# How many empty sets are there?

Thre are many sets with a single object, for example the set which only contains the statue of liberty or the set which contains my copy of Catch-22.

But how many sets are there that contain nothing?

On the face of it there seems to be just the one. The empty set is unique in its particularity as it contains nothing to distinguish itself.

On the other hand one could say that there are many empty sets, and that they are all identical - as in isomorphic - but not identical - as in they are exactly the same. To make this more concrete, consider Jamila & Jemima each carrying a basket, and in Jamilas basket there is an emerald, and in Jemimas basket there is a ruby. So the contents of their baskets are not identical. If they empty out their baskets. The contents of the basket are now identical (isomorphic) but the two baskets are not identical.

Which is the more accurate view?

Is it correct to say in formalised ZFC that there is just one empty set?

Is it correct to say that in the categorical set theory, say ETCS, that there are many empty sets but they are all isomorphic?

I am less interested in the 'formal' parts of the question as opposed to the conceptual arguments about uniqueness or non-uniqueness of empty 'sets' - where sets shouldn't be thought of as in ZFC

It does turn out that in ETCS there are more than one empty set (and since they are intial) they are also all isomorphic to each other. See the anser to this question

• There is exactly one empty set according to the most usual axiomatization of set theory (viz. ZFC). That there is one is guaranteed by the Empty Set Axiom. That it is unique is guaranteed by the Extensionality Axiom. – Hunan Rostomyan Feb 3 '14 at 7:01
• I am not sure if it is quite right to say that their IS more than one empty set in ETCS. My impression is that their COULD be more than one empty set, but it COULD also be unique. I may be misunderstanding something however. – Baby Dragon Feb 6 '14 at 5:43
• In ETCS you have an initial object that plays the role of an empty set. Because the definition is based on a universal property (for any object A there is exactly one morphism from the initial object to A), an initial object is unique up to isomorphism. Since in category theory you are anyway interested to uniqueness up to isomorphism, you speak of the initial object. – FWE Aug 13 '18 at 10:22
• ..though it is correct that in a concrete structure (world / interpretation / model) there might be several initial objects, however all guaranteed to be isomorphic. – FWE Aug 13 '18 at 10:26
• – user9166 Aug 21 '18 at 3:56

## 5 Answers

In ZFC we have two axioms that settle that question:

Empty Set. There is a set that contains nothing.

Extensionality. If sets A and B have exactly the same members, then A = B.

The Empty Set Axiom allows us to conclude that there is an empty set. Suppose there are two empty sets A and B. Vacuously, every member of A is a member of B (since A has no members), and vice versa. Therefore by the Axiom of Extensionality it follows that A and B are the same set. These axioms (existence and extensionality) thereby guarantee that there is exactly one empty set (usually denoted by '∅').

I'm not at all familiar with ETCS, so I won't comment on that part of the question.

Mozibur has already given a satisfying answer to the conceptual question, so I'll quote:

The empty set is unique in its particularity as it contains nothing to distinguish itself.

Since ZFC distinguishes sets by their contents, any two empty sets will be indistinguishable because neither can contain anything that the other doesn't. In the baskets analogy, since the two empty baskets have different locations we want to say that they are two distinct empty baskets. But since for ZFC sets are not located in space-time, the two baskets are identical because it cannot, in the language of ZFC, be said something true about one that's false about the other.

Extensionality, in ZFC, trims the universe by identifying any two things that have the same members, allowing us to unambiguously name such things as the empty set ∅, the intersection of two sets A ∩ B, the ordered pair of two sets (A,B), and so on. In a universe with many empty sets, the definition of '∅' would get more complicated because we would have to identify it with the class of all empty sets, and that complication would crawl all the way up the definitional hierarchy.

• Ok - great, that deals with the question in the framework of ZFC? What about about the conceptual question? Do you have any thoughts on that? – Mozibur Ullah Feb 3 '14 at 9:32
• My response turned out to be pretty long, so I've added it to the post instead. – Hunan Rostomyan Feb 3 '14 at 18:30
• @MoziburUllah Conceptually speaking, you could create a system with distinguishable empty sets, but it wouldn't be ZFC any more. The big question would be what would you gain from making that shift. – Chris Sunami supports Monica Feb 3 '14 at 18:56
• @FedericoRazzoli Exactly. 'A' and 'B' are different names for the same set; and the extensionality axiom says why we consider them identical. – Hunan Rostomyan Mar 6 '18 at 17:17
• @FWE good point. Thanks very much for the edit. – Hunan Rostomyan Aug 15 '18 at 17:33

Since the mathematical point has been made above, I'll just comment on the ontological side: One could ask the same question about everything, from the number 1 to human beings. Is there just the one number 1, or are there many isomorphic mathematical objects with its properties? Is there just one me, or are there many other isomorphic (upon some agreement of which physical objects are isomorphic) me's?

Since I doubt that one can come up with a reason for having a multiplicity of such objects (except possibly when talking about modal identity), it is probably best to have just one such object in one's ontology.

• This is a good point - in fact, Frege specifically argued to the effect that there is just one number 1, and that this need to categorically define the number 1 (and other mathematical objects) was what led him to his use of Hume's Principle as a key defining axiom for his formalization of second order arithmetic: en.wikipedia.org/wiki/Hume's_principle . Of course, as we now know, the same strategy struggled with BL5 and set theory... – Paul Ross Feb 3 '14 at 19:51
• @Addem I don not agree. An empty set might just be seen as a base object of your universe. A manyfold of empty sets (in the sense of things, that do not contain themselves other things) might therefore come up quite naturally (see e.g. the example of the sets of cars below). – FWE Aug 15 '18 at 11:54

There is only one empty set.

Two sets are considered as different by ZFC if one contains an element not within the other. This comes from the extensionality axiom of ZF.

The answers above give you the mathematical reason within ZFC for the uniqueness and existence of the empty set.

For this from an intuitive point of view, you can use the analogy with a box.

A set is not a box, but the content of the box.

So, you can have two different empty boxes, but their content is the same : the "empty content".

• A set is a box, for example: the empty set is the empty box (we call it "{}"); it contains nothing. If two boxes contain nothing, they're still different boxes, because, for example, you can use one to store your old books in it, and you can burn the other one. If two sets both contain nothing, you cannot "do" anything with one that you won't simultaneously be doing with the other. – Hunan Rostomyan Feb 3 '14 at 18:42
• A set is not a box; it is the content of the box. – Mauro ALLEGRANZA Feb 3 '14 at 20:13
• I guess we don't have to agree about that point. What did you think about the rest of my comment? – Hunan Rostomyan Feb 3 '14 at 20:24
• @HunanRostomyan I think your argument supports Mauro's point — two empty boxes might be different, but the contents of two empty boxes are the same. Also, unlike physical boxes, sets are "immutable" — if you "do" anything to a set, you get a different set. – augurar Feb 3 '14 at 20:28
• @HunanRostomyan You might be interested in math.stackexchange.com/questions/63910/… – augurar Feb 3 '14 at 20:48

Question 1. Is it correct to say in formalised ZFC that there is just one empty set?

ZFC requires a unique empty set, i.e. for ZFC there is just one empty set. This is due to the Extensionality axiom of ZFC, which identifies two sets whenever they have the same elements - formally:

∀ x ∀ y [∀ z [z ∈ x ↔ z ∈ y] → x = y]

and the Null Set axiom, which requires the existence of an empty set (i.e. a set containing no other set) - formally:

∃ x ∀ y [y ∉ x]

References.

Wikipedia.Axiom of extensionality

nLab.Axiom of extensionality

nLab.ZFC

Question 2. Is it correct to say that in the categorical set theory, say ETCS, that there are many empty sets but they are all isomorphic?

ETCS requires at least one and allows several initial objects alias empty sets. This is because an initial object 0 (that can be thought of as representing an empty set) is defined by a universal property, namely that for any object x there is exactly one mormhism from 0 to x - formally:

∀ x ∃! 0 → x

(the arrow here is a morphism, not a deduction) and is thus defined up to (unique) isomorphism. However since the initial objects are isomorphic one speaks of the initial object.

Two initial objects 0 and 0' must be isomorphic: since 0 is initial it exists 0 → 0', and since 0' is initial it exists 0' → 0. Then the composition 0 → 0' → 0 must be the identity on 0 (since 0 is initial, there can by definition just be one mormhism 0 → 0 but this has to be the identity, which is given in any category for any object). Analogue argumentation shows that 0' → 0 → 0' must be the identity on 0'. So 0 and 0' are isomorphic (what we just showed is the definition of two objects being isomorphic: a and b are isomorphic if there are two morphisms f: a → b and g: b → a and gf = id: a → a and fg = id: b → b.

References.

Wikipedia.Initial and terminal objects

nLab.Initial object

nLab.ETCS

Example (Sets of cars). If you would regard sets of cars, then any car, that is not containing other cars (so excluded loaded car transporters or cars with matchbox cars inside), is an empty set. ZFC regards all those cars as identical - thus there is only one car that is not containing other cars. ETCS allows to distinguish them, but regards them as isomorphic.

Question *. Which is the more accurate view?

My personal opinion. Asking for accuracy is perhaps not the right approach here. One could instead ask which concept to work with. This depends on your criteria:

(1) In case you are interested in the study of well-orders and the cumulative hierarchy, you might want to work with ZFC.

(2) In case you are interested in mathematics apart from that and how sets are used there, you should probably have a look at category theory and ETCS rather than ZFC.

References.

nLab.Cumulative hierarchy

Leinster.Rethinking set theory

• +1 Good answer. If you added references for the extensionality axiom of ZFC and for more information on the isomorphism in ETCS this would improve the answer because it would give the reader a place to go for more information. This would also make the answer not an opinion but a reporting of what others claim. – Frank Hubeny Aug 13 '18 at 15:30