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I think the answer you are looking for might be this: There is no syntactic way to identify theorems across different formal systems. If, that is, you look only at the symbols that comprise the theorem then the only "content" is their relation to the axioms of your formal system via the deductive system that allows you to prove it (the theorem) from them (...

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Yes. This has to do with the reason we construct formal systems in the first place: generally, we have a particular intuitive notion in mind and want to construct rules that give structure to that notion. For example, we have an intuitive notion of natural number, and we devise the axiom a+S(b) = S(a+b) in Peano Arithmetic to reflect (part of) what we know ...

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Yes, you can do this! The case you're thinking of is very similar to Tarski's original idea of defining a Truth predicate that was Materially Adequate, and a paper by Henryk Kotlarski, Stanislav Krajewski and Alistair Lachlan in 1981 showed that we can conservatively define a Truth predicate over the sentences of Peano Arithmetic. Their trick is to ...

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Here's an example to argue against the doubt that the "same" theorem cannot be proven in multiple mathematical systems: take p -> p for p a propositional variable. It can be proven in intuitionistic propositional logic, classical proposition logic, classical first order logic, Robinson's arithmetic, Peano's arithmetic, ZFC, in fact just about any formal ...

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The OP's question has been discussed here. I agree with the claim in the (linked) paper that the philosophy of mathematics can contribute to mathematics to the degree that it affects the practice of mathematics. Mathematicians are pragmatists in the sense that they see the foundational questions classically considered in the philosophy of mathematics as ...

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Godels theorems are about the expressive power of formal languages; and only tangentially touches on the nature of truth; for mathematicians, it's importance is in prompting the growth of a new field: model theory. Though we don't have a world where 2+2=5; though if we did, we should ask what does it mean? Is it also the case that 2+2=4? So that 4=5? And ...

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