[JadeNB]

> And if we treat zero as not a number, it would make division much easier to define. I wrote that sentence as a joke but now I wonder if maybe it’s true. Does addition really need to have an identity?

It probably doesn't, but, if you want to allow negative numbers, then addition is partial unless you have 0. It's perfectly reasonable to disallow negative numbers—historically, negative numbers had to be explicitly allowed, not explicitly disallowed—but it does mean that subtraction becomes a partial operation or, phrased equivalently but perhaps more compellingly, that we have to give up on solving simple equations for x like x + 2 = 1.

[dullcrisp]

Well you did say you were okay with set intersection being partial (or I guess also set difference for the more direct analogy). Maybe not everything needs a solution. (Plus we’ve just gone from division being partial to subtraction being partial…but when I say that I begin to suspect that this argument has been made a lot before and we decided that the negative numbers get to stay. I don’t have anything against them personally but they’re probably less natural than the empty set being a set.)

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.” But that sounds like a mouthful.

[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.