever we can prove should not be questionned by future extensions, if any are physically possible, of the concepts of calculability or provability.
Now some results make deep assumptions that are not always obvious to interpret. In the case of Gödel incompleteness result, one major aspect is that logical formulae, theorems and proofs can be encoded as integers. This means that our logical deduction systems are fundamentally denumerable entities (like Turing machines). If it turned out that a breakthrough in physics allowed us to deal effectively with non-denumerable systems, then results relying on this denumerability would be in question. This is precisely the case for Gödel incompleteness result, as stated (possibly it could come back in another form).
I addressed this denumerability aspect in my answer to the question on the physics provability of Church-Turing thesis. At the time this answer was entirely based on my informal understanding of these issues. Intending to improve a bit on my answer, I looked for some litterature, and the topic is currently actively researched. While my knowledge of this literature remains more than shallow, it seems that my intuition was correct, that a proper handling of the fundamentally discrete (or denumerable) character of calculability, and provability, is essential in the current state of the art, for deriving Church-Turing thesis from the laws of physics, and that continuity or real numbers are a major issue.
One approach I have looked at (I am limited to papers in open access on the web) relies on assuming a specific property of the physical world, presented as dual of the limitation on the speed of light and information, which is a limitation on the space density of information, both limitations together ensuring density limitation in space-time. The translation of this new law in physical terms can actually be subtle to account for various existing physical laws. This apparently excludes unregulated use of real numbers.
If this limited density law is actually verified, I think it would also mean that Gödel theorem is also a consequence of physical laws.
Whether such a limited information density law is actually verified is another matter. If it is not, then doors remain open for an extension of the concepts of calculability and provability.
In such a case, assuming that our mathematics are otherwise consistent, all provable results would remain provable, but we might be able to prove new theorems that were true but not provable in the classical denumerable setting.
So the answer to the question whether a theory of Everything would evade Gödel's incompleteness theorem is very much dependent on what Everything is, since it actually determines the context in which calculability or provability have to be defined. Would a theory of Everything include a law limiting information density.
Note that even with Gödel's result being valid, there could be a possibility of a
theory of Everything, in which all true facts regarding the universe
would indeed be true. It is just that you would not be able to prove
it (so that the universe would keep some mystery for us to wonder in
starry nights).
On the other hand, there could be no such theory of everything. But, what would the ultimate definition of a theory if physics allowed us to question the discrete nature of the language in which they are expressed ?
For the rest, other than taking 42 as ultimate answer, I can only suggest leaving the matrix to get the Truth about our world, or reading Simulacron-3.
Best Answer
No, there is not nor can there be a similar statement in physics. That is because we can know all there is to know about the mathematical systems we construct; after all, we have set them up ourselves (but then Gödel's incompleteness theorem tells us that there can be features of certain systems that remain unknowable; sorry for butchering what the theorem really says, btw).
Physics, on the other hand, ultimately attempts to model reality. The problem there is that we have fundamentally no way of knowing reality in itself; we cannot even be sure there is such a thing as reality although we take that as a fundamental axiom of physics. Thus, all we can do is propose models for the reality we experience. We cannot know what the relationship of such models to reality as it is might be.