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Then I'm curious: is it possible to know when we know everything there is to know?

Good question. I don’t know!

The answer is no. There are statements for which knowing if they are decidable is undecidable.

I know that \exists undecidable statements, I just don't know if this specific statement is decidable...

Note that being able to do this would imply P=NP.

You should read godel escher bach by hofstadter.