Homeworks:

• 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
  

Source of Problems:

Links

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