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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: 0s | 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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drdeca ◴[28 Jun 25 19:37 UTC] No.44407506[source]▶
No individual number is uncomputable. There’s no pair of a number and proof in ZFC that [that number] is the value of BB(748). And, so, there’s no program which ZFC proves to output the value of BB(748). There is a program that outputs BB(748) though, just like for any other number.
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cvoss ◴[29 Jun 25 01:34 UTC] No.44409666[source]▶
I think your mistake is your claim that BB(748) is a natural number. For you to know that, you would necessarily have to know an upper bound for the number of steps it takes for the BB-748 machine (whichever machine it is) to halt. But you definitely don't know that.

Related: It's incorrect to claim that each machine either halts or doesn't halt. To know that that dichotomy holds would require having a halting problem algorithm.

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1. drdeca ◴[29 Jun 25 16:02 UTC] No.44414135[source]▶
I don’t know it in a constructive sense, sure.

It’s still true though. I’m not wrong.