Set Theory and Beyond

We have already talked about properties, but we didn't define them. I'm not going to that right now either, but I remind you that a property (or a predicate) is some definite way to separate sets to those which have that property and to those which do not. For example, if \(P(x)\) is a property and \(a\) is a set, then \(a\) might have the property \(P\), written as \(P(a)\), or it might not have the property \(P\), written as \(\neg P(a)\). Now, for any set \(A\), we may look at the property \(S_A(x)\) asserting "\(x\) belongs to \(A\)". In this way, every set can be seen as a property, but the other way around is not true. An example of the property which doesn't come from a set is given by so-called Russell's paradox.  Namely, consider the property \(R(x)\) asserting that "\(x\) does not belong to \(x\)". I claim that this property does not come from a set, or more precisely, that there is no set \(r\) such that the property \(R\) is the same as t...

 I promised to talk about boolean algebras and I'll do that now. Instead of going for the full generality, let's stick with the picture drawn last time. We'll fix a set \(U\) which is not empty and look at its subsets. Just for today, we'll call these subsets objects . So saying that \(X\) is an object just means that \(X\subseteq U\). There are three binary operations between objects that we are interested in. "Binary" here means that the operation takes two objects as its arguments and returns a third one as the result. As you guessed it, those binary operations are \(\cup,\cap,-\). We'll also consider a unary operation \({^\complement}\), i.e. one that takes a single argument \(X\) and returns the \(X^\complement:=U-X\). Note here that this operation ("the complement") only makes sense if we fix an "all-enveloping set" \(U\). Otherwise, the complement would be something like \[\{x : x\not\in X\}\] which is not a set! In addition to ...

I've already told you what the union of two sets is. To remind you, if \(x\) and \(y\) are sets, then \(x\cup y\) is the set whose elements are exactly all the elements of \(x\) together with all the elements of \(y\). We established existence of this union set by appealing to the Axiom of Union. Two other basic set operations that you've probably heard about are intersection \(\cap\) and set difference/minus \(-\). However, these two are a bit different from the union and their existence follows from the Axiom of Comprehension. Let \(x\) and \(y\) be two sets. Their intersection should consist exactly of those sets which belong to both \(x\) and \(y\). We can say this also in another way: the elements of the intersection of \(x\) and \(y\) are exactly those elements of \(x\) which are also elements of \(y\). Maybe you've caught onto what I'm trying to suggest: the intersection of \(x\) and \(y\) is the set \[x\cap y:=\{z\in x : P_y(z)\}\] where \(P_y(z)\) is the proper...

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 \(\emptyset\) 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...

Today we continue building our mathematical universe, one set at a time. But first, a clarification is in order. Last time, I introduced two axioms that we expect to hold in our universe and I said that based solely on them, we can't prove that any other set exists except the empty one. What I meant by this is that the universe of sets which has in it only the empty set does satisfy these two axioms. Of course, we can't possibly represent all of Mathematics in such universe, so we must exclude it from consideration. We do so by introducing new axioms, which then restrict the universes which are acceptable for our work. So, you should think of what we are currently doing is trying to zero in on the "right universe". I will just put it out there as a teaser for future posts that the questions of meaning of this "right universe" have caused whole bunch of debates, both among mathematicians and philosophers. But now, back to our work. I'll just go ahead and ...

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 t...