Just to be clear. This implies that it's possible to write some specific statement (using PM axioms) that contradicts itself? Is there a readable example of this statement without using Gödel numbers but instead just with the axiom statements?
Everything gets qualified with "assuming ZFC consistent" or "assuming Peano consistent" because any inconsistent theory proves any statement. More of a proof technicality than anything too profound.
There is a construction of a model of Peano arithmetic, so it is consistent, as long as you accept the system used in the proof: https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof
Not sure if this sheds light on the parent commentator's question ... the terminology can be quite tricky.
[Gödel 1931] proposition I'mUnprovable for reasons
explained elsewhere in this discussion.
Fortunately, the rules on orders of propositions make it
impossible to construct proposition I'mUnprovable in PM.