I had just begun learning about set theory and I was wondering - especially after searching Gödel's incompleteness theorem - whether anyone tried proving that any of Hilbert's OPEN (i.e. unsolved, as of 2018) problems are independent of some chosen axioms (e.g. ZFC), like the Continuum Hypothesis?
Edit: I am asking specifically for those problems that are still unresolved, e.g. 8th, 12th, or 16th.