Is there a analysis conjecture proven to be unprovable or a proof is non-existence?
So, is it once a math history milestone
There is a large number of statements in analysis that have been proved independent of the axioms of ZFC. For a partial list, see this. A fair number of them are questions that had already been asked, and worked on seriously, long before they were proved independent.
Added: ZFC is Zermelo-Fraenkel set theory, with Axiom of Choice added. It is currently, and has been for quite a while, the standard "background theory" for most of mathematics.