Model Theory, Spring 2001 Problem Set 2: due Feb. 12

Marker: 2.5.17 (in your notes); 3.4.12, 3.4.13, 3.4.14 being handed out.

1. Let p ∈ S(A) be an algebraic type. I.e there are only finitely many realizations of p. Show p is principal (i.e. isolated). More generally, show that if tp(a_{1}, … a_{n}/A) is algebraic then each a_{i} ∈ acl(A).

2. Let T be an uncountably categorical theory. Show every uncountable model of T is ℵ_{0}-saturated. (This should be very easy assuming Morley’s theorem. How much of it can you get from the preliminary results?)

3. Show that if T is ℵ_{1}-but not ℵ_{0}-categorical, the prime model of T is minimal – has no proper elementary submodel.

4. Let T be any complete countable theory. Prove that for any cardinal κ, there is a model M of T with cardinality κ such that every definable subset of M is either finite or has cardinality κ.

5. T′ is said to be an inessential extension of T of T′ is the theory of (M, x)_{x∈X} where X is a subset of M. Show that if |X| < κ and T is κcategorical then T′ is κ-categorical.

6. (Uncountable languages). Show that if T is the theory of vector spaces over an uncountable field F then T is |F |^{+}-categorical but has a model of cardinality |F | which is not homogeneous. (This contrasts with a result we may prove: every countable model of an ℵ_{1}-categorical countable theory is homogenous– indeed ℵ_{1}-homogenous. )