This question is somehow of a follow up to to this other one, and it's something that has bugged me for a while.
I understand the notion that there's no "absolute truth" in math, in the sense that every theorem follows from an assumed set of axioms. The typical example is euclidean vs non-euclidean geometries.
The question I linked has an comment asking for clarification on the notion of an absolute truth. The way I understand the distinction between an non-absolute (claim of) truth and an absolute one is this: A non-absolute claim C is really just a short way of claiming: ZFC (or some other set of axioms we all agree upon) implies C. And an absolute claim C just claims C and nothing else.
So here is the thing that bugs me, if the claim 1+1=2 is just a short way of claiming "ZFC implies that 1+1=2" then is this claim itself absolute, or is it really claiming something like "XYZ implies (ZFC implies 1+1=2)"? Or, to put it another way, if you prove to me that 1+1=2 using ZFC axioms, and then I ask you to prove that your proof is correct, which axioms would you use?
I guess one option would be for XYZ to be just ZFC itself, so that "ZFC implies (ZFC implies 1+1=2)", but that's just equivalent to "ZFC implies 1+1=2", which would make the claim an absolute one, given the distinction I made above. The alternative is to have XYZ refer to some other set of axioms, but then either this claim would be absolute, or we'd need yet another set of axioms, ad infinitum.
Is there a simple answer to this, or is this a topic I can dig some more about? I honestly don't even know how to google this specific question (as opposed to the one I linked). Or am I just missing something obvious and the question makes no sense?