Set Theory and Beyond

I'll write this post in form of a dialogue between Master (M) and Apprentice (A). A: Say, O Noble Teacher, what are those "relations" that mathematicians always talk about? M: Why, my dear Apprentice, you have already seen one kind of relations! A: How so, O Exalted One? M: Well, $\in$ is a relation and $V$ is also a relation! These are class-sized relations. A: Oh, I see! But how does their "relation-ness" manifest? M: In the case of $\in$, it divides pairs of sets $(x,y)$ into those for which $x$ belongs to $y$ and into those for which $x$ does not belong to $y$. So, it tells us whether the statement ''$x\in y$" is true or false for any particular pair of sets $(x,y)$. In this sense, they are no different from predicates or classes. Class-sized relations are the same thing as predicates. A: I seem to understand now! $\in$ is a binary predicate, so a class-sized relation. $V$ is a class, which is the same thing as a unary pr...

So far, I listed Axioms of the Empty Set, Extensionality, Pairing, Union, Comprehension, Induction, Replacement, and Infinity. This is not the complete list making up the Mathematical Theory of Everything, we have to add two more axioms to the list. These axioms are called Powerset  and Choice . Axiom of Powerset is relatively easy to state: for every set $x$ in the universe, there exists a set whose elements are exactly all subsets of $x$ . As usual, Extensionality implies that such object must be unique and we denote it by $P(x)$. So, the defining characteristic of the set $P(x)$ is that a given set $y$ belongs to $P(x)$ if and only if $y$ is a subset of $x$. The other axioms don't imply existence of the powerset. More precise explanation of this will have to wait, but for now, we can kind of see that the set $P(x)$ is "larger" than $x$. If $a$ is an element of $x$, than $\{a\}$ is a subset of $x$, so an element of $P(x)$. But there are oth...

We now want to learn how to count. This is how it usually goes: we start from $0$ and then we keep passing to the successor, that is, $0$ is followed by $1$, which is in turn followed by $2$, which then gives its place to $3$, and so on. Since these objects are essential for mathematics, one would rightly expect them to appear in the universal theory of mathematics, i.e. Set Theory. Well, this is what they are: $$0:=\mathrm{\varnothing }$$ $$1:=0\cup \{0\}=\{0\}$$ $$2:=1\cup \{1\}=\{0\}\cup \{1\}=\{0,1\}$$ $$3:=2\cup \{2\}=\{0,1\}\cup \{2\}=\{0,1,2\}$$ and it continues. The problem, though, is that it continues indefinitely... This means that I can't possibly just list all definitions and have you know what the natural numbers are. A more sophisticated approach is required! The starting point is $0$, which is just empty set. Now, to get new natural numbers, you pass to the successor. The way I wrote my definitions above was to suggest what the successor operation  should be. In the prev...