No, a human could not. Similarly, No human nor computer can name all the real numbers.
Luckily, it turns out that there were only ever be a finite number of questions ever asked, which require only finite answers. Similarly, the halting problem is solvable on finite memory machines, aka the only kind that exist.
That's not exactly what I said though. I didn't say "run yourself." What I said was, "Run this code with this input, and then do something different." A human could certainly do that. A strong AI should certainly be able to.
Hence it can't be defined in terms of code and input.
Not a problem for humans. My code and input are massively inaccessible. Modeling them, especially modelling all the input I've received--could involve modeling a big chunk of the universe. Whether or not they're even deterministic is an open philosophical problem. I'm pretty safe from being asked to "run" myself.
Not so for a simple Strong-AI chatbot, though. If it's just a program a computer can run, if you can enumerate all the input . . . then by definition, it can't handle a simple request that any human would be able to handle. Not strong.
Luckily, it turns out that there were only ever be a finite number of questions ever asked, which require only finite answers. Similarly, the halting problem is solvable on finite memory machines, aka the only kind that exist.