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by srcreigh 715 days ago
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.

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).