Yes and then you can do self-referential statements again which leads you to the conclusion that Gödels theorems are provable only in a system that cannot prove its own consistency.
From the common sense the whole situation looks paradoxical, indeed. To prove consistency of some theory we have to use some stronger theory, to prove consistency of that theory we need even stronger theory and so on.
Perhaps, the best way to realize why there is no paradox is the following:
Our memory is finite. As well as our thinking. But the total number of true facts about mathematics is infinite. By constructing theories we are trying to compress the infinite number of true facts into some finite form. Godel's theorem says it's impossible. And it looks quite natural from this perspective.
Honestly it would be weird if the incompleteness theorems don't apply to themselves.
proposition I'mUnprovable into foundations makes the
foundations inconsistent.