Weekly Log:
Week 1 (1/17/24 – 1/19/24)
Our first meeting of the semester was cancelled due to a winter storm! To get back in the set theory groove, I read the article Believing the Axioms by Penelope Maddy. One thing I really appreciated about it was its emphasis that the axioms of ZFC aren’t divine laws bestowed upon us by a higher power, or necessarily intrinsic or obvious. ZFC is a very human construction that came about more or less as result of “historical accident”. Of course, ZFC is a powerful, useful, and valuable system, but it’s important to remember that alternative axiomatic systems (like Anti-Foundational Set Theory!) aren’t necessarily secondary or inferior.
I also spent some time going over Chapter 1 of Peter Aczel’s Non-Well-Founded Sets. The idea of non-foundational set theory is super interesting to me, but I still would like to work on my base knowledge of set theory a lot more before giving it my full attention. I worked through just about all of Henle’s An Outline of Set Theory last semester, and this year I’ve got my sights set on Jech’s Set Theory.
Week 2 (1/22/24 – 1/26/24)
This week I spent some time going through the first chapter Jech’s Set Theory, most of which was friendly review. I also wanted to refresh myself on trans-finite induction, and went over some theorems I’d seen last semester that use the concept.
Here is a pretty simple trans-finite induction proof that I did as an exercise. This theorem about the von Neumann hierarchy of sets comes from Henle’s An Outline In Set Theory.
During my review I also encountered the Hydra Game, a single-player game that’s an fun intersection of graph theory and ordinal set theory. I spent some time studying the game and this proof that “all hydras die in finite time”, which relies on transfinite induction.
Week 3 (1/29/24 – 2/2/24)
My main accomplishment this week was finishing chapter 1 of Jech’s Set Theory. I’m currently working through the exercises; I’d like to get most of them done and make progress on chapter 2 before the end of next week.
One exercise I spent a lot of time chewing on was 1.2: proving there is no set which has its own powerset as a subset. I saw two ways to argue this, namely cardinality and Foundation. Invoking Foundation yielded a very straightforward proof, but made me wonder if this statement also holds in non-foundational systems (I believe it does), and what a proof in such a system might look like. I’m also not sure how cardinality arguments might be different in a non-foundational system (if at all). Food for thought!
Week 4 (2/5/24 – 2/9/24)
It’s been a hectic week with my first batch of exams, but I did have some fun wrestling with one problem suggested to me by Dr. Boney; showing that Vω is a model of ZFC without Infinity. As I was working on it, I approached it from a very set-theoretic approach; focusing on just showing that for each axiom of ZFC, the sets with the properties required by the axioms were present within Vω. For the most part, these proofs felt pretty straight-forward to me, with the exception of Replacement and Choice.
In our weekly meeting, Dr. Boney emphasized the more model-theoretic side of the problem, and dug into a bit of the nuance of what it really means to be a model of ZFC. This perspective was one that I hadn’t really seen talked about a lot in the set theory books I’ve been reading, so it was exciting to learn more about.
Week 5 (2/12/24 – 2/16/24)
This week I spent most my time working through the beginning of chapter 2 of Set Theory. Most of the material thus far has been re-establishing properties of well-ordered sets and ordinals that I’ve been exposed to before, but it was great practice to sharpen-up on the definitions and important theorems.
Week 6 (2/19/24 – 2/23/24)
This week, Logic@TXST had a guest speaker, Dr. Andy Zucker from the University of Waterloo! He spoke to us on his work on Ramsey Theory, which from what I could gather seemed to be an interesting blend of Logic with Graph Theory. Though it’s not a field that I can say I’ve had much experience with or been drawn to in the past, it seems I keep encountering exciting areas of Logic which rely on Graph Theory as a backbone (Anti-Foundational Set Theory being another). I think it’s a field I should look more into; perhaps I’ll take Graph Theory next semester!
Week 7 (2/26/24 – 3/1/24)
Kept plugging away on the ordinals chapter of Set Theory this week, and I encountered the concept of trans-finite sequences for the first time. If we think of a sequence as a function with domain ω, then a trans-finite sequence is simply a function with domain of an ordinal larger than ω. A pretty straight-forward concept, but one I’d never seen before, and that I thought was novel and intuitive!
My friend and fellow Logic@TXST researcher Austin also spoke at our department’s student seminar this week on ultraproducts! I’d heard of the concept in passing before but had never really dug into it, so it was fun to see them built from the ground-up and refresh some basic Model Theory concepts along the way. Also, the talk did a great job of laying out the basics of ultrafilters, which was a Set Theory concept I hadn’t worked with before, but found quite interesting. It seems to me that thinking of an ultrafilter as “deciding what the “majority” of an infinite set looks like” provides a nice intuition on their nature.
Week 8 (3/4/24 – 3/8/24)
This week in Set Theory, I ran into Jech’s definition of transfinite recursion. This is a concept I hadn’t seen before, and honestly am still wrestling with to fully understand. I know the motivation is to create the machinery for the definition of ordinal arithmetic which appears on the next page, so there is at least a concrete application for me to reference.
After bringing it up with Dr. Boney, I definitely felt that I had a clearer picture of how this arithmetic machinery is built, but I still feel shaky on understanding Jech’s abstract definition of recursion, and how I might use this definition in other contexts. Perhaps I’ll consult some outside sources and try to re-tackle this portion of the chapter in the near future.
Week 9 (3/11/24 – 3/15/24)
Spring break! I caught up on homework and then slept a lot.
Week 10 (3/18/24 – 3/22/24)
I was pretty busy this week! For starters, the Texas chapter of the MAA had their annual meeting at Texas State this weekend, so I decided to attend. I got to chat with some faculty and students, check out some Texas graduate programs, got a cool T-shirt, and saw a lot of great talks. My favorite by far was by Dr. Karen Lange about some foundational problems and definitions in Computability Theory. She focused on computable and non-computable binary trees, and also introduced the field of “Reverse Mathematics”; an alternative approach where instead of using axioms to prove theorems, mathematicians start with a set of desirable theorems, and then deduce the axioms needed to prove them.
I had never heard of Reverse Mathematics, but it totally captured my interest! One of my main motivators for studying Set Theory is so that I can become more well versed in axiomatic alternatives to ZFC, so it seems like a perfect field for me to dive into. At Dr. Lange’s recommendation, I got my hands on a copy of Reverse Mathematics by John Stillwell. I’m excited to get started on it!
Back in Set Theory, I came across a rather interesting exercise: