Theory of everything vs Godel's incompleteness theorem.

lugita15

If a theory of everything happens to include a sufficient amount of facts about integers that it meets the conditions of Godel's theorem, then there will indeed be truths in the language of the theory which cannot be proven by the theory, but these will be truths about the integers, not truths about the physics of the theory. So in particular, Godel's theorem will NOT lead to unprovable properties of electrons.

Demystifier

I don't think that in a single theory one can make a clear separation between "mathematics" and "physics".

lugita15

But all Godel's theorem says, essentially, is that no axiomatic theory can prove all the (first-order) truths about the integers. So as such, it has absolutely no bearing on whether the theory has gaps concerning physics. Now you may be wondering how e.g. the set theory ZFC is susceptible to Godel's theorem, even though it only talks about sets, not integers. The reason is that you can represent integers in terms of sets, for instance defining zero as the set of the empty set. Thus Godel's theorem tells us that no axiomatic theory can prove all the truths about sets either.

So similarly, you can imagine some weird hypothetical TOE in which integers are represented not as themselves, which would be ho-hum physics wise, but rather in terms of he physical content of the theory. For instance, you can have an infinite number of particle types, each type corresponding to one natural number. Then various properties of numbers would correspond to various properties a particle can possess, and Godel's theorem would tell us that there is some particle property such that the proposed TOE cannot tell which particles possess it and which don't. But this is a rather far-fetched scenario, and it would only become an issue if the theory proved enough facts about the natural numbers for Godel's theorem to apply, and depended strongly enough on all knowing all the truths of number theory, both of which are very unlikely possibilities.