Richard: Boris's step (1) claims that BBB(n) is bounded by n+c for some constant c. It is not obvious to me either that this is true or that this is false. c is the size of a universal Turing machine which, in addition to simulating its argument, keeps track of how many steps it simulates. BB(n) is the maximum number of steps that any halting TM of size n performs, therefore there is some particular TM of size n that performs BB(n) steps. Give that TM as input the the aforementioned UTM, and it will compute BB(n). The uncomputability of BB prevents you from finding the right TM, and from proving it correct if it's handed to you by a magic light in the sky. But the TM still exists even if you'll never know which it is.