I have problems understanding Gödel's incompleteness theorem. I presume I have a misunderstanding of some phrase or I have to look closer at the meaning of some detail.
Gödel's second incompleteness theorem states that in a system which is free of contradictions, this absence of contradictions is neither provable nor refutable.
If we would find a contradiction, then we would have refuted the absence of contradictions. Gödel's theorem states that this is impossible. So we will never encounter a contradiction. Doesn't that mean that no contradiction exists? (If one existed, we could encounter it.) So this seems to be a proof that no contradiction exists. Thus, we proved the absence of contradictions, which contradicts the second incompleteness theorem.
This is a contradiction which I can't solve.
It does not state that the absence of contradictions is not refutable. However "Gödel's second incompleteness theorem states that in a system which is free of contradictions, this absence of contradictions is not provable" is accurate.
It may be the case that ZFC for example is consistent and also that "ZFC is inconsistent" is a theorem of ZFC.