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Well, OK, I suppose you could add something like 'BB(8000) is 6,' but once you go beyond ZFC, you're going beyond anything required for computation as we know i

Well, OK, I suppose you could add something like 'BB(8000) is 6,' but once you go beyond ZFC, you're going beyond anything required for computation as we know it.

Kranar 4y ago
No this is also wrong. You would not be able to add an axiom like 'BB(8000) is 6' or just any arbitrary value without introducing an inconsistency. The reason for the independence of BB(8000) from ZFC is precisely because there is some non-standard model of arithmetic for which a Turing Machine halts in that non-standard model but does not halt in the standard model. The only axioms you could introduce to ZFC that would not introduce an inconsistency would be one that eliminates that non-standard model of arithmetic without also eliminating the standard model of arithmetic. https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic https://en.wikipedia.org/wiki/Non-standard_model_of_arithmet... None of this requires going beyond any kind of notion of computation and any value that is computed for one such extension would necessarily have to be the same value among all consistent extensions of ZFC. It's not like you could have one extension of ZFC where BB(8000) is X and another extension where BB(8000) is Y without one of those extensions being inconsistent. There are plenty of extensions to ZFC that add new axioms for the sake of exploring niche mathematical ideas. ZFC is nice in that it's easily motivated and can serve as a well understood foundation whose proofs can be stated without reference to additional hypotheses. It also satisfies an unbelievably broad set of mathematics, but mathematicians studying set theory often extend ZFC with additional axioms such as large cardinal axioms, Neumann/Bernays/Godel (NBG) set theory is another common extension, Kelley Morse (KM) set theory. The latter two extensions are even able to prove the consistency of ZFC. As I referenced in Scott's quote, BB(8000) is almost certainly computable by adopting one of the large cardinal axioms.

Well, OK, I suppose you could add something like 'BB(8000) is 6,' but once you go beyond ZFC, you're going beyond anything required for computation as we know it.

Kranar 4y ago

No this is also wrong. You would not be able to add an axiom like 'BB(8000) is 6' or just any arbitrary value without introducing an inconsistency. The reason for the independence of BB(8000) from ZFC is precisely because there is some non-standard model of arithmetic for which a Turing Machine halts in that non-standard model but does not halt in the standard model. The only axioms you could introduce to ZFC that would not introduce an inconsistency would be one that eliminates that non-standard model of arithmetic without also eliminating the standard model of arithmetic. https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic https://en.wikipedia.org/wiki/Non-standard_model_of_arithmet... None of this requires going beyond any kind of notion of computation and any value that is computed for one such extension would necessarily have to be the same value among all consistent extensions of ZFC. It's not like you could have one extension of ZFC where BB(8000) is X and another extension where BB(8000) is Y without one of those extensions being inconsistent. There are plenty of extensions to ZFC that add new axioms for the sake of exploring niche mathematical ideas. ZFC is nice in that it's easily motivated and can serve as a well understood foundation whose proofs can be stated without reference to additional hypotheses. It also satisfies an unbelievably broad set of mathematics, but mathematicians studying set theory often extend ZFC with additional axioms such as large cardinal axioms, Neumann/Bernays/Godel (NBG) set theory is another common extension, Kelley Morse (KM) set theory. The latter two extensions are even able to prove the consistency of ZFC. As I referenced in Scott's quote, BB(8000) is almost certainly computable by adopting one of the large cardinal axioms.