Model-theory exercises

Math 406

2004.12.30

Problem 1. Suppose L = {P }, where P is a singulary predicate. Let 𝔄 be a finite L-structure, and suppose 𝔅 ≡ 𝔄. Prove 𝔅 ≅ 𝔄.

Problem 2. Let L = {P_{n} : n ∈ ω}, where each P_{n} is a singulary predicate. Let T be axiomatized by the sentences

∀x P_{0}x,
∀x (P_{n+1}x → P_{n}x),
∃x ¬(P_{n}x → P_{n+1}x),
∀x ∀y (¬(P_{n}x → P_{n+1}x) → (¬(P_{n}y → P_{n+1}y) → x = y)),

where n ∈ ω. Prove that T is complete.

Problem 3. Let Σ and Σ′ be sets of sentences of some L such that

Σ ⊨ ¬σ ⟹ Σ′ ⊭ σ

for all sentences σ of L. Prove that Σ ∪ Σ′ has a model.

Problem 4. Suppose Φ and Ψ are 1-types of a consistent theory T. Prove that T has a model in which both Φ and Ψ are realized. (Use only Compactness and the appropriate definitions.)

Problem 5. Assuming T is ω-categorical, prove that T has only countably many complete types.