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BusyBeaver(6) Is Quite Large

(scottaaronson.blog)
271 points bdr | 1 comments | | HN request time: 1.196s | source
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Scarblac ◴[] 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 ◴[] 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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boothby ◴[] No.44408600[source]
Individual numbers can be uncomputable! For example, take your favorite enumeration of Turing machines, (T1, T2...) and write down a real number in binary where the first bit is 0 if T1 halts and 1 otherwise, second bit is 0 if T2 halts... clearly this number is real and between 0 and 1, but it cannot be computed in finite time.
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1. Scarblac ◴[] No.44411344[source]
That's a number in R, obviously most of them are uncomputable (there is a countable number of Turing machines).

But for every natural number n there is a trivial Turing machine that just prints n and then halts.