Hal Ruhl wrote:
To answer these questions it seems necessary to inject information into your theory beyond what may already be there - the sentences - ...
Right. This indeed follows from Goedel's incompleteness.
Here you appear to me to be saying that your theory is indeed subject to random external input.
"Random" because we do not know if the set of sentences is consistent in its current state and if incomplete it can be added to. How can it be added to in a manner that is consistent with the existing state? .
We can choose whether a Godel statement should be judged true or false by consulting our "model" of arithmetic. See this post of mine on the use of "models" in mathematics from the thread "Something for Platonists" (you can see the other posts in the thread by clicking 'View This Thread' at the top):