Well I've searched this forum and didnt see this topic before, so I was wondering what the general consensus is on the impact of Godel's theorem (if any) on a possible TOE? I'm still wading through it all so just wanted some other opinions....
Re: Godel's incompletenss theorem Godel's theorem is a mathematics foundation subject. It has nothing to do with physics (TOE).
Re: Godel's incompletenss theorem Mmmm but put simply physics is based on mathematics, it is written in the language of mathematics. Stephen Hawking has addressed the problem, and has conceeded that it does apply; http://www.hawking.org.uk/index.php/lectures/91 So why has it nothing to do with physics?
Re: Godel's incompletenss theorem The basic argument that Hawking was making on this subject a few years back is that even if we do discover a theory of everything, physicists would still have a lot of work to do, because due to Goedel's incompleteness theorem, it would be impossible to ever discover all of the consequences of the theory of everything. Thus it's basically an argument that a discovery of a theory of everything, if it occurs, would still leave physicists with lots of work to do.
Re: Godel's incompletenss theorem I thought Godel's Incompleteness Theorem was only applicable to axiomatic systems. Does Physics have such systems or only pure mathematics?
Re: Godel's incompletenss theorem I think the expectation is that a theory of everything, whatever that may be, would be an axiomatic system.
Re: Godel's incompletenss theorem What would justify it being an axiomatic system? Pure Mathematics relies on the notion of axioms, Physics requires experimentation.
Re: Godel's incompletenss theorem The point is that axiomatic systems are not arbitrary. Some sets of axioms produce well-behaved mathematical structures, some do not. So the sorts of mathematical structures available is limited in some sense. The question, then, is which of these mathematical structures is isomorphic to reality. If we ever do manage to eliminate all but one mathematical structure as describing our reality, then that will be our theory of everything. And that one mathematical structure will probably be an axiomatic system.
Re: Godel's incompletenss theorem There is a good book on this kind of thing: Franzen, Godel's Theorem: An Incomplete Guide to Its Use and Abuse. There are lots of different reasons why Godel's theorem is not relevant to the search for a TOE: Physics is not axiomatic system. We don't have a TOE, so we don't know whether Godel's theorem would apply to it, even assuming that it could be made into an axiomatic system. Godel's theorem only applies to certain types of axiomatic systems. For example, it does not apply to elementary Euclidean geometry, which can be proved to be consistent. It is possible to prove that one axiomatic system is equiconsistent with another, meaning that one is self-consistent if and only if the other is. If we had a TOE, and we could make it into an axiomatic system, and it was the type of axiomatic system to which Godel's theorem applies, then it would probably be equiconsistent with some other well known system, such as some formulation of real analysis. Any doubt about the self-consistency of the TOE would then be equivalent to doubt about the self-consistency of real analysis -- but nobody believes that real analysis lacks self-consistency. Finally, there is no good reason to care whether a TOE can't be proved to be self-consistent, because there are other worries that are far bigger. The TOE could be self-consistent, but someone could do an experiment that would prove it was wrong.
Re: Godel's incompletenss theorem But does the incompleteness theorem show that a TOE is not possible, even in principle?
Re: Godel's incompletenss theorem How can it show it's impossible if we don't know if a TOE is an axiomatic system? At this point it isn't really applicable.
Re: Godel's incompletenss theorem Even if TOE is an axiomatic system, why should it be incomplete? There are complete systems for example the Euclidean geometry.
Re: Godel's incompletenss theorem Arithmetic is incomplete. We probably need arithmetic in the TOE, no?
Re: Godel's incompletenss theorem We don't have one yet, so we don't know. But, for example, the standard model has never been stated as an axiomatic system according to the definition used in Godel's theorems. The definition used in Godel's theorems is extremely strict. For example, here is a formal statement of Euclidean geometry: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.9012 Physicists never want or need to do anything with this level of formality. -Ben
Re: Godel's incompletenss theorem PhysDrew: I take it that you are quibbling over what "everything" in "theory of everything" means. Who cares? Certainly not physicists. All this discussion of Euclidean geometry is a bit of a red herring, for at least two reasons. #1: Even Euclidean geometry, at least where it intersections algebraic geometry, is incomplete or inconsistent. The Pythagorean theorem does it in. #2: Physics uses real numbers and more complex number systems, derivatives, all that. That places the mathematics used by physicists smack dab in the middle of a system of sufficient power so as to be subject to Godel's theorems. Once again, who cares? Physicists? No.
Re: Godel's incompletenss theorem It's not immediately obvious to me that we do. I know that sounds nuts, but this kind of thing is not necessarily intuitive. You would think that since the reals are a bigger, fancier mathematical system than the natural numbers, then since arithmetic is incomplete, the reals would have to be as well. But that's not the case. The elementary theory of the reals is equiconsistent with the elementary theory of Euclidean geometry, which is provably consistent. It's quite possible that a TOE could be expressed in geometrical language, without the use of any arithmetic.
Re: Godel's incompletenss theorem This is incorrect. See http://en.wikipedia.org/wiki/Complete_theory for the relevant notion of completeness and incompleteness. This is also incorrect. See #18. The informal notion of "power" you have in mind is not the appropriate concept for discussion Godel's theorems. E.g., the reals and the complex numbers are equiconsistent, because you can model the complex numbers using the reals: http://en.wikipedia.org/wiki/Model_theory
Re: Godel's incompletenss theorem Show me a proof of the Pythagorean theorem that does not involve multiplication distributing over addition, then. Any theory based on the reals is, as far as I know, subject to Godel's theorems. If you want to do physics without talking about measurement you are not really doing physics in my mind.