No, that's a misconception. If you add BB(748)==‹any N but the correct number›, you get an inconsistent system that will either claim that a machine doesn't halt even though it does (e.g. the real BB champion), or that some machine does halt at N steps, which you can then disprove by enumerating all the turing machines of the relevant number of states for N steps, and showing that no machine halts at that step.
Either way, the only BB axiom you can add without blowing up ZFC is the correct one.
Nope, that’s the crazy thing. busy beaver numbers are simple arithmetical constructions. There is a fact of the matter about each value of BB(n)! We just can’t ever know more than a small handful of them.
No, that's a misconception. If you add BB(748)==‹any N but the correct number›, you get an inconsistent system that will either claim that a machine doesn't halt even though it does (e.g. the real BB champion), or that some machine does halt at N steps, which you can then disprove by enumerating all the turing machines of the relevant number of states for N steps, and showing that no machine halts at that step.
Either way, the only BB axiom you can add without blowing up ZFC is the correct one.
Nope, that’s the crazy thing. busy beaver numbers are simple arithmetical constructions. There is a fact of the matter about each value of BB(n)! We just can’t ever know more than a small handful of them.