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According to Gödel's Incompleteness theorems, there exist problems in any sufficiently powerful, consistent system of arithmetic that are undecidable form the axioms of said system.

*What known problems exist in physics(if any), exhibit Gödel Undecidability?

And a sub-question, why are most certain that a ToE can be derived under these conditions?(This question is different from the others in that I am asking what is the motivation of unification given such limiting restraints, regardless, this is not my primary question.)

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    $\begingroup$ Luckily we have experiment to prove things for us $\endgroup$ – InertialObserver Mar 8 '19 at 2:28
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    $\begingroup$ possible duplicate of physics.stackexchange.com/q/14939 $\endgroup$ – niels nielsen Mar 8 '19 at 2:29
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    $\begingroup$ Possible duplicate of Does Gödel preclude a workable ToE? $\endgroup$ – Jon Custer Mar 8 '19 at 3:32
  • $\begingroup$ @InertialObserver Not sure that comment is sensical, given that this question is tagged mathematical-physics, and consequentially is about the rigorous underpinnings of the subject and has nothing to do with experimental physics. $\endgroup$ – hisairnessag3 Mar 8 '19 at 4:14
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I believe this comic from XKCD captures the problem quite succinctly: XKCD

The kind of mathematical rigor that one seeks with Gödel's theorems is so utterly distant from the level of rigor used in science that it is hard to make the connection. Gödel's theorems start to tackle the questions that arise as one approaches infinity.

When it comes to things like Theories of Everything, I don't think anyone has any illusions of what that means. If they come up with a Theory of Everything which proves consistent for the next billion years or so, they're pretty darn happy.

Practically speaking, Gödel's theorems also have very strict definitions of what "sufficiently powerful" that they entertain. For example, it only applies to axiom systems written in first order logic. Gödel's theorems don't apply to higher order logic. In practice, few people are truly striving to reach an axiom system in first order logic. That's really more the realm of philosophy, not science.

And once you reach said philosophical boundary, you arrive at the deeper conundrums, such as that of Plato's Cave. Science does not in any way prove anything, if yo are using the definition of "prove" typically found in math or philosophy. Science has a different meaning that it uses when it says "prove." If you get to the point where Gödel's theorems start getting in the way, the greater question of whether science can prove anything to a sufficient degree to invoke Gödel's theorems comes into play. There have been a few places where I know Gödel's theorems have come into play.

That being said, Gödel's theorems have showed up a few times. The most famous I know of is in the undecidability of spectral gaps. Models used to determine if a model is gapped or gapless demonstrate undecidable behavior. But that's not so much a statement about reality as it is about the models we use to make sense of reality.

And beyond that, there's always mathematical ways to get around Gödel's theorems. My personal favorite is the work of Dan Willard. He explored proof systems with numbers where multiplication is not total (Gödel's theorems assume multiplication is a total function). He starts with "Everything," and then uses division to work downward, rather than starting with "one" and using addition and multiplication to work your way up. In doing so, he carefully sidesteps the ability to use the diagonal lemma at the heart of Gödel's theorems, so they don't apply.

So even if you wanted to reach the level of mathematical rigor we are talking about, and found a way around the issue of science proving things, there is still a workaround. All you need to do is make sure your Theory of Everything is defined using a number system that cannot prove multiplication to be total, and Gödel's theorems dissapear into a poof of logic... literally.

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  • $\begingroup$ Great answer, detailed and complete. $\endgroup$ – hisairnessag3 Mar 8 '19 at 5:34
  • $\begingroup$ One thing though, this answer is in contradiction with your comments on higher-order logic: philosophy.stackexchange.com/questions/7239/… $\endgroup$ – hisairnessag3 Mar 8 '19 at 5:37
  • $\begingroup$ My wording is a bit fuzzy, but not in contradiction with that answer. Godel's theorems only apply in First Order Logic. However, if you have a higher order logic system which can be reduced to FOL, then that particular system is still subject to Godel's theorems. That's where the linked answer was going. The fuzzy bit I left out is that Godel also proved that it is impossible to prove a self referential axiom system in higher order logic unless it was reducible to FOL. $\endgroup$ – Cort Ammon Mar 8 '19 at 15:26

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