[JadeNB]

> Well you did say you were okay with set intersection being partial (or I guess also set difference for the more direct analogy).

Good point!

> I don’t have anything against them personally but they’re probably less natural than the empty set being a set.

An interesting idea, which history supports: 0 was considered as a number before negative numbers were, and we still usually consider only "natural sets" and not "negative sets" (except for Schanuel: https://doi.org/10.1007/BFb0084232).

> I might be reading too much into what you’re saying about the empty set though and you just mean we could use the word “set” to mean non-empty set and then say something like “set-theoretic set” to mean what we now mean when we say “set.”

Right, or a different word entirely, just like we refer to 1 only as a number that's not prime, not as a "number-theoretic prime." But, anyway, the analogy was just the first one that sprang to mind; it doubtless has many infelicities that could be improved by a better analogy, if it's not just a worthless idea overall.

[dullcrisp]

Yeah I guess what I got stuck on is that we don’t currently have a word for “a set that’s not a set” (I guess a class?) like we do for a number that’s not a prime but I think I was just lacking linguistic imagination.