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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?

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Godel's theorem only says for some fixed, recursively defined, axiom system there are statements you can't prove or disprove. A consequence of this is that you can add it (or its negation) as an axiom to get a new equiconsistent theory which can prove (or disprove) it. That shouldn't matter for physics because you can just add new axioms when you want. There's no reason a result in physics must be proved in terms of some fixed axiom system (like ZFC say) –  sperners lemma Nov 4 '12 at 11:03
    
On the meeting with Roger Penrose he said mostly the same things about the seeking of the "Theory of Everything". –  m0nhawk Nov 4 '12 at 12:42
    
Related post on Phys.SE: physics.stackexchange.com/q/14939/2451 –  Qmechanic Feb 10 '13 at 19:50

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up vote 15 down vote accepted

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.

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@Sperners Lemma's comment should really be promoted to an answer. For indeed, it is a fairly gross misunderstanding of what Gödel's theorem says to summarize it as asserting that "there exist mathematical results that cannot be proven", for the reason he briefly indicates.

And incidentally, though it is a quite different issue, a Theory of Everything in the standard sense surely doesn't have to entail that every physical truth could be "proven". Let's bow to the wisdom of Wikipedia which asserts

A theory of everything (ToE) or final theory is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle.

So NB a ToE is a body of laws which (if we assume that they are deterministic) will imply lots of conditionals of the form "if this happens, then that happens". But a ToE which wraps up all the laws into one neat package needn't tell us the contingent initial conditions, so (even if it is deterministic) need not tell us all the physical facts.

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It seems reasonable to assume an isomorphism between arithmetic and experiment ($2+2=4 \cong 2kg + 2kg = 4kg$, etc.) - wouldn't that make Godel's theorem apply? (There are some set of weights which when added together represent the experiment's own Godel number, etc.) –  Xodarap Nov 4 '12 at 15:05

He can't. What would it even mean to "prove" a physical result? A ToE would condense all of the physical laws we've observed into a unified (and preferably compact) form, but as an empirical statement it would not imply, nor be implied (or even affected) by, results in mathematics. No mathematical axiom system has anything more to do with reality than any other one, including e.g. ones to which Godel's theorem does not apply, ones in which Godel's theorem cannot be proven, or even ones which are just inconsistent.

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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.

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I saw this and thought this argument would not be complete without mentioning the work of david wolpert http://arxiv.org/abs/0708.1362

He shows that a theory of everything is impossible.

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I am with Hawking on this one, though perhaps for different reasons.

It is not particularly difficult to show that some theoretical aspects of nature encode arithmetic. That is to say, as an example, that the theory of electrodynamics "contains" a copy of the natural numbers, together with addition and multiplication.

That is enough to satisfy Godel's theorem, to show that the theory of electrodynamics is "incomplete".

Of course, I have only described an argument that a single theory is incomplete, not that every theory is incomplete.

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