Actually, Gödel's theory is fairly accessible compared to, say, the General Theory of Relativity. All you have to know that the process of proving a statement is a fairly mechanical, step-by-step process. At that point, "statements that can be proven true" and "statements that be proven false" are two fairly defined sets. At that point, "there are some true statements that cannot be proven true" is moderately clear.…
i don’t know if it’s accurate, but i get the sense godel liked finding pathologies. for example, his “godel metric” as a solution to einstein’s general relativity showcased “unphysical” possibilities but was still a legitimate solution.