The broad multiverse conception of truth (based on ZFC) is then simply the view that a statement of set theory is true simpliciter if it is provable in ZFC.
Notice that the mere fact that if φ is provable then so is φ+ is not enough to ensure that φ → φ+ is provable, so the axioms of typical ambiguity are prima facie nontrivial and we should not be surprised that they turn out to be strong.
If G is provable, then it is true and so not provable.
More exactly, Gödel showed that, if the system is consistent, then p is not provable; if it is ω-consistent, then ∼p is not provable.
The proof of this theorem consists essentially of a formalization in arithmetic of the arithmetized version of the proof of the statement, "If a system is consistent, then p is not provable"; i.e., it consists of a derivation within number theory of p itself from the arithmetic sentence that says that the system is consistent.
In sum, if F is consistent, then GF is not provable in F. For this first half, the assumption of the simple consistency of F suffices.
(GL) claims that if PA manages to prove the sentence that claims soundness for a given sentence A, then A is already provable in PA.
Hence, if one can find a model of $\varphi$ and also a model of $\neg \varphi$, then $\varphi$ is neither provable nor disprovable in ZFC, in which case we say that $\varphi$ is undecidable in, or independent of, ZFC.
If this is read arithmetically, the direction from left to right is just the formalized version of Gödel's second incompleteness theorem: if a sufficiently strong formal theory T like Peano Arithmetic does not prove a contradiction, then it is not provable in T that T does not prove a contradiction.
A finitary consistency proof of the kind envisaged by Hilbert would have accomplished this: if ideal mathematics proves a real proposition, then this proposition is already provable by real (i.e., finitary) methods.
