I don't see why the original version of Gödel's first incompleteness theorem (before Rosser's improvement, I mean) had to include the assumption of $\omega$-consistency in order to show that $F \nvdash \neg G_F $ — as opposed to the other direction, $F \nvdash G_F $, where consistency suffices anyway, by my understanding.
Could someone point out the flaw in the following (sketchy) argument?
(1) The provability relation for a formal system F is weakly representable in F, i.e. a predicate $PRV_F$ exists s.t. $F \vdash A \Leftrightarrow F \vdash PRV_F(A)$
(2) We define sentence $G_F$ s.t. $F \vdash G_F \leftrightarrow \neg PRV_F(/G_{F}/)$
(3) To show: If F is consistent, then $F \nvdash \neg G_F$ (second part of Incompleteness I)
Assume $F \vdash \neg G_F$.
Then by (2), $F \vdash \neg \neg PRV_F(G_F)$, so $F \vdash PRV_F(G_F)$.
Then by (1), $F \vdash G_F$, so F is inconsistent (and $\omega$-inconsistent as a result as well) from assumption that $F \vdash \neg G_F$.
Best Answer
The issue is already present in step 1. Consistency is not a strong enough assumption on the theory to guarantee that the provability relation is representable.
For example, if $T$ is taken to be $\text{PA} + \lnot\text{Con}(PA)$, which is not $\omega$-consistent, then we have $T \vdash \text{Pvbl}_T(\phi)$ for all $\phi$, but $T$ is consistent, so for many $\phi$ we also have $T \not \vdash \phi$. Thus we do not have the whole scheme $T \vdash \phi \Leftrightarrow T \vdash \text{Pvbl}_T(\phi)$, as claimed by (1).
The source of confusion may be that you seem to be quantifying over predicates when you write "a predicate $PRV_F$ exists s.t. $F \vdash A \Leftrightarrow F \vdash PRV_F(A)$". In fact we construct a specific predicate $\text{Pvbl}$ in order to derive scheme (1) from the question.
In order to show that $T \vdash A$ implies $T \vdash \text{Pvbl}(A)$, which is part of showing that $\text{Pvbl}$ does represent the provability relation, we rely on $\text{Pvbl}$ being the actual provability predicate, so that we can transform any derivation of $A$ into a derivation of $\text{Pvbl}(A)$.
Similarly, in showing that $T \vdash \text{Pvbl}(A)$ implies $T \vdash A$, we use the $\omega$-consistency of $T$ to know that, if $T \vdash \text{Pvbl}(A)$ then there is actually a natural number $n$ that codes a derivation of $A$ from $T$.
You may be thinking of general results that show that certain theories are able to represent every computable set, or weakly represent every r.e. set. It is true that the representability of the provability relation follows from these general results. But the proofs of these general results also assume that the theory is $\omega$-consistent, or at least $1$-consistent, not just that the theory is consistent.