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?
You could compare the situation involving the empty set that’s confusing you to an analog involving the set $\{2,3\}$:
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.
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!)
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).
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.
See also
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'.
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$\in$$x_0$)
Thus,
$\exists$y$\forall$z(z$\in$y $\iff$ z $\neq$ z $\wedge$ z$\in$$x_0$)
Equivalently, $\exists$y$\forall$z(z$\in$y $\iff$ z $\neq$ z)
As z $\neq$ z $\iff$ z $\neq$ z $\wedge$ z$\in$$x_0$
Informally,
y = { z$\in$$x_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.
$\exists$$y_1$$[\forall$x(x$\in$$y_1$ $\iff$ x$ \neq$ x) $\wedge$ ($\forall$$y_2$$\forall$x(x$\in$$y_2$ $\iff$ x$ \neq$ x) $\rightarrow$ $y_1$ = $y_2$)]
Proof:
Fix $y_1$,$y_2$ s.t.
$\forall$x(x$\in$$y_1$ $\iff$ x$ \neq$ x)
$\forall$x(x$\in$$y_2$ $\iff$ x$ \neq$ x)
We prove that $y_1$ = $y_2$
$\forall$x(x$\in$$y_1$ $\iff$ x$ \neq$ x $\iff$ x$\in$$y_2$)
and so $\forall$x(x$\in$$y_1$ $\iff$ x$\in$$y_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)
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.
is particularly misleading.
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.
