I've been looking for something like this for a long time now. I hope Arbital can be the platform that does it well.

Do you think we could have the actual names of the rules as subheadings or as footnotes? Like, at the end of the "Multiplication doesn't care about the grouping terms" we could write "Mathematicians call this the associative property of multiplication".

I thought so too when I wrote it up; I put it there as a placeholder for a Wikipedia-style initial definition once we find one that's more suitable, because I'm having a hard time thinking of one.

I'm meaning to write there, "different authors have wildly different conventions about what constitutes a whole number". How could that be made clearer?

I'm thinking we should compile a list of sentences or passages that a Math 1 person should be able to read, so people can diagnose for themselves which level of math they're at rather than relying on purely a sentence to describe them.

Is $N$ itself called $\omega$, or just the usual ordering of it?

So effectively all order relations are partial order relations?

Hmm... is this a corollary so much as a converse or an addendum? It would be a corollary (by being the contrapositive) if the statement were "Only extraordinary claims require extraordinary evidence".