Which Concepts to master before studying about large cardinals?

Question

I am a undergraduate student interested in large cardinals since I've watched a youtube video called "How to count past infinity", and the concept of inaccessible cardinal that is introduced in the video have made me feel extremely curious about other "larger" Cardinals.

Since then, I have tried to research about large cardinals by myself (My university's only set theory class did not teach me about them), I have looked up the book "The higher infinite", some other papers that could be found on the internet, and even wikipedia pages. However, I have been making very little progress, and I feel like I'm not understanding the concepts because of my low mathematical maturity.

For example, Some books introduced the "filter", and I don't quite understand for what use did someone define a "filter", and of course the usage of it.

So my questions are:

(1) Am I going through the struggles that everyone else have also experienced it?

(2) My university's set theory class only taught me up to definitions and some theorems about functions and relations, am I missing some other "basics" about set theory to learn large cardinals?

(3) If I have did missed some knowledge about set theory, where should I start now, and how should I improve my mathematical maturity?

Thank you.

Answer 1

A prerequisite for learning about large cardinals is having a basic understanding of axiomatic set theory. Typically this means knowing about the axioms of ZFC, about well-orders and ordinals, about cardinalities, and about transfinite induction.

Based on the comments it seems that your university's set theory course covered naive set theory: presumably this included an informal notion of what a set is, the definition of a relation and a function, notions like surjectivity, injectivity, bijectivity, and the power set. These are good basic skills to have (for any mathematician!) but does not really get you started on the mathematical field of set theory.

You can compare the difference between naive and axiomatic set theory to the difference between high school, where you use real numbers all the time without really knowing what they are, and your first real analysis course, where you (probably) "define" the real numbers to be an ordered field where every bounded above (below) set of numbers has a least upper (greatest lower) bound.

Thus, if you want to learn about large cardinals, you will have to learn some axiomatic set theory first, or at least in parallel. I do not have a particular recommendation, but the answers to Textbooks on set theory seem to have some opinions on the matter. :-)

Edit: definitely do not forgo reading Noah Schweber's answer, even if (like at the time of this edit) it has fewer upvotes. He's much more of an expert than me!

Answer 2

Mees de Vries' answer is actually a lower bound on the prerequisites for studying large cardinals. Not only is a solid background in axiomatic (specifically, $\mathsf{ZFC}$-based) set theory necessary, you will also need a solid understanding of the basics of model theory in order to appreciate what the various large cardinal axioms are really doing. For example, a basic early exercise is that $\mathsf{ZFC}$ (if consistent) cannot prove the existence of inaccessible cardinals. There are two ways to prove this:

• Showing that $\mathsf{ZFC}$ + "There is an inaccessible cardinal" proves $Con(\mathsf{ZFC})$ and then invoking the second incompleteness theorem.

• Observing that, $\mathsf{ZFC}$-provably, the cumulative hierarchy up to the least inaccessible (assuming such exists) is a model of $\mathsf{ZFC}$ + "There are no inaccessible cardinals."

Each of these requires a decent understanding of what models and proofs are and how they interact. And things only get more complicated from here; the "large" large cardinals (such as measurable cardinals) really only make sense against the backdrop of model theory and in particular the notion of elementary embeddings. The fastest responsible route to large cardinals in my opinion basically consists of a pair of semester-long classes (ideally taken separately so as to not overload the student), one in general logic/model theory including compactness and Lowenheim-Skolem theorems and the other in axiomatic set theory.