Does Gödel's Incompleteness Theorem really say anything about the limitations of theoretical physics? Stephen Hawking believes that Gödel's Incompleteness Theorem makes the search for a 'Theory of Everything' impossible. He reasons that because there exist mathematical results that cannot be proven, there exist physical results that cannot be proven as well. Exactly how valid is his reasoning? How can he apply a mathematical theorem to an empirical science?
Hawking's argument relies on several assumptions about a "Theory of Everything". For example, Hawking states that a Theory of Everything would have to not only predict what we think of as "physical" results, it would also have to predict mathematical results. Going further, he states that a Theory of Everything would be a finite set of rules which can be used effectively in order to provide the answer to any physical question including many purely mathematical questions such as the Goldbach conjecture. If we accept that characterization of a Theory of Everything, then we don't need to worry about the incompleteness theorem, because Church's and Turing's solutions to the Entscheidungsproblem also show that there is no such effective system.  

But it is far from clear to me that a Theory of Everything would be able to provide answers to arbitrary mathematical questions. And it is not clear to me that a Theory of Everything would be effective. However, if we make the definition of what we mean by "Theory of Everything" strong enough then we will indeed set the goal so high that it is unattainable. 

To his credit, Hawking does not talk about results being "unprovable" in some abstract sense. He assumes that a Theory of Everything would be a particular finite set of rules, and he presents an argument that no such set of rules would be sufficient.
Since "Theory of Everything" needs to be expressed in some kind of proof, which will be partially mathematical and partially observation, it is not clear, how such proof can be validated without the component of mathematics. 

Quantum computing might solve this problem for us, as it is able to operate using mathematical constants that are not verifiable or provable after they are executed, just like some theories in mathematics, that we can observe.