>>13617719>It was proven that such true yet unprovable statements existNo it wasn't.
>Was it proven that they are infinite in number?It was proven that there are infinitely many statements in a formal system strong enough to talk about arithmetic that are true but unprovable (if you assume the consistency of the system). If T is such a system, then Con(T) is unprovable if it's true. Then you can form a new system T'=T+ Con(T) and then if T was consistent, so would T', and T' doesn't prove Con(T') = Con(T + Con(T)) which is a different statement from Con(T). Thus Con(T + Con(T)) is also unprovable in T. You can keep doing this, defining recursively T^(i+1)=T^(i) + Con(T^(i)), and then the statements X_n = Con(T^n) are all unprovable assuming Con(T).
