Incompleteness theorem and theories of everything - a mathematician's view

[Whovian]

Right, 27 words of advice: I only recommend reading this if you're up for a long, probably controversial post regarding mathematical philosophy and philosophical mathematics and their relation to theories of everything.

Definition: A theory of everything is a system consisting of a finite of true statements, rules of statement and deduction included, in which any conceivable meaningful physical law can be stated, from which any true ones can be proven via the rules of deduction, and in which no meaningless physical laws can be stated. Purely mathematical statements like "1+1=2" are not considered physical laws, and as such the final condition does not exclude the possibility of us being able to state them.

I'll be assuming throughout this post that the Universe is consistent; i.e. there are no meaningful physical propositions which are both true and false. (I'll admit I'm hand-waving a bit here by not properly defining meaningful; to be frank, I haven't a satisfactory definition. We can use "observationally falsifiable" for now, but it's not clear that a theory of everything won't be able to result in observationally unfalsifiable statements. For example, metaphysical questions like "what came before the Universe" would not be considered meaningful.)

In this case, we can try to invoke Gödel's first incompleteness theorem on our theory of everything. To my understanding the theorem can be stated as follows:

Any formal axiomatic system which can describe basic arithmetic, if it is consistent, must also be incomplete (meaning we can construct "undecidable" statements which cannot be formally proven nor disproven.)

The given definition of a theory of everything was meant to conform with the definition of "formal axiomatic system" (or whatever phrase mathematicians are using nowadays.) Therefore we can try to apply the first incompleteness theorem to our theory of everything, which presents three possibilities:

1. You don't need to know arithmetic to be a physicist.
   
2. There is no theory of everything; we can construct meaningful statements which cannot be decided in any candidate for a theory of everything.
   
3. In order to conform to the definition of "theory of everything" given, we need to include any and all necessary mathematical prerequisites; the incompleteness could be in the mathematical prerequisites rather than the physically meaningful questions. I'll have to grudge my way through Science Without Numbers, ignoring as best as I can that Hartry Field is a bit of a jerk, to see if this possibility can be excluded.
   

Anyways, I feel like this post, especially the definition of "theory of everything," is becoming a bit of a mess. I wanted to open up discussion on this idea; feel free to criticize my reasoning (I tried conforming as best I could to mere first-order logic for my deductions,) my definition of "theory of everything," or anything else about this post. I'm frankly just a mathematician who dabbles occasionally in philosophy, so I'm not too familiar with philosophical conventions.

Notes regarding various philosophies of mathematics: For accessibility, I'm presupposing the existence of a "real world" independent of mathematics. This may bug mathematical monists. I'm also presupposing, as the Platonist I am, the absolute truth of mathematics; this may bug others. I have attempted to construct this post in a manner which does not assume that the philosophy being used is correct or justifiable, but only that it's valid and consistent. I am, however, also assuming that any conceivable theory of everything can be described mathematically, but it may be arguable that the mathematics required is not abstract, and is as such ontologically uncontroversial. If you have no idea what any of this paragraph means, don't worry; it's pretty much a fairly unimportant side note.

Finally, I do apologize for any and all offense to philosophical beliefs I may cause.

EDIT: I should note that these in no way reflect the views of the mathematical community and I doubt I'm a typical mathematician when it comes to my philosophy, but it's too late to change the title now.

[ZetaX]

The "real world" should probably not be seen as the theorems following from your axioms, but as a model of them. This is much closer to how physicists use it, too. In particular, Gödel's theorem does not apply anymore that way.

Furthermore, the undecidable statements could be things like "does this nucleus of Pu-238 decay in the next year¿" that are governed by quantum randomness. In other words, we already have some loopholes to 'hide' undecidable statements in.