I was thinking that something could be true and undecidable (as for the halting problem).
I'm somewhat on thin ice on the mathetematical formulation, but I suppose something could be undecidable but true with respect to all standard models of a theory, while not true for some non-standard model.
For instance, there may be statements acting on infinite sets (such as N) that may not be compressed into a recoursive formulation, which means that a proof of the valididity of the statement (if it is true) would require an infinite number of steps.
> I'm somewhat on thin ice on the mathetematical formulation, but I suppose something could be undecidable but true with respect to all standard models of a theory, while not true for some non-standard model.
The term "standard model" is interesting, because it pushes the problem a bit. So, let us take the natural numbers. Sure, we can prove in set theory that some of the undecided statements of Peano arithmetic are true for the standard model. But seen from the outside this means that we have changed our axiomatic system from Peano arithmetic to set theory. But there are still undecided statements – even arithmetic ones – in this new system! These will hold in some models of set theory, while being false in some other.
So, saying “Oh, I mean the standard ℕ, not any of those other ones” does not get us off the hook, because it raises the question: “The standard ℕ... in which model of ZF?”
I'm somewhat on thin ice on the mathetematical formulation, but I suppose something could be undecidable but true with respect to all standard models of a theory, while not true for some non-standard model.
For instance, there may be statements acting on infinite sets (such as N) that may not be compressed into a recoursive formulation, which means that a proof of the valididity of the statement (if it is true) would require an infinite number of steps.