Is everything provable as true, false, or undecidable?

I will prove:

There does not exist an algorithm that, for any statement P in the language of set theory, classifies P as being provable, disprovable, or independent of the axioms of ZF

Suppose you had an oracle $M$ that solves this problem. You can use this to create an algorithm that enumerates the theorems of a complete extension of ZF:

• Let $A$ be a tautology
• For every statement $P$ in the language of set theory:
  • Use $M$ to classify the status of $A \Rightarrow P$
  • If $A \Rightarrow P$ is provable in ZF, output $P$
  • If $A \Rightarrow P$ is independent of ZF, then set $A = A \wedge \neg P$

The idea is that the algorithm iteratively builds a list $A$ of new axioms to add to ZF to make it complete; the oracle $M$ is used to ensure we never select an inconsistent set of axioms.

Gödel's incompleteness theorem says that there does not exist an algorithm that can do this; consequently there is no algorithm for determining the output of $M$.