Gödel's original proof of the First Incompleteness theorem relies on Gödel numbering. Now, the use of Gödel numbering relies on the fact that the Fundamental Theorem of Arithmetic is true and thus the prime factorization of a number is unique and thus we can encode and decode any expression in Peano Arithmetic using natural numbers.
My question is, how can we use a non-trivial result like the Fundamental Theorem of Arithmetic in the meta-language that describes Peano Arithmetic, when, the result actually requires a proof from within Peano Arithmetic itself, not like other trivial things we believe to be true (i.e. existence of natural numbers and the axioms for addition and multiplication which we want to interpret in the natural way - Platonism)?
I understand we could instead use a different way of enumeration, e.g. a pairing function and the Chinese Remainder Theorem or simply string concatenation, but, then the need for a proof of uniqueness when encoding and decoding remains and in general, I am interested in the structure of Gödel's original proof.
Basically, I have two ideas of how it might be possible to resolve this:
Prove the Fundamental Theorem of Arithmetic within a different sound (?) system.
Maybe there is nothing needed to be done to 'resolve' this, because I am just misinterpreting something and it is actually acceptable to use provable sentences of PA in the meta-language.
Any ideas about this?
EDIT: I have realized how to make my question less confusing:
Say PA proves FTA. Then if we only assume PA is consistent, that does not rule out the possibility of FTA being false. Now, if FTA is false, then PA and the meta-language too includes a false statement and thus the whole proof is useless.
How is this resolved? Is it maybe related to the fact the originally we actually assume $\omega$-consistency and obviously, for each natural number $n$ separately its unique prime factorization can simply be found algorithmically?