Pete  L. Clark
Return to Pete's home page

 Math 8900II -- Introduction to Model Theory and its Applications, TuTh 2:15 - 4:30, Boyd 303

Instructor: Assistant Professor Pete L. Clark, pete (at) math (at) uga (dot) edu

Course webpage: (i.e., right here)

Office Hours: By appointment.

The ground rules are: (i) please give me at least 24 hours notice. (ii) Please send me an email the night before a morning appointment or the morning of a later appointment to remind me that we are meeting. (iii) If I do not show for an appointment (empirically, the chance that I will fail to show seems to be about 5-10%), feel free to call me on my cell phone. Probably I'm not too far away. (iv) If we do book an appointment, please do show up or call or email to let me know you're not coming!

Course text: The primary text will be the lecture notes posted below. However, I will mention that both when learning this subject seven years ago and in preparing for this course, I have found David Marker's GTM Model Theory: An Introduction to be extremely helpful. I currently have an extreme aversion to the major mathematical publishing companies, so I cannot in good conscience recommend that you buy any math book with your own money (exception: Dover books are great!). However, were this not the case I would recommend that you buy Marker's book. Also on Marker's webpage are some lecture notes which cover similar topics; these are also recommended.

Course Prerequisites: The only real prerequisite is the mathematical maturity and experience of a second (or strong first) year PhD student. Most of the applications will come from algebra, algebraic geometry or number theory -- in part because I think these will be of most interest to UGA graduate students -- so some prior exposure to these areas will help you appreciate the results. Little specific technical knowledge is required: for instance, we will see some results about affine algebraic varieties, but under the guise of "zero loci of systems of polynomial equations". Especially, no background in mathematical logic is assumed (really; if it helps, I know very little myself!). Of course I expect a knowledge of logic at the level required to understand and write proofs at the advanced undergraduate level and beyond. For instance, you should be familiar and comfortable with the use of quantifiers.

Course Content: Here is a rough outline:

Chapter 1: Languages, structures, sentences and theories
Chapter 2: Big Theorems: Completeness, Compactness, Lowenheim-Skolem
Chapter 3: Complete and model complete theories
Chapter 4: Categoricity: a condition for completeness
Chapter 5: Quantifier elimination: a condition for model completeness
Chapter 6: Ultraproducts

If time permits...

Chapter 7: The Ax-Kochen Theorem

Course Notes

Chapter 1 (9 pages): (pdf)
Chapter 2 (11 pages): (pdf)
Chapter 3 (12 pages): (pdf)
Chapter 4 (7 pages): (pdf)
Chapter 5 (12 pages): (pdf)
Chapter 6 (13 pages): (pdf)

The lecture notes include exercises. But some of the more important exercises -- as well as a few extras -- are reproduced here, for your convenience.

Exercise 0: Give some examples of integral domains R such that (i) -1 is not a sum of two squares in the fraction field of R, (ii) R is not a UFD, and (iii) R[i] = R[t]/(t^2+1) is a UFD. (Note that condition (i) is equivalent to the fact that R[i] is a domain whose fraction field is a quadratic extension of the fraction field of R.) Example: let k be a field in which -1 is not a sum of 2 squares, and put R = k[x,y]/(x2 + y2-1).

Exercise 1: Let K be the algebraic closure of a finite field and n a positive integer. Show that an injective polynomial map P: Kn -> Kn is surjective. (Hint: reduce to the case of finite subfields of K.)

Exercise 2: Does the conclusion of Exercise 1 hold with the roles of "injective" and "surjective" reversed? If not, where does the proof break down?

Exercise 3: For each fixed positive integer n, show that the class of fields K having the property that each injective polynomial map P: Kn -> Kn is an elementary class. In more concrete terms, find a collection of sentences in the language of rings such that a field has this property iff it satisfies all of these sentences.

Exercise 4: Show that, for every positive integer n, the rational field (Q) is not in the elementary class of fields of Exercise 3. Is the real field (R) in this class?

Exercise 5: Let X be a finite L-structure and let Y be an L-structure that is elementarily equivalent to X.
a) Show that Y is also finite, of the same cardinality as X.
b) (Harder) Show that in fact Y is isomorphic to X. (Hint: Your sentence should be an n!-fold conjunction of other sentences, one for each bijection from X to Y.)

Exercise 6: Let L be a language. Let Th(L) be the set of all L-theories, and let 2^{L-Str} be the collection of all classes of L-structures. (Throughout this problem, we are going to ignore the unpleasant fact that 2^{L-Str} is not a set. If this bothers you, replace it by the set of all sets of isomorphism classes of L-structures of cardinality bounded by some large cardinal kappa.)
a) Define maps Phi: Th(L) -> 2^{L-Str} and Psi: 2^{L-Str} -> Th(L) as follows: for an L-theory T, Phi(T) = C_T is the class of all models of T. For a class C of L-structures, we define Psi(C) to be the the set of all sentences which are true in every element X of C. Show that -- pushing the set-theoretic issues aside -- the pair (Phi,Psi) is a(n antitone) Galois connection.
b) Show that the induced closure operator on Th(L) sends a theory T to the set of all (syntactic = semantic, by the Completeness Theorem) consequences of T, i.e., to the set of all sentences P such that T \models P. Show that the induced closure operator on 2^{L-Str} sends a class C of L-structures to the minimal elementary class which contains C: let's call this the elementary closure of C.
c) Reinterpret the theorem "If a theory has arbitrarily large finite models, then it has infinite models" as a statement that a certain class of L-structures is not equal to its elementary closure.
d) What is the elementary closure of the class of all finite abelian groups?

Exercise 7: Let T be a satisfiable theory.
a) (Lindenbaum's Theorem) Show that T is contained in a maximal satisfiable theory (i.e., a theory which is not properly contained in any satisfiable theory). Suggestion: use Zorn's Lemma plus Compactness.
b) Characterise the set of theories T which are contained in a unique maximal satisfiable theory.