To elaborate a little further on why BB(30) does have a true value. If BB(30) truly equals some particular numeral m, then there actually does exist a Turing m

To elaborate a little further on why BB(30) does have a true value. If BB(30) truly equals some particular numeral m, then there actually does exist a Turing machine M that has no more than 30 states and does halt with m 1's on its tape. Not only does this machine really exist, but the fact that it halts in this state is true Delta_0 statement, and thus it is trivially provable, you don't even need induction. It is provable in Robinson Arithmetic, it is provable in Peano Arithmetic, and it is provable in ZFC. Thus we have that RA proves "BB(30) ≥ m" and PA proves "BB(30) ≥ m" and ZFC proves "BB(30) ≥ m", etc. By transitivity each of these systems also proves "BB(30) ≥ m₀" for every numeral m₀ less than m. But if m₁ is a numeral larger than m, it is no longer the case that RA proves "BB(30) ≥ m₁" and PA doesn't prove "BB(30) ≥ m₁" and (assuming ZFC is sound) ZFC doesn't prove "BB(30) ≥ m₁" Thus we "can tell" what the true value of BB(30) is because it is the greatest value in which RA (or PA or (presumably) ZFC) proves "BB(30) ≥ m" for the numeral m denoting that value. Equivalently the value of BB(30) is the least value m for which RA, PA, or any other sound system, does not prove "BB(30) ≠ m". We "can tell" this value even if we might not be able prove it in PA, or ZFC, or "ZFC + there exists a Mahlo cardinal". Indeed each different proof system likely can extend the range of BB values that are provable. If you add an axiom "BB(30) = m" for some particular numeral m as an axiom to ZFC and it is the wrong value of m, you will either get an inconsistent system if "m" is too small, or you will get an unsound system if "m" is too large. In case "m" is to large, the unsound system will wrongly prove that certain Turing machines halt that do not actually halt. Specifically it will falsely prove that the Turing machine that searches for (a Turing machine with no more than 30 states that halts with "m" 1's on it tape) will halt, when such a machine does not actually halt.