Math 506: Model Theory
Math 506 is an introduction to Model Theory, a part of mathematical logic. At the beginning of the course, we will follow Dave Marker's Model Theory: An Introduction. I will post my notes from each day of class (see links below). Homework will be posted below as well. Grades in our course will be assigned based on performance in the weekly homework (70 percent) and the final exam (30 percent).
Office hours are a great time to come and ask about concepts as well as work on problems as a group. We will set fixed office hours after the first week. Until then please make an appointment.
Monday, January 13: we covered the basics of languages, structures, interpretations, embeddings, isomorphisms, and terms. We followed the notation of Marker, section 1.1. My notes are just an outline of what I planned to talk about - as the semester goes on, inevitably, I'll have to write down more detailed notes in order to give reasonable lectures, but this is an indication of how standard the material is. I also distributed homework #1, and I want to make a few remarks about the assignment. As usual, please let me know if you have any questions about the material.
Note that assignment 1 is due on January 22, rather than the Januart 20 (when there is no class because of MLK). Also, note that there will be no class on January 17.
Wednesday, January 15: we continued with the basic definitions from first order logic. My notes are a little more detailed, but since I am following Marker closely, they are still quite sketchy. Again, we are still in section 1.1 of Marker's text, though my presentation diverged just a bit. For instance, we defined homomorphism of structures, which Marker does not use in this part of the text - in places where I diverge from the text, I try to give more detail in the notes. For a presentation which uses homomorphism quite a lot more, see Hodge's texts (Model Theory and A Shorter Model Theory). We also introduced a notion we will deal with for the rest of the semester, satisfaction. We inductively defined what it means for a structure to satisfy a formula. This lead naturally to a hint on part of the last homework problem of assignment 1 - to prove that Z and ZxZ (as additive groups) are not elementarily equivalent. One way of stating our hint - think about sentences which talk about the cosets of 2G (for G=Z and G=ZxZ). Note that the hint in this language will work well for the more general parts of the problem.
We also mentioned that there are very (!) hard problems about groups and elementary equivalence.
Wednesday, January 22: We finished up some basic definitions and defined (again) elementary equivalence, substructures, theories, the theory of a structure, and proved some basic results which you should know - that quantifier free formulas have the same truth value in a structure and its substructures and that isomorphism is implies elementary equivalence. More though, you should know the proofs of these results (basically both are inductions on the complexity of the formula). The proof techinque is one that is repeated throughout the beginning of model theory. Following the results, we defined elementary classes, and we gave some examples. We also explained why at this point it is a bit hard to show that any particular thing is not an elementary class, but we will have tools soon. The notes are still pretty sketchy, since every good model theorist knows these types of proofs by heart (see 1.2 and 1.3 of Marker's text).
Friday, January 24: We continued with definitions and ultimately defined what it means for a subset of a model to be definable. This is one of the most important notions from our course, and it appears in Marker's text in section 1.3. Before getting to this we covered some of the examples from 1.2. I am going to diverge from the text in the coming lectures and do the compactness theorem (chapter 2) next (before topics like interpretability, which occurs in chapter 1). I will post combined notes for Friday along with the Monday notes.
Monday January 27 - Friday January 31: We covered the compactness theorem in detail. Our approach was to avoid the notion of a formal proof entirely, and work with the notion of satisfiability in detail. Ultimately our proof involved what is called a Henkin construction (by the way, you can use such an argument to prove various things beyond compactness).
The notes are more complete than past weeks, but still involve a lot of places where I omit the proof in the notes, but did it on the board in class. You can find complete details in Marker's text, Chapter 2, section 1. Note that I distributed HW 2, and you should certainly be working through it, since it is rather long.
Monday, Feb 3: We covered Vaught's test, and then looked at applications for algebraically closed fields, torsion free divisible abelian groups, and a couple other examples. We actually went through the detail of proving the Ax-Grothendieck theorem. We have now covered material through 2.2 of Marker's text.
Wednesday Feb 5 -Friday Feb 7: After the easier part of Lowenheim-Skolem, we proved Tarski-Vaught, a criterion for a substructure to be an elementary substructure. This criterion simplifies many proofs (basically, doing this one induction now saves you from repeating essentially the same argument many times) in applications like Lowenheim-Skolem and proving things like unions of elementary chains are elementary extensions - by the way we proved both of those things. We are in the middle of chapter 2 of Marker's text - see the notes for specific results we covered.