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BusyBeaver(6) Is Quite Large

(scottaaronson.blog)
271 points bdr | 1 comments | 28 Jun 25 16:53 UTC | HN request time: 0.231s | source
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Scarblac ◴[28 Jun 25 17:29 UTC] No.44406478[source]▶
It boggles my mind that a number (an uncomputable number, granted) like BB(748) can be "independent of ZFC". It feels like a category error or something.
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tromp ◴[28 Jun 25 19:20 UTC] No.44407378[source]▶
What makes BB(748) independent of ZFC is not its value, but the fact that one of the 748-state machines (call it TM_ZFC_INC) looks for an inconsistency (proof of FALSE) in ZFC and only halts upon finding one.

Thus, any proof that BB(748) = N must either show that TM_ZF_INC halts within N steps or never halts. By Gödel's famous results, neither of those cases is possible if ZFC is assumed to be consistent.

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woopsn ◴[29 Jun 25 00:54 UTC] No.44409464[source]▶
Does the fact that BB(k)=N is provable up to some k < 748 mean that all halting problems for machines with k states are answered by a proof in ZFC?
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1. ◴[29 Jun 25 15:17 UTC] No.44413798[source]▶