@Him Why are you trying to lecture mathematicians about Gödel's theorem? Anyone with a math major has at least heard about this theorem. It's so funny that you just went there and said: "there is a whole subject called "metamathematics" that rigorously studies the limits and properties of mathematical systems" when this is - basically - common sense for mathematicians.
@Him BTW, Gödel's theorems say something VERY precise and people tend to think they can understand it via vague phrases such as: "Godel's result is a statement about "what mathematicians can possibly ever know" as opposed to "what mathematicians currently do know""
There are entire books devoted to this issue, such as this one.
Discussion on answer by Mikhail Katz:…
Discussion on answer by Mikhail Katz:…