Many such lists exist. For example: https://en.wikipedia.org/wiki/List_of_statements_independent...
My earlier point is not that we could know for sure that our struggles are due to being up against such a wall--just that our ability to suspect as much is relevant to how we allocate our time.
What do you mean by this? The Goedel sentence of a system is logically independent from the system by the standard meaning of "logical independence".