My Thoughts on Teaching Model Theory

I have been realizing recently that many of my math ideas in their “ultimate” forms seem highly suited for formulations in model theory. However, when first trying to study some of the resources I found online for model theory, I didn’t even understand that well what the subject was about or why certain definitions were made, let alone appreciate the overall power of the subject. I think there should be a different approach to teaching model theory, incorporating better motivation and more fully reflecting the incredible power and scope of the subject.

Here, we discuss a plan for teaching an introduction to model theory.

Prerequisites

Initially, I had conceived of model theory as suitable after a prerequisite of universal algebra; however, it seems that universal algebra may not be a subject that is generally well-understood, so going via it as a prerequisite may not be the best general approach unless someone is already familiar with it. However, abstract algebra doesn’t seem to pose as much of a difficulty, so that may still be fine as a prerequisite — in fact, it would serve as a better source of examples to motivate homomorphisms and isomorphisms, which in turn would be good for justifying the inclusion of free variables. Requiring universal algebra, on the other hand, may restrict the audience unnecessarily.

Also, some of my previous attempted explanations had been more like explanations of abstraction in general as opposed to the specific abstract subject of model theory — I hadn’t adequately clarified what differentiates model theory from other abstract subjects like abstract algebra and universal algebra. Illuminating this would require a different, or rather an additional, approach.

What I eventually came to think is that model theory is best taught after introductory mathematical logic. Indeed, model theory is in fact a branch of mathematical logic, and this is what differentiates it from abstract algebra: it deals generally with logical statements, not with specific sets of axioms, as algebraic structures would. Illuminating that would help clarify the scope and formulation of model theory a lot more.

Come to think of it, we can even introduce universal algebra after model theory, as essentially a special subset of first-order model theory. Or alternatively, we could use those connections between universal algebra and model theory to enhance the learning experience for those who do have background in universal algebra. Actually, we can do both, yielding more value both for people who didn’t know universal algebra previously as well as for people who did. Thus, we can say that universal algebra is recommended, but not required.

Thus, the prerequisites we would assume are (1) Intro to Mathematical Logic and (2) Abstract Algebra, where we also recommend (but do not require) Universal Algebra.

Outline

Now, we outline what a model theory textbook table of contents would look like. (Note that this section is prone to constant fluctuation as my thoughts and learnings evolve.)

First, when I wrote out a sample exposition of this, I saw that it’d be a great way to introduce first-order logic too — in fact, motivation for first-order logic goes hand-in-hand with motivation for first-order model theory (which would also be the best model theory for an introduction.) This is my plan:

1. Formulations of Model Theory — Follow the below path to motivate and formulate the standard definitions of first-order logic and its model theory.
   1. Start from introductory (zeroth-order) mathematical logic and introduce variables into statements. An example could be that instead of the statement “Pam is wearing a green shirt,” we could have a generalized statement like “ is wearing a green shirt,” where could be someone from a larger set of people that we could assess this statement for. Then, the truth of “Pam is wearing a green shirt” is equivalent to the truth of “ is wearing a green shirt” for .
   2. Introduce free vs bound variables by saying that the occurrence of before was free, while the occurrence of in “there exists such that is wearing a green shirt” is bound. Introduce existential quantifiers along with this. Mention that we can even have statements (introduce the technical term “formula”) with both free and bound statements, e.g., “there exists such that is wearing the same color shirt as ”.
   3. Introduce the concept of a model, both from examples like this one as well as those that arise in standard math (e.g., the real numbers and common algebraic structures); along with this, introduce the concept of formulas where all variable occurrences are bound, or sentences.
   4. Build formulas “from the ground up” and introduce languages, relations, and operations. Write the first full definition of a model of a set of sentences. (Mention that if we remove relations and work with just operations then this is exactly algebra, and illuminate the ensuing connection with abstract algebra and universal algebra.)
   5. Discuss examples of theories that are not first-order (or at least not obviously) — for example, “there exists such that .” Clarify the nuances of first-order logic carefully with the counterexamples.
   6. Introduce homomorphisms and isomorphisms, generalizing the concept from abstract algebra to maps that preserve language symbols.
   7. Illuminate the appearance of free variables in model theory by showing that isomorphism is equivalent to what can be informally described as “truth equivalence of all formulas, with appropriate substitution”; this is essentially a formalization of a common guiding intuition from abstract algebra. (I myself first figured this out in my exploration on formalization of equivalency of axiomatizations.) Further expound on the difference between formulas and sentences by discussing elementary equivalence and the fact that isomorphism is in general not the same as elementary equivalence.
2. Major Results of Model Theory — Discuss and prove the major results of first-order model theory.
   1. Right now, I need to learn more in order to fill out this section.
3. Towards Higher-Order Model Theory — Discuss the beginnings of model theory of non-first-order logics, using examples of non-first-order theories. Maybe incorporate the idea that philosophically the scope of mathematical logic would be (possibly informally) defined to encompass any kind of logical statement, so that any theory could be expressed by some (suitably formalized) logic; then, its full scope, model theory can be thought of as the study of models of any kind of theory.
   1. Right now, I need to learn more to fill out this section. What logics are typically in scope for an introductory model theory exposition? Many-sorted logic? Second-order logic? I personally seem partial to many-sorted logic (which I learned about through an exploration that I will probably write about soon), but I should learn more and update this once I have a better understanding of both (1) the landscape of different common logics and (2) what is typical of an intro to model theory treatment.

Sidenote on the higher-order model theory part: I have recently been reading about things like abstract model theory and Lindström’s theorem, so the scope of the subject goes even beyond the model theories of particular logics. What an exciting area of math to learn about!

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EDIT: After taking another look at this, it seems that the “Major Results of Model Theory” and “Towards Higher-Order Model Theory” sections would just be copies of what is already included in standard model theory textbooks; unless I find that I think I can teach things from those sections better (once I learn them), I think it makes most sense to focus my expository endeavors in this subject for now on the first section, where I provide better motivation to reach the standard formulations of first-order model theory and offer a better bridge to some of the standard texts.