My question motivated by a part of this page from Saul Kripke's book Naming and Necessity, which is also viewable on google books. In the middle of the page he say something, which seems unnatural to me: Namely he implies that one mathematical theorem can be proved in two different formal systems. Later, in a slightly different context he starts talking about possible worlds and how there are difficulties to identify one referent (e.g. president Nixon) in another world.
Why doesn't it seem to be a problem for him to identify a to be proven statment, when changing from one formal system to another?
Independend of the reference to Kripke: If I show a mathematical identity in one formal system, then the whole deduction is based on a particular axiomatic framework. Any theorem should be unique and not to be found under another axiom system. So by this reasoning, if one considers one statement in one theory, it can not be proven in another. Clearly, there are many models of set and in this sense there are different abstractions of the same intuitively true relation 2+3=5, but proving it in one arithmetic doesn't imply anything about it in another, right? It (that is, the analog) should in priciple be re-proved there. Therefore, i.e. because there are different foundations (similar to there being different religions), it shouldn't be possible to distiguish one framework as better than another and to we wouldn't know how to infer any truth of the real world relation from the maths.
Is there a reason or possibility to identify a statement (which can be derived from a logical framework) to another statement in another framework?