DOUBLE EDIT: I didn’t actually describe the insight required in proving the completeness theorem and compactness theorem. He used the former to prove the latter (because all proofs are finite, and if something is inconsistent, it can’t have any models, so every statement must be proveable). I don’t know what his key insight was for the compactness theorem, as I’ve only proven it via the completeness theorem.
That logic could be aritmatized, i.e. you could use the simplest arithmetic system to mimic logical deduction, and hence could use the logic of the simplest arithmetic system to talk about itself, letting you get all those nasty self reference paradoxes. Doing that was highly non trivial. Of course, he also proved a bunch of other important theorems like the completeness theorem: if every model of a set of axioms implies some some statement T, then the set of axioms must prove that statement. And also the compactness theorem: a set of logical statements is consistent iff every finite subset is consistent.
I assumed that Richard meant the incompleteness theorems, of which the first is quite easy to boil down to one insight (the liar paradox but with provability).
EDIT: I am talking about first order logic here.
DOUBLE EDIT: I didn’t actually describe the insight required in proving the completeness theorem and compactness theorem. He used the former to prove the latter (because all proofs are finite, and if something is inconsistent, it can’t have any models, so every statement must be proveable). I don’t know what his key insight was for the compactness theorem, as I’ve only proven it via the completeness theorem.
That logic could be aritmatized, i.e. you could use the simplest arithmetic system to mimic logical deduction, and hence could use the logic of the simplest arithmetic system to talk about itself, letting you get all those nasty self reference paradoxes. Doing that was highly non trivial. Of course, he also proved a bunch of other important theorems like the completeness theorem: if every model of a set of axioms implies some some statement T, then the set of axioms must prove that statement. And also the compactness theorem: a set of logical statements is consistent iff every finite subset is consistent.
I assumed that Richard meant the incompleteness theorems, of which the first is quite easy to boil down to one insight (the liar paradox but with provability).