I don’t have the understanding, but apparently there are finitary statements which are independent of ZFC. This has been used to prove that BB(745) is independent of ZFC.
Furthermore, Scott aaronson jokingly conjectured that even BB(20) is independent of ZFC. That’s my reference here where we end up needing Aleph* for BB(18).
https://mathoverflow.net/a/26605
Furthermore, Scott aaronson jokingly conjectured that even BB(20) is independent of ZFC. That’s my reference here where we end up needing Aleph* for BB(18).