Set Theory and Beyond

It is time for us to start approaching the infinity! I'll start by addressing the issue of picking a subset of a set. If a set is finite, say the set $\{x,y,z,w,t\}$, I might look at its elements one by one and decide if I like them or not. So, I like $x$ and $y$, I don't like $z$, I do like $w$, but I don't like $t$. Is there in our universe the set of elements that I liked? Of course, it's just the set $\{x,y,w\}$ (which we went through the trouble of describing last time). So we know how to pick subsets of finite set. However, if our universe is to be sufficiently reach, it must have the set $\{0,1,2,3,\dots \}$ of all natural numbers, so how do you pick subsets of this set? Naively, you might try to just go through natural numbers one by one, saying again for each one if you like it or not. But are you ever going to finish your "like-dislike" thread this way? After all, we've been all told as children that you can't count (go through...

 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...