I think this interpretation is both intriguing and clarifying, thanks. It suggests that a good framing of the incompleteness theorem is:
Any sufficiently rich system S can be extended by multiple interpretations I, such that all of its provable theorems are true for all of I, but I contains interpretations who disagree on the truth of some unprovable statements of S
And if you try to narrow down the interpretations, you either don’t get far enough—because there are infinitely many possible interpretations, and no matter how many times you exclude half of them by adding a new axiom, still infinitely many possible interpretations remain—or you introduce some new concept—but then this new concept has its own multiple interpretations.
For example, if you try to make axioms for integers, it turns out that there are also some “nonstandard integers” which technically satisfy the axioms, even if they are obviously not the thing we meant. The traditional way out is to add an axiom like “the actual integers are the smallest set (from the perspective of set inclusion) of things that satisfy these axioms”, but now you have invoked the concept of a set, and if you try to define what that means, you just open another can of worms… and this process never stops.
I think this interpretation is both intriguing and clarifying, thanks. It suggests that a good framing of the incompleteness theorem is:
Any sufficiently rich system S can be extended by multiple interpretations I, such that all of its provable theorems are true for all of I, but I contains interpretations who disagree on the truth of some unprovable statements of S
And if you try to narrow down the interpretations, you either don’t get far enough—because there are infinitely many possible interpretations, and no matter how many times you exclude half of them by adding a new axiom, still infinitely many possible interpretations remain—or you introduce some new concept—but then this new concept has its own multiple interpretations.
For example, if you try to make axioms for integers, it turns out that there are also some “nonstandard integers” which technically satisfy the axioms, even if they are obviously not the thing we meant. The traditional way out is to add an axiom like “the actual integers are the smallest set (from the perspective of set inclusion) of things that satisfy these axioms”, but now you have invoked the concept of a set, and if you try to define what that means, you just open another can of worms… and this process never stops.