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A proof of a statement A may be valid in one formal system X but not capable of being written down in another formal system Y (and more strongly, there may be no proof at all that can be written down in Y that proves A or not-A, even if the statement A itself can be written down in Y).

It helps to be precise. Stating the incompleteness theorem imprecisely generally just makes people confused about what it could even mean. Strictly speaking, an incompleteness theorem has to be proven within a particular formal system X (the "meta-system"). It expresses--within X--that "for every consistent formal system Y expressible in X, there is a statement A expressible in Y such that there is no proof in Y of A or not-A". Here Y is the "object system".

The statement A is only unprovable within Y in particular. If it is expressible in X as well, it may be provable (or disprovable) within X. Indeed, one can trivially construct a formal system in which A is provable and which can express Y by adding A as an axiom to X.

The real, philosophically significant, non-particular-formal-system-specific import of the incompleteness theorem is not just that there are unprovable statements (which isn't really true, and is in fact kind of mathematically meaningless, outside of the context of a specific formal system, given that any statement or its negation can be taken as an axiom, and all we have to judge such choices is a non-mathematical notion of "self-evidency") but that (as the name suggests!) no formal system is complete--every formal system can express statements that it can't prove. It's all about that universal quantification.






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