> it is always possible to compute BB(k) for a specific and arbitrary k This is not true. There is a number N where BB(n) is incomputable for all n > N. This

> it is always possible to compute BB(k) for a specific and arbitrary k This is not true. There is a number N where BB(n) is incomputable for all n > N. This fact tie back to being able to solve the Halting Problem, in various clever ways if you knew BB(n). The value of BB(744) depends on whether the Riemann Hypothesis is true or false. This means that it cannot be computed in general. The Riemann Hypothesis being either true or false are both consistent with ZFC. The value of BB(748) depends on the self-consistency of ZFC.