[JadeNB]

> One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1".

This is true and compelling as things developed, but I think it's an explanation of where history brought us, rather than a logical inevitability. For example, I can easily imagine, in a different universe, teachers patiently explaining that we declare that the empty set is not a set, to avoid complicating theorems, proofs, and exposition by the endless repetition of "non-empty set."

(I agree that this is different, because there's no interesting "unique factorization theorem" for sets, but I can still imagine things developing this way. And, indeed, there are complications caused by allowing the empty set in a model of a structure, and someone determined to do so can make themselves pointlessly unpopular by asking "but have you considered the empty manifold?" and similar questions. See also https://mathoverflow.net/questions/45951/interesting-example....)

[tux3]

That's an interesting thought, but I think that'd break the usual trick of building up objects from the empty set, a set containing the empty set, then the set containing both of those and so forth.

That universe would be deprived from the bottomless wellspring of dryness that is the set theoretic foundations of mathematics. Unthinkable!

[JadeNB]

> That universe would be deprived from the bottomless wellspring of dryness that is the set theoretic foundations of mathematics. Unthinkable!

"Wellspring of dryness" is quite a metaphor, and I take it from that metaphor that this outcome wouldn't much bother you. I'll put in a personal defense for set theory, but only an appeal to my personal taste, since I have no expert, and barely even an amateurish, knowledge of set theory beyond the elementary; but I'll also acknowledge that set-theoretic foundations are not to everyone's taste, and that someone who has an alternate foundational system that appeals to them is doing no harm to themselves or to me.

> That's an interesting thought, but I think that'd break the usual trick of building up objects from the empty set, a set containing the empty set, then the set containing both of those and so forth.

In this alternate universe, the ZF or ZFC axioms (where C becomes, of course, "the product of sets is a set") would certainly involve, not the axiom of the empty set, but rather some sort of "axioms of sets", declaring that there exists a set. Because it's not empty, this set has at least one element, which we may extract and use to make a one-element set. Now observe that all one-element sets are set-theoretically the same, and so may indifferently be denoted by *; and then charge ahead with the construction, using not Ø, Ø ∪ {Ø}, Ø ∪ {Ø} ∪ {Ø ∪ {Ø}}, etc. but *, * ∪ {*}, * ∪ {*} ∪ {* ∪ {*}}, etc. Then all that would be left would be to decide whether our natural numbers started at the cardinality 1 of *, or if we wanted natural numbers to count quantities 1 less than the cardinality of a set.

[tux3]

I should apologize if I came off too colorful, I only meant it as a friendly jab - but my bias is showing :)

Appreciate the defense of set theory, I can't find a problem with it!

[JadeNB]

No apology needed! It's all in fun, and we might as well enjoy the discussion.

[murderfs]

A good example of this is the natural numbers. Algebraists usually consider zero to be a natural number because otherwise, it's not a monoid and set theorists want zero because it's the size of the empty set. My number theory textbook defined natural numbers as positive integers, but I'm not entirely sure why.

[mathgeek]

> My number theory textbook defined natural numbers as positive integers, but I'm not entirely sure why.

Since both the inclusion and exclusion of zero are accepted definitions depending on who’s asking, books usually just pick one or define two sets (commonly denoted as N_0 and N_1). Different topics benefit from using one set over the other, as well as having to deal with division by zero, etc. Number theory tends to exclude zero.

[JadeNB]

> commonly denoted as N_0 and N_1

Oh my, it had never occurred to me that one could disagree, not just about whether the natural numbers include 0 or don't, but also about how to denote "natural numbers with 0" and "natural numbers without." Personally, I'm a fan of Z_{\ge 0} and Z_{> 0}, which are a little ugly but which any mathematician, regardless of their preferred conventions, can read and understand without further explanation.

[mathgeek]

Yep, lots of ways to denote these sets. It’s not a disagreement but rather a preference (although certainly some folks will gladly disagree).

[ForOldHack]

Number theory includes zero as the identity element for addition, much as 1 is the identity element for multiplication.

I am totally assuming you knew this already.

[mathgeek]

For the sake of making an easy transition to the monoid, yes. Personally a fan.

[gus_massa]

Many (most?) results are easier to write if you allow the empty set. For example:

"The intersection of two sets is a set."

[JadeNB]

> Many (most?) results are easier to write if you allow the empty set. For example:

> "The intersection of two sets is a set."

Many results in set theory, yes! (Or at least in elementary set theory. I'm not a set theorist by profession, so I can't speak to how often it arises in research-level set theory.) But, once one leaves set theory, the empty set can cause problems. For the first example that springs to mind, it is a cute result that, if a set S has a binary operation * such that, for every pair of elements a, b in S, there is a unique solution x to a*x = b, and a unique solution y to y*a = b, then * makes S a group ... unless S is empty!

In fact, on second thought, even in set theory, there are things like: the definition of a partial order being a well ordering would become simpler to state if the empty set were disallowed; and the axiom of choice would become just the statement that the product of sets is a set! I'm sure that I could come up with more examples where allowing empty sets complicates things, just as you could come up with more examples where it simplifies them. That there is no unambiguous answer one direction or the other is why I believe this alternate universe could exist, but we're not in it!

[chongli]

I don’t see why it’s a problem that the empty set cannot be a group. The empty set, being empty, lacks an identity element. Thus all groups are non-empty.

The same is true for any structure which posits the existence of some element. Of course it cannot be the empty set.

[JadeNB]

> I don’t see why it’s a problem that the empty set cannot be a group. The empty set, being empty, lacks an identity element. Thus all groups are non-empty.

It's not necessarily a problem that the empty set cannot be a group. (Although the only reason that it cannot is a definition, and, similarly, the definition of a field requires two distinct elements, which hasn't stopped some people from positing that it is a problem that there is then no field with one element.)

The problem is that there's a natural property of magmas (sets with binary operation), namely the uniquely solvability condition I mentioned, that characterizes "group or the empty set," which is more awkward than just characterizing groups. Or you may argue, fairly, that that's not a problem, but it is certainly an example where allowing the empty set to be a set complicates statements, which is all that I was meaning to illustrate. Hopefully obviously, without meaning seriously to suggest that the empty set shouldn't be a set.

(I remembered in the course of drafting this comment that https://golem.ph.utexas.edu/category/2020/08/the_group_with_... discusses, far more entertainingly and insightfully than I do, the characterization that I mention, and may have been where I learned it.)

[chongli]

If you don’t allow the empty set to be a set then you break the basic operations of set theory. For example, to show two sets are disjoint you compare their intersection with the empty set.

In an alternative axiomatization (without the empty set) you’re going to need to create some special element which belongs to every set and then your definition of disjoint sets is that their intersection is equal to the trivial set containing only the special element. What a clumsy hack that would be!

[JadeNB]

> If you don’t allow the empty set to be a set then you break the basic operations of set theory. For example, to show two sets are disjoint you compare their intersection with the empty set.

You certainly can do that, but it's not the only way. Even in this universe, I would expect to show that concrete sets A and B are disjoint by showing x ∈ A → x ∉ B, which makes perfect sense even without an empty set.

> In an alternative axiomatization (without the empty set) you’re going to need to create some special element which belongs to every set and then your definition of disjoint sets is that their intersection is equal to the trivial set containing only the special element. What a clumsy hack that would be!

Rather, in this alternate universe, intersection is partially defined. Again, even in this universe, we're used to accepting some operations being partial!

[dullcrisp]

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? Maybe we just saw that multiplication has an identity and got a bit carried away. I’m not too sure about this negative number business while we’re at it. Could be that we just took a wrong turn somewhere.

[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.