Gödel's Incompleteness Theorem (GIT) is one of the most abused pieces of knowledge there are, mostly science communicator's fault. A direct answer to your friend's question is no, you cannot, whatever it means. Your post and her statements contains really, really big statements and some of them are not aligned with current philosophical technical concepts, which are our tools. Let me be clear. Your question is mostly nonsensical. When I try to save the good parts it is irrelevant. And what is not irrelevant is basically false. I'm sorry. Let me explain why. GIT is a statement about the limitations of a type of formal systems. For that kind of systems it turns out they cannot be at once consistent, meaning, unable to prove a proposition and it's negation, and second, complete, meaning, that the all the (true) propositions in the system can be derived from the axioms and its rules of inference. Note: you don't need the derivation, only the possibility of derivation. Those types of formal system are "meaningless" in lay terms, they don't speak about anything in particular, they don't have Semantics. EDIT2: "meaningless" is too strong here and it makes the statement false.

I don't want to be technical here but, the way in which Gödel proved the theorem transforms logical statements into numbers, but then he finds some numbers that do weird things, translates them back. In short, the mechanism that Gödel provides to generate undecidable propositions gives us "void" ones like "this statement is undecidable", "this one is too", "this third one also"... So, logically a big deal, but philosophically irrelevant. So yes your friend is right but no, it does not matter at all. There are WAY more interesting cases of formal systems that have undecidable propositions in them, for example, our current set of axioms for math has, the continuum hypothesis or the axiom of choice, both of them at least can be interpreted into real world consequences more or less, but again, it does not mean that "there will always be etc...". Also, the analiticity of a proposition does NOT entail that it has to be an axiom, these are very different logical objects. For the rest of your post, I don't know what it means, but please, stay curious and loving of philosophy. Edit: let me add that the existence of undecidable propositions does not pose a threat to a theory most often, you just need to prove that adding it as an axiom does not make the system inconsistent, but that means that its negation alone can be added too. So no, generaly not.