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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: 0.205s | 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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1. edanm ◴[29 Jun 25 10:38 UTC] No.44411963[source]▶
> 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.

Isn't it more accurate to say that any proof that BB(748) = N in ZFC must either show that TM_ZF_INC halts within N steps, or never halts?

Meaning, it's totally possible to prove that BB(748) = N, it just can't be done within the axioms of ZFC?