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Just because your quasi-mathematical model breaks, that doesn't necessarily mean this has anything to do with reality. We have high confidence that mathematical models in physics and chemistry often do have consequences and say something about reality. Economics has an interesting track record here as well. In the realm of ethics, politics, and ideology? Ehhh...not so much.

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Ideology has almost nothing at all to do with mathematics, you got that right.

It's interesting that contradiction, that is inconsistency, is a hint that a mathematical system does not model something in reality, because we don't want to believe such a thing can exist. But it is about as hard to work out the kinks of inconsistency from of a mathematical model as it is from ideology, law, ethics, and other such systems that we construct to guide our behavior. So I would say that METAmathematics has a lot to say about us and our ability to construct consistent systems-- it's really hard.

Freedom is one such example since it cannot belong to everyone when competing interests are at play. That might be a strange way to slice it though, since the language we use to describe freedom is probably not as well defined as we think.

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