Axiom of Powerset

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 other elements of $P(x)$ as well: for example, $x-\{a\}$ is an element of $P(x)$ as well. Also, if $b$ is another element of $x$, then $\{a,b\}$ is yet another element of $P(x)$. If $x$ has 5 elements, then $P(x)$ has 25 elements; if $x$ has 10 elements, then $P(x)$ has 1024 elements. Hint: the second number is two to the power of the first one. Once we figure out how to count infinite sets, we'll see that this kind of jump in the number of elements persists there as well!

Okay, we now have all axioms except Choice. I'll delay for a bit introducing choice and I'll let us contemplate what we have built so far. The theory that we have so far (without Choice) is called ZF. So, we'll try to see what the universe described by this theory looks like and how we get the "usual math" in it.