It's more or less the same argument. They key additional ingredient in Godel's proof (besides some slight technicality on soundness vs. omega-consistency) is th

It's more or less the same argument. They key additional ingredient in Godel's proof (besides some slight technicality on soundness vs. omega-consistency) is the encoding of an arithmetic interpreter inside of arithmetic itself. That a bash script can run and capture the output of another bash script is pretty obvious, but the first order logic analogue is not nearly as neat and easy. The argument given here is a modern translation of Turing's work in his original paper introducing Turing Machines and the Halting problem, in which he devotes an entire section to the connection between his proof and that of Godel.