Sets with more than two elements

 We've been trying to describe a universe of sets which can code all of Mathematics. The basic principle of "set-ness" is the Axiom of Extensionality, asserting that sets are completely determined by their elements. So, knowing that sets are indeed "set-ish", we went on to establish what kinds of sets are actually there. We started by asserting the Axioms of the Empty Set and Pairing. Thanks to them, we now know that there exists a set $\mathrm{\varnothing }$ with no elements, that for any set $x$ there exists a set $\{x\}$ with only $x$ as an element, and that for any sets $x$ and $y$ there exists a set $\{x,y\}$ that has only $x$ and $y$ as its elements. This is very nice, you might say, but what about the set $\{x,y,z\}$ that has exactly $x$, $y$, and $z$ as its elements, where these three are some given sets? Well, it can be shown that the three axioms that we've introduced so far do not guarantee existence of such a set.

We might try to go on and introduce an "axiom of triples" which would give us this, but then there would be question of the sets of the form \(\{x,y,z,w\}), and so on. Hence, we chose a different path, introducing an axiom which is going to handle all of these sets at once, but much more as well. Now my readers should brace themselves for a bit of conceptual trauma. The Axiom of Union asserts that for any set of sets $X$, there exists a set $x$ whose elements are exactly the elements of the elements of $X$. I think that it can be challenging to wrap one's head around this when you see it for the first time, so let's take this apart. First of all, every set is a "set of sets", since all the objects that exist in our universe are sets. I included this redundancy as a conceptual help to the reader.

Now let's look at one set which we'll call Apples and another one which we'll call Oranges. If $z$ belongs to Apples, we'll say that $z$ is an apple, and if $z$ belongs to Oranges, we'll say that $z$ is an orange. Now, let's look at the set $X=\{\mathrm{Apples},\mathrm{Oranges}\}$ and let's determine the set $x$ which is provided to us by the Axiom of Union. If $z\in x$, then $z$ must be an element of an element of $X$, or in other words, an element of Apples or an element of Oranges. And also the other way around, if $z$ is an element of Apples or an element of Oranges, then $z$ is an element of an element of $X$, so it belongs to $x$. Well then, what are the elements of $x$? Exactly those $z$ which are either an apple or an orange! 😁

Let $X$ be any set (of sets) and let $x$ be the set whose elements are exactly the elements of the elements of $X$. Note that we then know exactly what are the elements of $x$ itself, so by Axiom of Extensionality, $x$ is the only set in the universe whose elements are exactly the elements of the elements of $X$. We can then introduce the following notation for $x$: we denote it by $\bigcup X$. Okay, now that we understand the axiom, let's see how they give us sets of the form $\{x,y,z\}$.

Let's look three sets $x$, $y$, and $z$. By Axiom of Pairing, we can form sets $\{x,y\}$ and $\{z\}$, and then also the set $\{\{x,y\},\{z\}\}$. Now we call Axiom of Union to help and get the set $\bigcup \{\{x,y\},\{z\}\}$, which I will call $\mathrm{Triple}$. Guess what, $\mathrm{Triple}=\{x,y,z\}$! How do I know this? Well, the elements of $\mathrm{Triple}$ are exactly the elements of elements of $\{\{x,y\},\{z\}\}$, or in other words, the elements of $\mathrm{Triple}$ are exactly the elements $\{x,y\}$ together with the elements of $\{z\}$. Since the elements $\{x,y\}$ are just $x$ and $y$ and since the only elementsof $\{z\}$ is $z$, we can conclude that the elements of $\mathrm{Triple}$ are exactly $x$, $y$, and $z$, which is another way of saying $\mathrm{Triple}=\{x,y,z\}$.

Let's introduce a bit of notation. Instead of writing $\bigcup \{X,Y\}$, I will be writing $X\cup Y$. This is the set whose elements are exactly the elements of $X$ together with the elements of $Y$, so called the union of $X$ and $Y$. Having in mind this notation, we have that 

$$\{x,y,z\}=\bigcup \{\{x,y\},\{z\}\}=\{x,y\}\cup \{z\}.$$

I'll just go on and define $\{x,y,z,w\}$ to be $\{x,y,z\}\cup \{w\}$ and define $\{x,y,z,w,t\}$ to be $\{x,y,z,w\}\cup \{t\}$, and so on. Indeed, we're now able to produce sets of any number of elements, as long as that number is finite, but not anything else. I haven't told you what being finite or infinite means and this is a theme for future posts. In fact, trying to understand the infinite is in the very heart of Set Theory and I hope I'll be able to transmit this to you in the future. For now, I leave you with an exercise for your thoughts: how would you describe infinity?