User blog comment:Upquark11111/An Explanation of Loader's Number/@comment-11227630-20171208141439/@comment-26454151-20171209043055

1. The accumulate variable is what I call theory. It is declared in the global scope, so it's automatically initialized as 0.

2. If the Y combinator can be assigned a valid type in λPω, then it exists. Sorry for the unsatisfying answer, I'll look into it when I have more time. I read somewhere that λ2 (System F) can type the Y combinator as PI a.(a->a)->a.

3. Derive(99) is actually not that small. Remember the way that theorems are represented? Because of the pair encoding that Loader defined, theorems are represented as power towers of 2. Also, the Derive function also does a search through every proof downwards, and adds those theorems onto the power tower. Small by googological terms, sure, but with slightly longer proofs, the resulting theorem can get ridiculously big.

4. I actually do have a proof of 0:nat with all inference rules listed next to each step. I'll translate that into Loader's bitstream format when I have more time.