it's just so damn satisfying

Today, I covered roughly 6 white boards with definitions, lemma, theorems, proofs, and scary set-theoretic notation, knowing that I could write in a compressed/abbreviated form while speaking the sentence aloud and everyone would either follow me or be confident enough to ask for clarification ("What does 'QED' stand for?"). After having built up the importance of completeness to them since the beginning of my 2nd year course last year, we were finally in a position to prove the completeness of K, T, D, B, S4, S5, and other systems. (Proof of the completeness of K, verbatim from what I wrote on the board: "trivial".)

I had to pause the students for a moment to reflect on this. This time last year, if they had come into a room with all of that scribbled on the boards, they would've fled in terror or said "there's no way I can do that, that's way too hard." Instead, one of them commented on how it seemed almost a let down, how easy it was to prove completeness results, though it's only in retrospect, after 6 chapters and almost an entire term's worth of work proving other results along the way, that it is so easy.

But that's how logic should be: If you set up the definitions right, the results should be almost automatic. The question, then, is getting the right definitions...