BTW: I tried constructing a new number that could not be computed by any other Turing machine using a variant of the diagonilization argument. Basically, enumer

BTW: I tried constructing a new number that could not be computed by any other Turing machine using a variant of the diagonilization argument. Basically, enumerate all Turing machines that generate numbers: Turing machine 1 Turing machine 2 Turing machine 3 ... Now construct a new Turing machine that produce a new number in which the first digit is the first digit of Turing machine 1, the second is the second digit of Turing machine 2, etc. Now add 1 (with wrap-around) to each digit. This will generate a new number that cannot be generated by any of the existing Turing machines. The bug with this argument (as ChatGPT pointed out) is that because of the halting problem, we cannot guarantee that any specific Turing machine will halt, so the constructed program will not halt, and thus cannot actually compute a number.