Set Theory and Beyond

 Once we have ordered pairs (which we got in the last post), we are ready to define all kinds of important mathematical objects. The most obvious one is probably the product  $A\times B$ of two sets $A$ and $B$. The definition is pretty straightforward: $$A\times B:=\{⟨a,b⟩:a\in A,b\in B\}.$$ However, we should ask ourselves if this is a set: as far as we know it, there is nothing preventing this one from being a proper class! So, as you might've guessed, we solve our problems by declaring new axioms. 😎 I'll now try to describe Axiom of Replacement , which should (together with our old axioms) imply that $A\times B$ is a set. Remember that we had the notion of a predicate $P(x)$. What we meant here was a property that a set may or may not have. Very similarly, we can consider a property $R(x,y)$ that the ordered pair of sets $x$ and $y$ may or may not have. One way to think about it is that we can look at some predicate $R^{\ast }(z)$ such that ...

 We've already introduced the notion of a pair of sets $\{x,y\}$. This object is symmetric insofar that it doesn't give a precedence to either $x$ or $y$. One way to formally state this is to observe that $$\{x,y\}=\{y,x\}.$$ This follows simply from Extensionality: both sets have same elements! On the other hand, it will be of no surprise to you that sometimes we need an object which pairs together $x$ and $y$, while keeping the information which one comes first. Let us denote this ordered pair by $⟨x,y⟩$. According to our guiding principles, this should be another set, just another object in $V$. This means that, even though we have a clear idea what the object $⟨x,y⟩$ is, we need to pick a concrete way to code it within the universe. I say "pick" because there is more than one way to do so and giving the precedence to one over another is basically arbitrary. So, let us make our choice and define the ordered pair of sets \...

We now know what $V$ is, the universe of all sets. We started this journey in order to build this universe and one of the guiding principles was that all objects in the universe are just pure sets. However, it is not completely clear what this means, so I went on to say that sets should be determined by their elements and nothing else. We had so far only one axiom that asserted this kind of "setness" of sets, that is the Axiom of Extensionality. Arguably, though, this is not enough. For example, nothing that I've said up to now can answer whether there exists a set $x$ which belongs to itself. Even more concretely, is there a set $x$ which is equal to $\{x\}$? I'm sure that you will find a prospect of such an object unusual, after all it would satisfy things like $$x=\{x\}=\{\{x\}\}=\{\{\{x\}\}\}=\cdots .$$ Furthermore, say that $y$ also satisfies $y=\{y\}$, is then $y$ equal to $x$? If they're not equal, then $y$ is an element of $y$ which is d...