When you say "provably", do you mean that in the mathematical sense or in a practical sense?
For the practical sense, other commenters have already replied.
For the mathematical sense, I would say it would be pretty surprising to me if BB(5) had been undecidable -- 5 states and 2 symbols is just so few with which to try to encode undecidable behavior.
However, it's worth noting that as a consequence of the incompleteness theorem, there must be some n such that standard mathematics cannot prove the value of BB(n). And in recent years various people have been working on finding such n and seeing how low they can get the number. The current record[0] is 745.
I expect that record can be probably be brought lower (more easily than the record for computing BB can be brought higher!), but even so, that's a lot of distance between 5 (the highest we know) and 745 (the lowest we know to be unknowable).
[0]This is the current record both for ZFC and for PA, in case you're wondering what I mean by "standard mathematics"... so it at least ought to be possible to bring it down further for PA! I think so far nobody's found a way to do better for PA than for ZFC, even though, like, that has to be possible, right?
For the practical sense, other commenters have already replied.
For the mathematical sense, I would say it would be pretty surprising to me if BB(5) had been undecidable -- 5 states and 2 symbols is just so few with which to try to encode undecidable behavior.
However, it's worth noting that as a consequence of the incompleteness theorem, there must be some n such that standard mathematics cannot prove the value of BB(n). And in recent years various people have been working on finding such n and seeing how low they can get the number. The current record[0] is 745.
I expect that record can be probably be brought lower (more easily than the record for computing BB can be brought higher!), but even so, that's a lot of distance between 5 (the highest we know) and 745 (the lowest we know to be unknowable).
[0]This is the current record both for ZFC and for PA, in case you're wondering what I mean by "standard mathematics"... so it at least ought to be possible to bring it down further for PA! I think so far nobody's found a way to do better for PA than for ZFC, even though, like, that has to be possible, right?