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Very interesting. I'm wondering if this has implications in physics ? Since "the book of nature is written in mathematical language", does this mean that attempts to find a unified theory of everything are doomed ? I'm thinking not necessarily, since we wouldn't need to prove all possible physical phenomena, only those that actually happen, but how do we know if they are among the one that can be proven ?

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The implications are really on the concept of language. It proves that there is no language that can be complete; as such, we cannot express a complete system, even if it existed. So a unified theory of everything might well exist, but you wouldn’t be able to describe it. Which is a nice philosophical riddle.

I think that if you allow some axioms, you can describe it. But I'm not sure if it will be called complete.

I think because "(next axiom)" is not defined, it will make your language not mappable to integer numbers and you avoid Godel proof.

A very comprehensive theory of physics including the physics of

computation can be developed. However, such a foundational

theory will be inferentially undecidable.

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