But that sentence isn't self-contradictory like "This is a lie", it is just self-referential, like "This sentence has five words". It does have a well-defined meaning and is decidable for all hypothetical consistent people other than hypothetical consitentified Stuart Armstrong.
You're right, I didn't think that one through, thanks!
I still think the interesting thing is the potential for writing down a mathematical statement humanity can't decide, not an English one that we can't decide even though it is meaningful, but I'll shut up about the question for now.
Building on the very bad Gödel anti-AI argument (computers's are formal and can't prove their own Gödel sentence, hence no AI), it occurred to me that you could make a strong case that humans could never recognise a human Gödel sentence. The argument goes like this:
Now, the more usual way of dealing with human Gödel sentences is to say that humans are inconsistent, but that the inconsistency doesn't blow up our reasoning system because we use something akin to relevance logic.
But, if we do assume humans are consistent (or can become consistent), then it does seem we will never knowingly encounter our own Gödel sentences. As to where this G could hide and we could never find it? My guess would be somewhere in the larger ordinals, up where our understanding starts to get flaky.