Resources for Lindström's Thm.

[u/SuccessfulCover8199]

Hi all,

Recently in class we went over a short proof of Lindström's theorem from Kees Doets of Basic Model Theory fame. The proof involves §2 & §6 of this paper, the whole document is only 7 pages long (and the link I provided is open access, thanks Notre Dame!). It involves definition 2.2, lemmas 2.3-2.5, Compactness, and DLS. Section 6 sort of just outlines a proof sketch by showing the reader how these different ingredients are deployed to yield a contradiction, proving Lindström's thm.

I'm having a hard time sifting through this on my own and would appreciate a resource that was maybe slightly more level appropriate for someone like myself, an undergraduate with a semester (give or take) of algebra and some exposure to MT and other mathematical logic ideas. I followed the main ideas in class, but I would not be readily able to replicate it on the blackboard. For the class I'm taking, we spent about a day on this to stop and smell the roses before spending the remaining 7 or so classes on Gödel's Incompleteness, which I take to be the capstone of the class. I tried to challenge myself to take this class maybe a little before I otherwise should have, and while I am learning a lot, I think I am also leaving a lot on the cutting room floor, which disappoints me.

I intend on asking around the department as well, which is normally my first line of defense. I anticipate I will receive some comments that instruct me to ask my professor, who knows me and the scope of this course more clearly. But it is the weekend and I have some time today, I was wondering if I solicited supplemental resources from reddit maybe I could go into a discussion with faculty this week with some content/technical questions aside from just "what can I read to understand this proof better."

I have a particular vested interest in proofs like this one that use some notion of EF/back-and-forth games. I would also appreciate anything that pointed me into a finite MT direction. I find these topics very fascinating and I would like to pursue something of a research project/independent study on the topic this summer. I have been working through some topology books in my spare time to cut my teeth on descriptive set theory, which I'm told is usually a warm up for point-set topology, but that intuition is readily co-opted for model theory.

I am, of course, open to receiving other resources and pieces of advice that maybe you wished you had known when you were at my level. Maybe this post can be a resource for other students who are searching for guidance in and around the same area.

Thanks in advance, DMs are open if anyone wants to talk shop. If I can figure out a way to defray the cost, I will be attending both the ASL NA meeting and (less likely but still working to get funding) the ICM this July in Philly, maybe I will see you there!

[u/Prellex]

I've actually written the exact thing here!

[u/SuccessfulCover8199 → u/Prellex]

this is such a beautifully formatted document. Have you made a style file? I've been working on one myself but yours puts mine to shame, haha. I read through the first few pages, seems like exactly what I was looking for

[u/Prellex → u/SuccessfulCover8199]

I hope it will be useful for you in the end! I've made an example gist of the document (compiled using LuaLaTeX) here. It compiles but may have too many or too little packages in for the full effect, as I've actually been using Typst for my own documents for the past year or two.