[pjungwir]

Actually Turing's paper "On Computable Numbers" distinguishes between "automatic machines" (a-machines) and "choice machines" (c-machines), where the latter can pause to ask for input. It seems to me this accounts for your I/O. (I think this is also how you'd add a RNG to a Turing machine.) His paper pretty much only considers a-machines, so I'm curious what is written about c-machines elsewhere. It's unclear to me how much c-machines "change the story." For instance you can't escape Godel's Incompleteness Theorem by adding a finite number of axioms to fill in all the unprovable statements, because there are infinitely many of them.