0.1 MORE EXERCISES

(1) Let T be a countable complete theory. Suppose that T is stable in ℵ_{0}. Prove that for every λ ≥ ℵ_{0} the theory T has a saturated model of cardinality λ.

(2) Let T be a countable complete theory. Suppose that T is stable in ℵ_{0}. If R[φ(x; a)] = α for an ordinal α then for every β < α there exists a formula ψ_{β}(x; b_{β}) such that R[ψ_{β}(x; b_{β})] = β.

(3) Let p be a type. Prove that for every automorphism f (of ℭ) we have that R[p] = R[f(p)].

(4) Use (2) and (3) to give an alternative proof to the fact that for countable theories, ℵ_{0}-stability implies R[x = x] < ω_{1}.

(5) If T has the order property then T is unstable.

(6) Let λ be an infinite cardinal, show that I(λ, P A) > 1.

(7) Let T be a complete first order theory. Prove that the following are equivalent.

(a) T is stable

(b) every countable complete T′ ⊆ T is stable.

(8) Show that T_{ind} is unstable and is model complete.

(9) Show that the rank function R[·] defined in class and the function from page 211 of my book are equal.

(10) Let ⟨G, ·⟩ be an infinite group we say that G is stable/superstable iff T h(⟨G, ·⟩) is. By H ≤_{def} G denote H is a definable subgroup of G.

(a) G is ℵ_{0} stable then

¬∃{H_{n} ≤_{def} G | n < ω} with 1 < [H_{n} : H_{n+1}]forall n < ω.

(b) G is superstable then

¬∃{H_{n} ≤_{def} G | n < ω} with [H_{n} : H_{n+1}] ≥ ℵ_{0}forall n < ω.

(c) G is stable then there is no {H_{n} ≤ G | n < ω} uniformly definable subgroups with H_{n+1} ⪇ H_{n}foralln < ω. Where uniformly definable subgroups stands for: There are φ(x; y; a) maybe with parameters from G and {b_{n} | n < ω} ⊆ G such that H_{n} = φ(G; b_{n}; a) for all n < ω.

(11) Use the previous exercise to prove the following:

Theorem 0.1.1 Let ⟨G, ∘⟩ be a group and suppose that h : G → G is a non trivial definable (in the language of group theory) homomorphism. If T h(⟨G, ∘, h⟩) is superstable and the kernel of h is finite then h must be surjective.

Moreover,

Theorem 0.1.2 Let ⟨G, ∘⟩ be a group and suppose that h : G → G is a non trivial definable (in the language of group theory) homomorphism. If T h(⟨G, ∘, h⟩) is ℵ_{0}-stable then h must be surjective.

Hint: Use Exercise 1.6.26.