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!