# Is algebra needed to really understand and/or enjoy model theory?

What are the desirable pre-requisites to be able to learn model theory well? In particular, it seems that connections to algebra are used heavily especially as examples. I would like to know if a good grasp of algebra is essential for a deeper understanding of model theory and whether a good grasp of analysis and topology would compensate for lack of the former.

Thank you.

• Knowing something never compensates not knowing something else. – Mariano Suárez-Álvarez May 18 '15 at 2:31
• Sorry for being unclear, I meant that analysis and topology might compensate in the following sense: in terms of examples and applications and also to have something intuitive and concrete to keep in mind when braving the more abstract concepts of model theory. – Student May 18 '15 at 2:45

Many people on this site might have a better answer than me. However, I will attempt to answer this question anyway.

Claim 1: Algebra is an unnecessary prerequisite for beginning model theory.

The classical results in model theory (e.g. compactness, completeness, Upward and Downward Löwenheim-Skolem) do not follow from classical results in algebra. One doesn't need to know much about fields, groups, rings, et cetra to adequately grasp these concepts. These theorems are important to understanding model theory and are central to the field, but do not require any formal algebra.

Claim 2: Algebra provides a large class of nice examples for model theory.

After learns the classical results in model theory (the ones listed above is not an exhaustive list), in my opinion, one should have examples they can play with. While again, algebra is not necessary, not understanding how model theory plays with these structures leaves out a huge class of examples. Some non-algebraic examples include The Random Graph and other infinite graph structures, Peano Arithmetic, Infinitely many equivalences classes all with infinitely many elements, and different types of linear orders. While one can think about how the classical results relate to these structures, one misses a huge amount of data by not understanding how they interact with algebraic structures. Understanding how Ultraproducts work with these structures will give you some idea about how they work, but understanding how ultraproducts work with algebraic structures provides a more complete picture and give much more useful examples (in my humble opinion).

Claim 3: Algebra has historically (and one might even argue this is still true today) guided some proofs in model theory.

When Morley proved his famous Categoricity Theorem, the background object Morley had in mind was the structure $Th((\mathbf{C}; +, \times, 0, 1))$. The concepts of 'algebraic closure' and 'transcendental element' are terms stolen from algebra. Furthermore, as model theory progresses into the future, classical algebraic concepts keep popping up in model theory. For instance, Consider Zil'ber's trichotomy conjecture. The fact is, algebra and model theory have much more in common that originally meets the eye.

Claim 4: Model Theory has potentially deep connections to Algebraic Geometry

This claim follows from the proof of the Mordell-Lang Conjecture by Hrushovski in all characteristics. This proof was a model theory proof of a algebraic geometry theorem (as well as the first one). This is an exciting time in the field of study and while you might not need to know/understand this result to study model theory, it is important to understand that results like this are possible (and extremely interesting). I would argue that learning how model theory can be applied to algebra further demonstrates a deep connection between the two fields (again, in my humble opinion).

Claim 5: However, again, Model Theory need not be applied to Algebra.

There are other areas of model theory which do not really touch algebra. In your question above, you mentioned analysis and topology. Topology, while not first-orderizable, does not really give good examples of structures to study for model theory (however, Model theorists from time to time use topology (see Stone Spaces)). Yet, one can use model theory to study something called 'Tame Topology'. In particular, O-minimality studies (usually) structures over the reals which are 'nice'. There are some beautiful foundational theorems for these structures (e.g. O-minimality is preserved under elementary equivalence, the cell-decomposition theorem - proven by Pillay, Knight, and Stienhorn). Furthermore, this is an active area of research (and potentially, mine). The results here need less of an understanding of algebra and more an understanding of topology, analysis, and model theory in general.

Claim 6: Unapplied Model Theory is also an active area of research.

Finally, I should say that not all of model theory is applied. There are interesting results in pure model theory which are still being produced. For example, Byunghan Kim has many results in simplicity theory, Malliaris and Shelah have amazing results in general classification theory (The Keisler Order has an infinite descending chain!!).

TL;DR - In my humble opinion, you should probably learn some algebra (at least, at some point).

• Isn't Boolean algebra rather useful for model theory? E.g., the theory of a reduced product $A^I/D$ being determined by the theory of $A$ and the theory of the Bookean algebra $2^I/D$ (Feferman-Vaught, I think)? – bof May 18 '15 at 5:02
• @bof: This is not an exhaustive list of what is necessary to do modern model theory. Surely, Łoś's theorem and the theory of ultrafilters is important. However, it is my impression that these are somewhat side notes to what you really should be knowing and the current direction of the field (As I perceive the current direction, which, you know, could be incorrect). But yes, Boolean algebras do come up in model theory. – Kyle Gannon May 18 '15 at 5:09
• Typo in the follow up of claim 5 : it is probably Pillay and not Pilly. – Pece May 18 '15 at 14:18
• @Pece: You're right! My bad. – Kyle Gannon May 18 '15 at 16:42

I have always felt that examples usually just confuse you (though not always), having always specific properties that are traps as they do not hold in general.

S.Shelah, The future of set theory

Indeed, the easiest (counter)examples in model theory are obtained by hand using basic combinatorial ideas. This holds for simple examples, as well as major innovative examples such as Hrushovski'e new minimal set. Actually, it is the latter that inspired the interest in algebraic structures such as Zilber's fields.

However, if you do not have the brain of Shelah, or you aim for a tenure position, you better prove to the rest of the word that what you do is somehow connected to mainstream mathematics. You get $10\times\#$citations.

In the '90s it was common to think that algebraic geometry was the nearest area. In the last years there are also other options. E.g.

1. Continuous model theory is gaining popularity. It has connections to functional analysis.

2. Descriptive set theory and functional analysis have interacted with model theory in many other ways. This is an examples and this is a second very different examples

3. There have been some major achievements that connect free groups and model theory (though no model theorist understand them at the moment). See work related to Zlil Sela.

4. Discrete combinatorial geometry has some surprising applications to model theory. See this for example Though this goes the opposite direction, still it helps in demonstrating that model theory is connected to mainstream mathematics.