2 Answers

  1. For the sake of decency, let us formulate Godel's theorem “If formal arithmetic is consistent, then there exists a non-deducible and irrefutable formula in it”.

    Godel's theorem doesn't forbid anything from the real world, because the universe doesn't have to work the way mathematicians want it to. We can accept as evidence an experiment, as for example was the case with Ohm's law(ordinary, school). But if we consider it from the point of view of theoretical or mathematical physics, then yes, we can regard Godel's theorem as the fact that an experimentally discovered phenomenon will appear that cannot be described in these postulates. But physics is not mathematics, no one forbids postulating other axioms and saying “In this case, we use them”. As a result, we will get a single “theory of everything”, which will indicate two systems of axioms where we can apply each system(for example, “If we describe Ivan Vasilyevich, then we use the system of axioms A, if we describe the behavior of Peter Higgs, then we use the system of axioms B), each system of axioms will indicate one formula according to which everything works that a smart theorist told her to describe.

  2. Godel's incompleteness theorems prohibit this theory from being an axiomatic first-order theory, which, in fact, was initially unlikely. But, generally speaking, Godel's results are like a reason to think (as they say, “Shtrilits was wary”) – and how will it be possible to prove the completeness of the theory of everything, that is, to prove that it is really “everything”? Apparently, by mathematical means in no way, because with higher-order theories it will not be easier, even if this hypothetical theory of everything is formalized within the framework of high-order axiomatic theories, which also no one promised. That is, the theory of everything will probably be based on a non-strict induction (“we did not notice anything that contradicts it, but, on the contrary, we noticed a lot of things that DO NOT contradict it”). Thus, we may one day formulate a theory of everything, but we will never strictly prove that it is really consistent and really everything.

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