As a child, I noticed that the proofs of mathematical theorems were esoteric knowledge, known only to a few adults. I struggled to follow even the simplest proofs, and hoped that one day I might learn to create a proof or two of my own. This was not only a high aspiration, but a dangerous one. I saw no reason why certain knowledge of a true fact would be accessible to humans via proof. Any-one who embarked on the quest to find a proof risked embarking on a doomed quest to find a non-existent proof.

For me Gödel's completeness theorem is the miracle. Every valid statement has a proof. Amazing!

Aim a little higher, every true statement, and there might not be a proof. It is no surprise to me that this is true. It is a big surprise to me that Gödel was able to prove it; ordinary proofs are hard to find, and proofs of the limits of provability presumably even more deeply hidden.

Non-standard models of arithmetic are weird. Theorems that are true of the standard model of arithmetic and false in some non-standard model must surely be convoluted and obscure. The first order version of the Peano axioms nail down the integers, not perfectly, but very well. Restricting one-self to theorems that are true in all models of them, even the weird, non-standard ones, feels like a very minor restriction. Gödel's completeness theorem raises the possibility of writing a computer program to find a proof of every theorem that isn't convoluted and obscure. Gödel completeness theorem is the really big deal.

Except it isn't. That computer program turns out to be one of those wretched tree search ones that soon bogs down. The real problem turns out to be the combinatorial explosion inherent in unstructured search through the Herbrand universe. One needs Unification and one needs a still missing ingredient to give search a sense of direction. The interesting questions are about the "sense of direction" that lets us find some of the deeply hidden proofs that do exist. Will LLM's help? The answer will be interesting, either way.