Set Theory

Set theory is widely used as the foundation for all mathematics in the sense that the most basic mathematical object is a set. At its core, a set is just a collection of distinct things, though in set theory sets are the only things that exist so that ultimately elements of sets must always be other sets. The properties of sets are defined by a handful of axioms. Initially set theory concerns itself with constructing all the mathematical concepts you know and love using only sets: the various types of numbers (natural, rational, real, etc.), functions, and even relations on numbers like < and >. These things are used all throughout every other field of mathematics. Then the theory goes on to define and explore more abstract topics like cardinal and ordinal numbers, which are not necessarily as widely used as the more basic concepts.

Of the topics that I have studied so far, I would say that set theory is probably my favorite. First, it is very neat to learn how everything in math can be built up using only sets. Then things like cardinal and ordinal numbers generalize the natural numbers in different ways. Cardinal numbers, for example, generalize the notion of the size of a set in a way that concludes that infinite sets can have different sizes so that there are different infinities! Ordinal numbers, which is probably my favorite single concept so far, generalize the natural numbers in a way that lets you count “past” infinity and keep going. These are also truly numbers in that you can define arithmetic on them in a natural way with regard to their meaning while preserving the usual arithmetic for the natural numbers. It is truly fascinating to learn about.

The text from which I am studying set theory is “Introduction to Set Theory” by Hrbacek and Jech, which I am finding to be an excellent introduction to the field. While some treatments are “naive”, this is axiomatic, which means it begins by stating the axioms that ground all of mathematics and then builds up from there. Really, the only prerequisites for this text are a solid understanding of logic and proofs, though basic high school algebra is also crucial later on when you start learning about natural numbers. I am about 60% of the way through this text. For various reasons I first took a break from it to study real analysis, then returned to it, then took another break to study topology. I plan to alternate chapters with this and topology going forward.

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