In this post I'll discuss some basic results of model theory. It may be helpful to read through my previous post if you haven't yet. Model Theory is an implicit context for the Heavily Advanced Epistemology sequence and for a few of the recent MIRI papers, so casual readers may find this brief introduction useful. And who knows, maybe it will pique your interest:

A tale of two logics

propositional logic is the "easy logic", built from basic symbols and the connectives "and" and "not". Remember that all other connectives can be built from these two: With Enough NAND Gates You Can Rule The World and all that. Propositional logic is sometimes called the "sentential logic", because it's not like any other logics are "of or relating to sentences" (/sarcasm).

first order logic is the "nice logic". It has quantifiers ("there exists", "for all") and an internal notion of equality. Its sentences contain constants, functions, and relations. This lets you say lots of cool stuff that you can't say in propositional logic. First order logic turns out to be quite friendly (as we'll see below). However, it's not strong enough to talk about certain crazy/contrived ideas that humans cook up (such as "the numbers").

There are many other logics available (second order logic AKA "the heavy guns", ω-logic AKA "please just please can I talk about numbers", and many more). In this post we'll focus on propositional and first-order logics.

I'm reviewing the books on the MIRI course list. After my first four book reviews I took a week off, followed up on some dangling questions, and upkept other side projects. Then I dove into Model Theory, by Chang and Keisler.

It has been three weeks. I have gained a decent foundation in model theory (by my own assessment), but I have not come close to completing the textbook. There are a number of other topics I want to touch upon before December, so I'm putting Model Theory aside for now. I'll be revisiting it in either January or March to finish the job.

In the meantime, I do not have a complete book review for you. Instead, this is the first of three posts on my experience with model theory thus far.

This post will give you some framing and context for model theory. I had to hop a number of conceptual hurdles before model theory started making sense — this post will contain some pointers that I wish I'd had three weeks ago. These tips and realizations are somewhat general to learning any logic or math; hopefully some of you will find them useful.

Shortly, I'll post a summary of what I've learned so far. For the casual reader, this may help demystify some heavily advanced parts of the Heavily Advanced Epistemology sequence (if you find it mysterious), and it may shed some light on some of the recent MIRI papers. On a personal note, there's a lot I want to write down & solidify before moving on.

In follow-up post, I'll discuss my experience struggling to learn something difficult on my own — model theory has required significantly more cognitive effort than did the previous textbooks.