> The definition of enumeration commonly used in computability does not require that you know when you are done enumerating. Sorry, but this is a non-sequitur.

> The definition of enumeration commonly used in computability does not require that you know when you are done enumerating. Sorry, but this is a non-sequitur. In order to find (by enumeration) the shortest program producing a certain output, you absolutely do need to know when you're done, and thus you do need to solve the halting problem.