A typical definition of model theory is “universal algebra plus relations.” However, the term “model” in natural language can typically be used quite generally, especially if we consider how people typically understand its meaning for designing and developing any kind of system. Thus, we could guess just based off terminology that in mathematics, a “model” could be defined, as generally as possible, basically as an “example” (set plus interpretation function) that satisfies any given set of axioms. Is this equivalent to the traditional formulation of model theory? In other words, does the traditional formulation of model theory capture the most general kinds of axioms that an “implementation” could satisfy?
I first started doing this exploration back in May 2021, where I wrote down some thoughts that I now think are wrong. At that point, I concluded that the standard definitions of model theory would indeed be the most general formulation, but now this seems questionable. Based purely on appearances, this is clearly not the case. For example, the standard definitions for topology and metric spaces are not amenable to model theory.
And in general, my terminology-based logic was flawed: in math, we have to be careful when interpreting terminology based off of natural language meanings. This is because often times terms that are borrowed from natural language are given technical meanings that may have more specific aspects to them, and we can’t fully understand the scope and definition of a term without incorporating these more specific elements in our learning. For example, besides model theory, this is true of representation theory.
This has implications in education. For maximum clarity, it’s important to distinguish what a subject is not as well as what it is. Part of this includes defining the scope of a subject or a particular term properly, and that necessitates avoiding false equivalencies or trivializations of non-obvious equivalencies. To do this, it is useful (especially for a technical audience) to be as specific and accurate as possible about the conditions that are part of the definition of a concept. For example, I have been working on an expository account of model theory, and this discussion will figure into my presentation there.
Despite what we have been saying, if we consider equivalencies that may not seem obvious, then there can be a real question here that could lead to real and interesting insights. The better question we can ask for this is: can we consider certain classes of systems that have axioms that “seem more general” than those expressible in the language of model theory, but where we can nevertheless still derive “non-obvious” ways to express those systems via model theory? Expressed more succinctly, what classes of systems are non-obviously amenable to model theory? Even if we can’t do this, can we still come up with general methods to apply model theory to more kinds of systems? An example of this kind of study would be my exploration on Topology as an Algebraic Structure; indeed, this question would be a generalization of the one asked there.
It remains to investigate this further.
EDIT (January 2023): The issue here is that I had only seen first-order model theory. This branch of model theory is indeed equivalent to “universal algebra plus relations.” It is known that first-order logic is not expressive enough for some theories: see https://www.math.ias.edu/~lurie/278xnotes/Lecture1-Overview.pdf. Essentially, mathematical logic by definition defines the general syntax of any kind of logical statement, and there are multiple possible such definitions that correspond to multiple possible logics with different expressivities. (For example, first-order logic is more expressive than propositional / zeroth-order logic, and higher-order logic is even more expressive.) Thus, by definition, model theory encompasses any kind of “structure that satisfies a given set of axioms.” If this weren’t the case, then model theory couldn’t be considered a branch of mathematical logic. So topology and all other mathematical structures are amenable to model theory, just not necessarily first-order.
The questions I asked are still interesting: for example, it still remains to figure out whether topology is amenable to first-order model theory, and it still remains to try to apply universal algebra to first-order model theory and possibly other branches of model theory, in the same way that we have been exploring the application of universal algebra to topology.
