Nowadays, having an idea about set theory is an absolute must for everyone who is considering looking into higher level Mathematics. This is mostly because of how the modern Mathematics is built – it is no longer about putting random thoughts together and formulating observations based on our intuition. Rather, it is formalized, in the strict sense of the word.
Everything a mathematician ever talks about has to be well-defined, every observation needs to be proven. Interestingly enough, the concept of a set appears to be very fundamental – so much so that it seems to be one of the most basic building blocks of Mathematics. Many, if not all of the theories created prior to the introduction of set theory (19th century) were later re-formulated in the language of sets. Constructing the natural numbers is just a small example of this.
No one shall expell us from the paradise that Cantor has created for us.
This famous quote by a german mathematician David Hilbert reflects the importance of set theory to Mathematics. Thanks to it, all fields of Mathematics have grown exponentially, and myriads of new ones were created. What is it about?
The set theory itself is very complicated – in fact, there is more than just one formulation of the theory, each one arriving at slightly different conclusions, thus creating a bit different mathematical universe than the other. All of them cover the “elementary” (high-school level and lower) Mathematics in the same way, though.
The central notion of the set theory is, unsurprisingly, the notion of a set. What is a set? Well, this is perhaps one of the trickiest questions.
The concept of a set is so natural to our minds, that we almost take it for granted. Indeed, the definition you will stumble upon in most of high school textbooks stems from our very natural way of thinking:
A set is a collection of mutually distinguishable objects satisfying certain property.
This definition forms a basis of what is known as the naive set theory. Naive – because it “dumbs-down” certain aspects of the theory, in order to avoid paradoxes. One of the most famous of such paradoxes is the Russell’s paradox: Suppose you have a rather stubborn barber, whose principle is to only shave men who do not shave themselves. Should the barber shave himself? Believe it or not, this has to do with a set-theoretical paradox, and it shows that not everything described by the above-mentioned definition is actually a set. See more on the Russell’s paradox on Wikipedia.
After all, if you think about it for a moment, a set actually is a collection in its essence. Therefore, we have defined a set by itself, and that’s not okay. This problem is solved via axiomatization, and the solution is very far from easy. Today, I’m not even sure if there is a comprehensible way of answering the question “What is a set?” directly. All I know is that it is a “real” object existing within our minds and, with a little effort, we are able to grasp it and work with it.
And that is what I would like to talk about in this mini-series: How do we define the operations on sets? How to create new sets? Are there any special kinds of sets? And most importantly – how the new concepts based on sets are introduced? I hope to cover all this with a sufficient level of clarity.
Now, even though much of it might seem to be very easy, if not primitive to you, I find that going over the fundamentals over and over again and re-thinking the ideas is very helpful. In addition to that, naive set theory provides a fantastic exercise of mathematical language and formalism!
If you wish to learn more about set theory right away, I suggest reading an overview on Set Theory on Wikipedia. It not only gives you an idea about its history and how it came to be, explains basic notions, but also talks about the amount of controversy it has brought into mathematical circles.
Should you have anything you’d like to add, don’t hesitate and post a comment! Same goes if you are curious about something specific. Let’s have a nice discussion!
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