Comment by cvoss

I think your mistake is your claim that BB(748) is a natural number. For you to know that, you would necessarily have to know an upper bound for the number of steps it takes for the BB-748 machine (whichever machine it is) to halt. But you definitely don't know that.

Related: It's incorrect to claim that each machine either halts or doesn't halt. To know that that dichotomy holds would require having a halting problem algorithm.