Pairing Sets

Today we continue building our mathematical universe, one set at a time. But first, a clarification is in order. Last time, I introduced two axioms that we expect to hold in our universe and I said that based solely on them, we can't prove that any other set exists except the empty one. What I meant by this is that the universe of sets which has in it only the empty set does satisfy these two axioms. Of course, we can't possibly represent all of Mathematics in such universe, so we must exclude it from consideration. We do so by introducing new axioms, which then restrict the universes which are acceptable for our work. So, you should think of what we are currently doing is trying to zero in on the "right universe". I will just put it out there as a teaser for future posts that the questions of meaning of this "right universe" have caused whole bunch of debates, both among mathematicians and philosophers.

But now, back to our work. I'll just go ahead and state the next axiom, the Axiom of Pairing: for any two sets, there exists a set which has them both as elements and has no other element. If we look at sets $x$ and $y$, say, then this axiom guarantees that we can find $z$ whose elements are just $x$ and $y$ and nothing else. But then, if some other set $w$ also has just $x$ and $y$ as elements, the Axiom of Extensionality ensures us that $w$ is equal to $z$. Thus, there is a unique set which has $x$ and $y$ as elements and has no additional elements. We call this set the pair of $x$ and $y$ and denote it by $\{x,y\}$. Note that we have never assumed that $x$ and $y$ are different from each other. And indeed, in the case when they are equal, we get the set $\{x,x\}$, which is also denoted by $\{x\}$.

Now, we have a procedure, an operation, which allows us to produce new sets from ones that we have. This operation is the operation of pairing and it works by taking sets $x$ and $y$ as arguments and returning $\{x,y\}$ as its value. Thus far, we have found only one set, the empty set $\mathrm{\varnothing }$. Well, let's plug it in in our operation of pairing. The operation takes $\mathrm{\varnothing }$ and $\mathrm{\varnothing }$ as arguments, and voilà, it gives us a new set $\{\mathrm{\varnothing },\mathrm{\varnothing }\}$! In accordance to our convention, this set is also denoted by $\{\mathrm{\varnothing }\}$, and it is really a new set: it is not equal to the empty set $\mathrm{\varnothing }$! Now, some people find this confusing, but here is what's going on: $\mathrm{\varnothing }$ has no elements, while $\{\mathrm{\varnothing }\}$ has one element, and that one element is equal to $\mathrm{\varnothing }$. Or if you like metaphores, $\mathrm{\varnothing }$ is like an empty bag, while $\{\mathrm{\varnothing }\}$ is like a bag which has inside another bag and that other bag is in turn empty.

Let's maybe practice a bit of notation? $\mathrm{\varnothing }$ has no elements, so whatever $x$ is, we have that $x\notin \mathrm{\varnothing }$. In particular then, $\mathrm{\varnothing }\notin \mathrm{\varnothing }$. On the other hand, $\mathrm{\varnothing }\in \{\mathrm{\varnothing }\}$ by the very definition of the set $\{\mathrm{\varnothing }\}$. Well, do you see now that these two sets don't have exactly the same elements? Also, note one more thing. If $z$ is some set, wherever it comes from, and if we somehow know that $z\in \{\mathrm{\varnothing }\}$, we then know exactly what $z$ must be: $z$ must be equal to the empty set. The reason for this is simple: the set $\{\mathrm{\varnothing }\}$ contains no other element except the empty set $\mathrm{\varnothing }$.

Since we now have sets $\mathrm{\varnothing }$ and $\{\mathrm{\varnothing }\}$, we can plug them together in our pairing operation and get another set: $\{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\}$. I say that this set is different from the previous two and I'll let you figure out why for yourself. 😉 In addition we can also pair $\{\mathrm{\varnothing }\}$ and $\{\mathrm{\varnothing }\}$ and get $\{\{\mathrm{\varnothing }\}\}$, or then pair $\{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\}$ and $\{\{\mathrm{\varnothing }\}\}$ and get $\{\{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\},\{\{\mathrm{\varnothing }\}\}\}$. We can continue pairing and pairing, and get many new sets (indeed, infinitely many, but we still don't know what it means to be infinite). However, all sets that we can produce with what we have so far are of the form either $\mathrm{\varnothing }$ or $\{x\}$ or $\{x,y\}$. Do you notice something weird? Hint: we can't produce a set that has three different elements!! Come back next time to learn how to transcend this inconvenience! 😁