# Math 770

# Fall 2023

Instructor: Uri AndrewsEmail: (My last name)@math.wisc.edu

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:

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:

§1.2:3,4,6

§1.3:1,2

§1.4:2,3

§2.1:5

§2.2:1,3,5,7 - Due Tuesday 10/10:

§2.4: 1,2,4

§2.5: 2,3 (for $L_{\omega,\omega}$),4

§3.1:2,7

§3.2:3,4,5,6

§3.3:2,5,7 - 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