(soft question) I'm giving a talk on model theory to undergrads in a few weeks, thoughts topics to include/exclude

Question

I am an undergrad and have been doing independent study on model theory for 6+ months now, and am slated to give a talk to undergrads later this month on model theory. I aim to give a soft introduction and maybe build some excitement for what it one of my favorite areas of math. I plan on spending a few minutes going over the basic definitions of languages, structures, formulae, theories, etc... and then show some basic results. I was hoping to use Ax's theorem about polynomial maps from $\mathbb{C}^n$ to $\mathbb{C}^n$ as the grand finale, or perhaps showing the one point compactification of the naturals is 1st order indistinguishable from the naturals. I'm worried the significance or interest of these fact will miss my audience. I was wondering if anyone know of a fun or exciting result I could use which are perhaps more obviously accessible? This is my first hour long talk, thank you.

Answer 1

My comment was getting long, so I moved it to an answer.

Michael's comment contains advice which is of course very applicable here -- you won't get very far in an hour with an undergraduate audience, assuming no background.

For more concrete advice on the topic: you will realistically have to dedicate about 20-25 minutes just to explaining the definitions of language, $L$-structure/sentence/formula/theory, etc., because you will likely need examples at every step. I am algebraically minded, so I really like using rings as an example; often my first examples of an $L$-sentence are $1+1=0$, $\forall x\, (x=0)$, and other such oddities.

I want to point out that the key reason why the Ax-Grothendieck Theorem is amazing is because it's a particular application of the Compactness Theorem. That result is really beautiful. With an introductory talk, showcasing applications of compactness is in my opinion the best use of your time.

Here are a couple of concrete ideas for topics:

1. Show there exists a non-Archimedean field $K$ such that $\mathbb{R}\equiv_L K$ (where $L$ is the language of rings).
2. Show that there is a countable model of ZFC; if you want your talk to be philosophically oriented, this can be a good starting point for that.