Model Theory, Spring 2001 Problem Set 4: due Apr. 9

Be prepared to discuss several of these questions and explain how you have begun them in class on Apr 2. The extra time will be work them out; there may be a few added.

A theory is small if there are only countably many n-types over the empty set (for every n).

1. Show that a small countable theory T is ω stable if and only if for every M ⊧ T and p ∈ S(M), p is definable over a finite subset of M.

2. If T is stable, E is a set of indiscernibles over A with |E| ≥ \overline{κ}(T), there is an E_{0} ⊆ E such that |E_{0}| < \overline{κ}(T), E − E_{0} is independent over E_{0} and all elements of E − E_{0} realize the same strong type over E_{0}.

3. Show that if T is a small superstable countable theory and I is a countable set of indiscernibles over ∅ then |S(I)| = ℵ_{0}.

4. Show that if α is an automorphism fixing A then for any a, b, if stp(a/A) = stp(b/A) then stp(αa/A) = stp(αb/A). Give an example where stp(a/A) ≠ stp(αa/A).

The type p is almost satisfied in A if every finite subset of p is realized in every model containing A.

5. (T stable) Show p does not fork over A iff p is almost satisfied in A. What goes wrong for simple T?