> 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!
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.
> 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.)
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!
> 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!
Rather, in this alternate universe, intersection is partially defined.
Yes, but then topology becomes a very tedious exercise because so many proofs rely on the fact that the empty set is contained in every topology, that the empty set is both closed and open, and that intersections frequently yield the empty set. With partially defined intersection you're forced to specially handle every case where two sets might be disjoint.
> Yes, but then topology becomes a very tedious exercise because so many proofs rely on the fact that the empty set is contained in every topology, that the empty set is both closed and open, and that intersections frequently yield the empty set. With partially defined intersection you're forced to specially handle every case where two sets might be disjoint.
Certainly this would be a good objection if I proposed to get rid of empty sets in our universe. (I don't!) But an alternate universe that developed this way would have either just accepted that topology was an inherently ugly subject, or worked out some equivalent workaround (for example, with testing topologies by {0, 1}-valued functions, of which we can take maxima and minima to simulate unions and intersections without worrying about the possibility of an intersection being empty), or else come up with some other approach entirely. (There is, after all, nothing sacred about a topology being specified by its open sets; see the discussion at https://mathoverflow.net/questions/19152/why-is-a-topology-m.... That's how history shook out for us, but it's hardly an inevitable concept except for those of us who have already learned to think about things that way.)
I am not claiming that this would be an improvement (my suspicion is that it would be an improvement in some ways and a regression in others), just that I think that it is not unimaginable that history could have developed this way. It would not then have seemed that the definitions and theorems were artificially avoiding the concept of an empty set, because the mathematical thought of the humans who make those definitions and theorems would simply not think of the empty set as a thing, and so would naturally have taken what seem to us like circuitous tours around it. Just as, surely, there are circuitous tours that we take in our universe, that could be made more direct if we only phrased our reasoning in terms of ... well, who knows? If I knew, then that's the math that I'd be doing, and indeed I see much of the research I do as attempting to discover the "right" direct path to the conclusion, whether or not it's the approach that fits in best with the prevailing thought.
I know that at one time we did mathematics without the number zero and that its introduction was a profound (and controversial) change. The empty set seems like a perfectly natural extension of zero as a concept. Perhaps the universe with no empty set also has no zero? Would be very interesting to see how mathematics would develop without either construct.
> "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!