Let's take something like Gödel's First Incompleteness Theorem.
I may not know anything about it. I might have heard something vague about it being about "the limits of mathematics". I might know somewhat more specifically that it has to do with statements that are unprovable, but not much more beyond that. I may know that it's about arithmetic. I may know that it's related to the Halting Problem. All of that is possible without remembering the exact wording of the theorem with all its conditions. It would be easy to forget, for example, that Presburger arithmetic is complete and that you need addition and multiplication for incompleteness. Or to remember that Gödel's original theorem requires omega-consistency[0] and that only Rosser's modification makes it work for general consistent theories. And even if you do remember the exact statement of the theorem - would you be able to reconstruct the exact proof (or at least one proof)? If so, in how much detail? Would you remember the exact trick involved in showing that Robinson Arithmetic can describe all mu-recursive functions? And in all of this, are you going to be sure you have no gaps or slight misunderstandings? At which point would you say you "know" Gödel's First Incompleteness Theorem?
[0] Case in point, I actually had to use google to verify that the condition really was called omega-consistency. This was the name I remembered, but I was only about 80% sure of it. And if there's any mistake in my explanations above, then that also wouldn't massively surprise me.
You haven't answered my question: at which point do you say you "know" Gödel's first theorem? Or for an easier example: at which point do you "know" Java? Do you need to understand all the intricacies of the class loader? Or at which point do yoh "know" English? Only when you know every word in the OED?
I think your definition of knowledge is entirely unworkable. It requires you to split every fact into a million little subfacts, it doesn't account for vague recollection, for uncertainty, etc., and it makes it so that basically almost nobody ever knows anything of value.
I think this totally flies in the face of how people actually know things and how people apply their knowledge. The vague and fuzzy understanding that comes with being an expert in a field - IOW having absorbed the principles without remembering the minutiae - is exactly what constitutes valuable knowledge.
I may not know anything about it. I might have heard something vague about it being about "the limits of mathematics". I might know somewhat more specifically that it has to do with statements that are unprovable, but not much more beyond that. I may know that it's about arithmetic. I may know that it's related to the Halting Problem. All of that is possible without remembering the exact wording of the theorem with all its conditions. It would be easy to forget, for example, that Presburger arithmetic is complete and that you need addition and multiplication for incompleteness. Or to remember that Gödel's original theorem requires omega-consistency[0] and that only Rosser's modification makes it work for general consistent theories. And even if you do remember the exact statement of the theorem - would you be able to reconstruct the exact proof (or at least one proof)? If so, in how much detail? Would you remember the exact trick involved in showing that Robinson Arithmetic can describe all mu-recursive functions? And in all of this, are you going to be sure you have no gaps or slight misunderstandings? At which point would you say you "know" Gödel's First Incompleteness Theorem?
[0] Case in point, I actually had to use google to verify that the condition really was called omega-consistency. This was the name I remembered, but I was only about 80% sure of it. And if there's any mistake in my explanations above, then that also wouldn't massively surprise me.