# Why is the empty set described as "unique" when it is a subset of every set?

I am completely new to set theory, so please bear with me.

I have watched a youtube video that is an introduction to set theory and have read enough basic websites to have learned that the empty set is a subset of every set, and that it is unique.

I understand the first proposition, but not the second.

How can it be unique if there's one in every set? Unless the word "unique" means something different in set theory to everyday usage. Does it just mean its properties are unique?

• Please include the precise uniqueness argument that you do not understand. Lacking such the question is not well-defined. Commented May 8 at 20:16
• "How can it be unique if there's one in every set? " The empty set is a subset of every set, not a member of every set. It is vacuously true that for an arbitrary set $S$, every element in the empty set is also in $S$. Commented May 9 at 12:59
• It's somewhat philosophical when it comes to abstract concepts like sets. en.wikipedia.org/wiki/Problem_of_universals Is every "3" in the world all the same? In Plato's theory of forms, "threeness" is the essence of anything we call "three" in the real world.
– qwr
Commented May 9 at 16:05
• @chepner: what you say is true, but the OP's problem remains: the empty set is a member of many sets, so how can it be unique? Happily, we can refute this by a simple analogy: Brad Pitt has appeared in many films, so how can he be unique? Commented May 9 at 17:14

You could compare the situation involving the empty set that’s confusing you to an analog involving the set $$\{2,3\}$$:

1. $$\{2,3\}$$ is the unique set of primes that differ by exactly one.
2. That uniqueness is not diminished by the fact that it’s a subset of both $$\{2,3,25\}$$ and the set of all roots of $$x^3 - 5x^2 + 6x$$.
• Thanks. I see now that some of my confusion is down to me having only looked at examples which correspond to Venn diagrams. Commented May 8 at 19:46
• @ReneThomas That explains a lot! There was a question a while ago about how to indicate the empty set on a Venn diagram. My takeaway from that question and its answers is that Venn diagrams are a terrible way to reason about the empty set. Commented May 10 at 5:25
• Oh, I will have to try to look that up. I have been learning yesterday how it is possible to delineate subsets, where said subsets were specified when the original formation of the sets was being expounded. Commented May 11 at 7:53

Yes, unique in this context means “in terms of its properties.” The empty set is the only set that is a subset of every set.

• Thank you for making your answer so direct and succinct. I note that there do appear to be dissenting views among the answers given here, and perhaps some of that is due to differences between alternate varieties of set theory/theories. Commented May 9 at 19:15
• @ReneThomas If I had known this question would ride the HNQ list for days I'd have been less succinct! But I'm glad it's helped. Commented May 10 at 13:54
• I think uniqueness in this context is stronger, then "in terms of its properties". Empty set is unique on the nose. Group $\mathbb{Z}$ in the category of abelian groups is unique in terms of its properties, i.e. given by universal property providing uniqueness in this case, but it is not unique on the nose: $\{\dots, -1,0,1,\dots\}$ and $\{\dots, -5,0,5,\dots\}$ are different $\mathbb{Z}$'s. Commented May 10 at 16:34

In simple language, a set is defined by its elements: sets are equal if (and only if) they have the same elements.

An empty set has no elements.  This means that any empty set is equal to any other empty set, because they both have no elements.  And as all empty sets are equal, there's nothing to distinguish them; it's simpler to speak of only one empty set.

Also, note that (as Michael said in a comment) the empty set is a subset of all sets; it's not a member of all sets.  (It's possible for a set to contain other sets as members, so a set could contain an empty set — but that doesn't apply to many of the examples you're likely to come across, especially if you're just starting with set theory, so you'll probably want to forget about that for now!)

• Thanks. I have picked up on the distinction between subsets and sets that are members, but I am indeed trying to forget it! Commented May 8 at 19:48

How can it be unique if there's one in every set? Unless the word "unique" means something different in set theory to everyday usage. Does it just mean its properties are unique?

This has nothing to do with set theory in particular:

The weakest tennis player (assuming that exists) is included in the set of weaker players for every other player. In that sense, there is a weakest player for everyone. Nevertheless, there is only one weakest player (it is the same for everyone).

• Wow - that struck me like you just dropped a logic bomb on me! Commented May 8 at 19:52
• Nevertheless, there is only one weakest player (it is the same for everyone). This is a great analogy - except for tennis players. In individual contest sports, e.g. tennis, fencing, boxing, etc, the question of weakness is actually subjective and relative to the individual sportsperson's own natural game/style. A case might be made that in tennis A beating B and B beating C does not always allow us to conclude that A will beat C . . . . though it usually does. Sorry to be pernickety ! Commented May 9 at 13:13
• I was thinking of something like this too (without the good analogy). A special feature of the empty set is that it has a universal property, being a subset of every set. In contrast, there is no set with the universal property of being a superset of every set. So the idea that one set has a particular relation to every other set can be surprising and isn't always true. But rather than being contradictory or something it is a special and important property of the empty set. Commented May 10 at 12:43
• Isn't the universal set a superset of every set? Commented May 11 at 7:55

TL;DR Uniqueness in mathematics is indeed determined based on an objects properties.

The longer answer is that it follows from how set equality and uniqueness are defined for sets. Two sets are considered equal if and only if they contain all the same elements.

A mathematical object with a certain property is considered unique if any two objects with this property are necessarily equal to each other.

Its then pretty easy to show (perhaps even formally) that the set with the property of having no elements is unique.

The situation is made a bit less clear because you can define set theory using logic that includes a definition for equality or with logic that does not.

In the case of equality existing as part of the logic, the new set-based notion of equality is effectively defined to be compatible with the one from the logic by virtue of an axiom.

In the case of a logic without equality this axiom is replaced by a definition, where the set-based equality is the only one.

In both cases the resulting notion of set-equality behaves the same.

In relating this to the empty set being a subset of each set. Set A being a subset of set B just means all elements in set A are also in set B. Since the empty set does not have any elements, any set B will automatically also contain all 0 elements included in the empty set. So "being a subset" is just a relationship between two sets involving a comparison between their elements.

Also note that this notion of uniqueness is not exclusive to sets. In mathematics uniqueness is usually determined by an objects properties, not its relationship to other objects (like sets it might appear in).

Its similar to numbers: there is only a single unique number that is equal to the number 1. It may show up in many places and it might be an element of many sets (e.g. {1, 3} or {1, 6}), but in all cases it is the unique mathematical object equal to 1. As such all these occurrences of the number 1 are considered 'the same object'.

• Thank you. I am beginning to realise I haven't even thought through what a number actually is ! Commented May 8 at 19:49
• @ReneThomas if you are referring to the fact that the (eg natural) numbers can be constructed set theoretically using only sets, thats nice, but I wouldn't get too philosophical about that. I am pretty sure numbers predate set theory, if not in its modern incarnation, at least close. As it turns out mathematical structures can be modeled in multiple ways. You could model numbers using sets, or you coud do it without them, using something else. Commented May 20 at 14:20

Other's have given fantastic analogies and examples, but I haven't seen anyone formally go through why the empty set is unique.

Given that we have the Axiom of Set Existence and the Comprehension Axiom Schema

The existence of the empty set is given by the following theorem.

Theorem

$$\exists$$y$$\forall$$x(x$$\in$$y $$\iff$$ x$$\neq$$ x)

Proof:

By the Set Existence Axiom: $$\exists$$x$$(x=x)$$, fix $$x_0$$ s.t. $$x_0$$ = $$x_0$$

Consider $$\phi$$(z) := z $$\neq$$ z

Then, By the Axiom Schema of Comprehension

$$\exists$$y$$\forall$$z(z$$\in$$y $$\iff$$ $$\phi$$(z) $$\wedge$$ z$$\inx_0$$)

Thus,

$$\exists$$y$$\forall$$z(z$$\in$$y $$\iff$$ z $$\neq$$ z $$\wedge$$ z$$\inx_0$$)

Equivalently, $$\exists$$y$$\forall$$z(z$$\in$$y $$\iff$$ z $$\neq$$ z)

As z $$\neq$$ z $$\iff$$ z $$\neq$$ z $$\wedge$$ z$$\inx_0$$

Informally,

y = { z$$\inx_0$$ | z $$\neq$$z} = {z | z $$\neq$$ z} = {z| $$\phi$$(z) } exists ( is a set)

Why is such a y unique?

Informally, uniqueness means that if $$y_1$$ = {w| $$\phi$$(w) } and $$y_2$$ = {z| $$\phi$$(z) }, then $$y_1$$ = $$y_2$$ i.e. rather than " a empty set". We can refer to it as "the empty set".

We need the Axiom of Extentionality.

Theorem:

$$\exists!$$y$$\forall$$x(x$$\in$$y $$\iff$$ x$$\neq$$ x)

i.e.

$$\existsy_1[\forall$$x(x$$\iny_1$$ $$\iff$$ x$$\neq$$ x) $$\wedge$$ ($$\forally_2\forall$$x(x$$\iny_2$$ $$\iff$$ x$$\neq$$ x) $$\rightarrow$$ $$y_1$$ = $$y_2$$)]

Proof:

Fix $$y_1$$,$$y_2$$ s.t.

$$\forall$$x(x$$\iny_1$$ $$\iff$$ x$$\neq$$ x)

$$\forall$$x(x$$\iny_2$$ $$\iff$$ x$$\neq$$ x)

We prove that $$y_1$$ = $$y_2$$

$$\forall$$x(x$$\iny_1$$ $$\iff$$ x$$\neq$$ x $$\iff$$ x$$\iny_2$$)

and so $$\forall$$x(x$$\iny_1$$ $$\iff$$ x$$\iny_2$$), thus by the Axiom of Extentionality $$y_1$$ = $$y_2$$

We have verified the existence and uniqueness of the empty set.

With this Theorem $$\exists!$$y$$\forall$$x(x$$\in$$y $$\iff$$ x$$\neq$$ x), we can unambiguously denote the empty set with the defined term:

$$\emptyset$$ = The unique y s.t. $$\forall$$x(x$$\in$$y $$\iff$$ x$$\neq$$ x)

• +1: Nice to have an answer that works it through in terms of axiomatic set theory. I think however that is can be improved: It would be nice to include the axioms of the axiomatic set theory used explicitly in the answer, or at least formally state the axioms used explicitly, instead of just naming them (especially, because the presentation and form of the axioms is not the same everywhere, even different presentations of ZFC use different, though equivalent, forms of the axioms), and at least link to Wikipedia on ZFC for reference. Commented May 9 at 12:34

Since no one yet has explicitly said this:

Even though the word "unique" is grammatically used in this way, uniqueness is not a property a set has or does not have.

The statement

There exists a unique empty set.

should be regarded as an abbreviation of

There exists an empty set and all empty sets are equal to each other.

Only after this has been established, we can speak of "the" empty set, whence the phrase

The empty set is unique.

• Surely one can assert that $\varnothing$ refers to a unique entity in the set-theoretic superstructure without being misleading; see my answer. Commented May 9 at 12:09
• @MikhailKatz I don't see that the word "unique" adds any meaning (both here and in your answer). What would a non-unique entity be and how could $\emptyset$ possibly refer to a non-unique entity? If your point is that there is only one empty set, not just up to isomorphism, but in a strong logical sense, we don't have any disagreement. Commented May 9 at 18:13
• Stefan, then we are in agreement :-) Commented May 12 at 6:58

A set is a collection of objects that are well-defined, distinct, usually but not necessarily homogeneous and unique, i.e. no duplicates of the same object may exist in a set.

The set of subsets of a set is defined to include the empty set, also known as the null set.

As such, every set of any kind or kinds of object will share one subset, i.e. the null set, with all other sets regardless of the type(s) of objects in the set concerned.

In that sense, the null set is a unique subset of all sets unlike another set of one or more elements which may only be a subset of a limited number of sets.

To add to the other answers, maybe seeing the proposition "The empty set is unique" written out in formal logic can also help. It is equivalent to $$\forall y :x \subseteq y \iff x = \emptyset.$$

Any two sets with the same cardinality are isomorphic. That is, we can map each element of one set to a unique element of the other set, and vice-versa.

The relationship between any set X and any set Y that both happen to be empty (cardinality of 0) is even stronger than an isomorphism. It is an equality. It may sound a bit strange, but every element in a set X with cardinality 0 is the same as every element in any set Y with cardinality 0. ("Every" in this case applying to "none"!). Thus, if both X and Y are sets with cardinality 0, they are the same set. Thus, there is only one set with cardinality of zero. That is what the expression "the empty set is unique" is pointing to, although perhaps not as clearly as one might like.

The uniqueness of the empty set $$\{\;\}$$ does not merely mean that it is the unique entity with its properties. It means something stronger: it is literally a unique entity in the set-theoretic superstructure built in the usual way, for example via the von Neumann construction.

In this construction, one starts with the empty set $$\{\;\}$$ which is usually interpreted as the integer $$0$$, then constructs the set $$\{ \{\;\} \}$$ which is interpreted as $$1$$, then $$\{\, \{\;\} ,\{\{\;\}\}\,\}$$ interpreted as $$2$$, etc.

The natural numbers are defined as the least inductive set. Then one introduces the rationals via one of the usual constructions, and the reals via, for example, Dedekind cuts.

The superstructure is developed using the power set operation (based on the appropriate set-theoretic axiom).

In the entire superstructure, there is literally a unique empty set $$\{\;\}$$ and not merely a unique object with its properties, or up to isomorphism or equivalence, etc.

In general English, when we say that something is "unique", we may mean that there is only one of them in the universe. But we might also mean that it is the only thing in the universe with this particular set of properties, even though there may be many instances of the thing.

Like someone might rationally say, "Carbon is unique because ..." and then describe some chemical property that no other element has. This does not mean that there is only one atom of carbon in the universe, but that however many atoms of carbon there are, carbon is the only ELEMENT with these properties.

In this case, it's ambiguous in what sense the empty set is unique. One could question whether there are many instances of the empty set, one for each situation where the empty set arises, or if there is only one empty set and there are many references to it. Like presumably you are a unique person. There is only one person in the history of the world who is you. But there could well be many references to you, like your name on your drivers license, the identification of you on this post, any number of times people have talked about you, etc.

Personally I think what the writer you cite meant was that the PROPERTIES of the empty set are unique, that an empty set is different from any other set because, etc. That doesn't preclude there being many instances of the empty set.

• It depends on the choice of set theory. In ETCS, there really is only one empty set, up to unique isomorphism. This is because ETCS is isomorphism-invariant: it can't tell the difference between isomorphic objects. Commented May 8 at 17:24
• Thanks, Jay. People on here are so canny at coming up with illustrative examples . Commented May 8 at 19:53