Model Theory, Spring 2001 Problem Set 3: due Mar. 2/Mar 16

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

1. Let p ∈ S(B) be an algebraic type. Suppose B is not contained in the algebraic closure of A ⊆ B. Prove p forks over A.

2. Let p ∈ S(B) and suppose B is contained in the algebraic closure of A ⊆ B. Prove p does not fork over A. (This may get clearer after a few more lectures.)

3. ϕ(\overline{x}, \overline{y}) has the order property if there exist \overline{a}_{i}, \overline{b}_{i} for i < ω such that ϕ(\overline{a}_{i}, \overline{b}_{j}) iff i < j. Prove that if ϕ has the order property then T is unstable. Think about the converse; if you really want to prove it, you will need a combinatorial lemma that can be found in the appendix to Shelah. Try to conjecture the lemma.

4. M is a conservative extension of N if for every \overline{m} ∈ M, and every ϕ(\overline{x}, \overline{y}) the subset of N (to an appropriate power): ϕ(M, \overline{m}) ∩ N is also given by ψ(\overline{x}, \overline{m}^{∗}) for an appropriate ψ and \overline{m}^{∗} ∈ N. Show T is stable iff every elementary extension of a model of T is a conservative extension.

5. Show that if d mapping ϕ(\overline{x}, \overline{y}) - an L(T) formula to dϕ(\overline{y}, \overline{m}) which defines t(\overline{c}/M) is a Boolean homomorphism from the Boolean algebra of L(T)-formulas to the Boolean algebra of L(T)-formulas with parameters from M. Find an example to show it is essential that M be a model.

6. Show that p = t(\overline{a}/M\overline{c}) is a coheir of t(\overline{a}/M) iff p is in the topological closure of \{t(\overline{m}/M\overline{c}); \overline{m} ∈ M\}. (\overline{m} and \overline{a} have the same length.)

Definition: The theory T does not have the finite cover property if for every finite set of formulas Δ, there is a k such that a Δ-type p is consistent if every k-element subset of it is consistent.

7. Show that if T is stable without the finite cover property there for every formula ϕ(\overline{x}, \overline{y}) and every A with |A| ≥ 2 there is a pair of formulas: (dϕ(\overline{y}, \overline{z}), χ(\overline{z})) such that:

1. If χ(\overline{c}), ϕ(\overline{x}, \overline{y}) iff dϕ(\overline{y}, \overline{c}) defines a complete consistent type over A.

2. for every A and p ∈ S(A), there is \overline{c}_{p} ∈ A with χ(\overline{c}_{p}) and dϕ(\overline{c}_{p}) defines p.

(I will hand out copies of 4-5 pages from my book which gives the background on definability.

(Aside: If T does not have the finite cover property then T is stable but I didn’t want to confuse this argument with that additional fact.)