Gödel's famous incompleteness theorem implies, in particular, that there are statements unprovable in $\mathsf{ZFC}$. This implies that we could never hope to settle the truth of every mathematical statement using a reasonable (r.e.) axiom system, shattering Hilbert's program. However, in the 90 years since, many examples have been found of statements which are independent of the $\mathsf{ZFC}$ axioms.
Is it true (or possible) that any statement in $\mathsf{ZFC}$ (or another r.e. system $\mathsf{S}$) could be proven either:
- true;
- false;
- independent of $\mathsf{ZFC}$ (resp. $\mathsf{S}$)?
This would allow a weak sort of realisation of Hilbert's program: for any statement, we can either prove/disprove it, or show it is independent of $\mathsf{ZFC}$.
No.
The set of statements that is provably provable or provably refutable in $\mathsf{ZFC}$ is recursively enumerable but not recursive. If you could prove that the set of all statements independent of $\mathsf{ZFC}$ was determinable, it also would be recursively enumerable, but that's the complement of all statements that are either provably true or provably false, so the latter set would then be recursive, which it's not.