Given what I see in these discussions, I suspect your use of the word "spontaneously" is a critical issue for you, but also not for me.

None of us exist in a vacuum*, we all react to things around us, and this is how we come to ask questions such as those that led Gödel to the incompleteness theorems.

On the other hand, for "can a program prove it?", this might? I don't know enough Lean (or this level of formal mathematics) myself to tell if this is a complete proof or a WIP: https://github.com/FormalizedFormalLogic/Incompleteness/blob...

* unless we're Boltzmann brains, in which case we have probably hallucinated the existence of the question in addition to all evidence leading to our answer