Math 770

Fall 2023

Instructor: Uri Andrews
Email: (My last name)
Office: 723 Van Vleck Hall
Textbook: "A Shorter Model Theory" by Wilfred Hodges
Office Hours: Wednesday 1-3PM on zoom. Drop me an email to let me know when you'll stop by and I'll send a link.


Homeworks will be regularly assigned and posted here with due-dates. The numbers below refer to section and problem numbers in Hodges. If you do not yet have the textbook and need to know what the problems are, please drop me an e-mail.
  • Due Tuesday 9/26:
  • Due Tuesday 10/10:
    §2.4: 1,2,4
    §2.5: 2,3 (for $L_{\omega,\omega}$),4
  • Due Thursday 11/9:
    §5.1: 1,2a,4,5,6,7,9,10
    §5.2: 1,2,6,9
    §6.2: 3,4,5,6,7,9
  • Due Tuesday 11/28 (Base set: Non-starred questions. Starred questions form the expanded homework set.):
    §6.1: *1, 2, 3
    §6.3: 1, *2, 3
    Is Th$(\mathbb{R},<,Z)$ $\aleph_0$-categorical where $Z$ is a unary predicate naming $\mathbb{Z}$?
    Is Th$(\mathbb{R},<,Q)$ $\aleph_0$-categorical where $Q$ is a unary predicate naming $\mathbb{Q}$?
    §6.4: 1, *2, *4, *8,
    *Find an example of an $\aleph_0$-categorical theory which does not have QE. (Problem 1 is a hint here -- A Fraisse construction builds something ultrahomogeneous, which therefore has QE. But a reduct of something ultrahomogeneous might not have QE.)
    Let $\mathcal L$ be $\{U_i\mid i\in \omega\}\cup {E}$ where $E$ is binary. Let $\mathbf K$ be the collection of all finite $\mathcal L$-structures $\mathcal A$ where $\forall x (U_{i+1}(x)\rightarrow U_i(x))$ for each $i$, $E$ is symmetric and irreflexive, and so that for each $i$, $\mathcal A$ satisfies $\neg \exists x \exists y (xEy \wedge \neg U_i(x))$. Show that the Fraisse generic for $\mathbf K$ is not $\aleph_0$-categorical and does not have QE.
    §7.4: *3, *4, 11bc, 13
  • Not due, but recommended for those prepping for quals. Also, the grader said he's willing to look it over and give feedback if you want. (Base set: Non-starred questions. Starred questions form the expanded homework set.)
    §5.4: 2, 4, 5*, (the grader refers to this as "the evil problem" -- it's not truly all that horribly evil, though it is certainly on the naughty list.), 6*
    §5.5: 1, 4
    §7.1: 2, 4, 8,
    §7.2: 4, 7*,
    §7.3: 1, 3*, 5*, 8 ("companionable" means "has a model companion")
    §8.1: 1
    §8.2: 4

Source of Problems:

A good source of problems can be found in the "M" section of old qualifying exams. They can all be found here. If you are a math graduate student, do consider taking 773 and the logic qual. After this course, you're most of the way there!


  1. Andres Caicedo's notes on proving compactness via the ultraproduct construction.