You can not know what you don't know, and learn gobs of new stuff at the same time.
That's the natural state of most learners.
More succinctly:
I can still teach Simpson's Rule in a Calculus class to a young man or woman who subscribes to Creationism, or the Flat Earth theory. None of their false knowledge affects their innate ability to take in new knowledge.
https://blog.research.google/2024/01/can-large-language-mode...
Negation in the training data, exhaustion, or unknowable unknowns like future events is about an LLM will ever be able to say it "doesn't know" reliably.
LLMs have no concept of truth or correctness really, they are just probabilistic appending to output to match learned ideal patterns.
I think they do have those concepts, like they have many others but they are not bound to them in any way.
But at their core, they're going to generate the most plausible continuation, even if that isn't very plausible (like non-existent functions). I think this can be improved (playing with the temperature in self hosted models depending on the nature of the task e.g.), but it doesn't seem fully solvable with current LLMs.