Mathematical Theory of Everything

One hears very often phycists speaking about the "theory of everything", which is something of a holly grail for them. To quote Wikipedia: "A theory of everything, final theory, ultimate theory, unified field theory or master theory is a hypothetical, singular, all-encompassing, coherent theoretical framework of physics that fully explains and links together all aspects of the universe. Finding a theory of everything is one of the major unsolved problems in physics." Well, the thing is, in Mathematics, we do in fact have such a theory! (Even though it gets comparably less publicity.) More precisely, we have a master theory that gives singular, all-encompassing, coherent theoretical framework of mathematics which links together all aspects of the MATHEMATICAL universe. (The other part, that it "fully explains" everything, that's work in progress. 😄) This theory is called Zermelo-Frankel Set Theory with Axiom of Choice, or in short, ZFC. Let me try to explain what kind of picture it paints. 

We imagine a universe of sets. All objects of this universe are sets, which means that an element of a set is itself a set. Before we give examples, we need to mention what relations a set can have with another set. To begin with, there are two basic ones, which we denote by $=$ and $\in$. Now, $=$ is just the standard equality, which is to say that every set is equal to itself and no other set. The other one, $\in$, is the relation of "being an element of". So, we might write $x\in y$ to say that $x$ is among the elements of the set $y$. Of course, $x$ might not be an element of the set $y$, in which case we will write $x\notin y$. There are many other additional relations between sets, but all of them can be obtained from these basic two.

These two relations are mutually connected in a way that is articulated as the Axiom of Extensionality. This axiom (or principle) asserts that a set is completely determined by its elements. Another way to express this principle is to say that two sets that have exactly the same elements are equal to each other. The intuition that is expressed in this way is that there is nothing else to being a set other than the elements that it contains. For some people, this idea is so obvious that it deserves the label of an axiom. For my readers who are not familiar with the notion of an axiom, let's just say that it's a kind of a statement that does not require any additional justification. These statements were first discovered by ancient greek mathematicians in their practice of geometry.

There is another basic principle of ZFC, which is called the Axiom of an Empty Set. This axiom asserts that there exists a set which contains no elements. Now, if $x$ and $y$ are two sets that contain no elements, then they have exactly the same elements (i.e. none), so by the Axiom of Extensionality, they must be equal. We have just proved our first theorem: there exists a UNIQUE set which contains no elements. For those who are unfamiliar with the notion of a theorem, it is a (true) statement that can be justified based on the accepted axioms (and previously proved theorems). For our first theorem, we used both of the mentioned axioms to justify it.

Having established that there exists a unique set with no elements, we are in position to name that distinguished object. We call it the empty set and we denote it by $\mathrm{\varnothing }$. This is our first definition: it introduces a new notion based on existing ones. We are also able to introduce a new relation on sets, in addition to the two basic ones: the relation of "being an empty set". The difference between this relation, call it $P$, and the previous two ($=$ and $\in$) is also in that $P$ is unary, while $=$ and $\in$ are binary. What is meant here is that $P$ refers to only one set, while $=$ and $\in$ refer to two sets. Going back to the relation $P$, we say that ''$x$ is an empty set", and write it (only for today) as $P(x)$, if and only if $x$ has no elements.

We have seen in this post the basic building blocks of ZFC and I plan to introduce other ones in the posts to come. As I said at the beginning, this theory can represent all of mathematics, so what is awaiting us in the future are many fascinating constructions and truths, hopefully culminating with me showing you how to code all standard mathematical objects as sets, and much more! However, what we can do for now, we have already done: we can't show that any other set except the empty one exists. This means that new axioms are needed and you have something to look forward to. :)