I The Smallest Non-Computable Busy Beaver Number: Is it Knowable?

Thread starter johnkclark
Start date Mar 21, 2018
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The first four Busy Beaver numbers are known to be finite and computable, specifically 1, 6, 21, and 107, while the 7918th Busy Beaver number is finite but not computable, as proven by Scot Aaronson. There is interest in determining the smallest non-computable Busy Beaver number, but it remains uncertain if this number can be identified. Discussions highlight the challenges posed by incompleteness in reasoning systems, suggesting that certain Busy Beaver numbers may elude proof of their properties within established mathematical frameworks. The possibility of narrowing the gap between computable and non-computable Busy Beaver numbers is acknowledged, yet it requires significant advancements in understanding and encoding. Ultimately, the existence of the Busy Beaver function raises profound questions about computability in mathematics.

Mar 25, 2018

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stevendaryl said:

I don't know why you keep saying it. ... If you reject set theory and logic, then I don't see the point in your posting in this forum.

There is nothing to add.

As long as there is no common base, which has to be first-order logic unless stated otherwise, I can prove that a discussion is meaningless. Repeating false statements doesn't add facts. Furthermore a distinction like

johnkclark said:

But I tend to think truth is more important than proof.

doesn't make sense in mathematics, because mathematics doesn't search truth, only logical consistent conclusions, which we call proofs, and is philosophically highly undetermined here. For those interested in truth, I refer to https://en.wikipedia.org/wiki/Truth and the links therein - some of which led to other logical systems like constructivism.

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