Lehr- und Forschungsgebiet
Mathematische Grundlagen der Informatik RWTH Aachen
Prof. Dr. E. Grädel, R. Rabinovich
WS 2010/11
Mathematical Logic II — Assignment 10
Due: Monday, January 10, 12:00
Exercise 1 3 + 5 Points
LetA⊆B be twoτ-structures for a signature τ. Prove the following statements.
(a) If for all finite setsC ⊆A and all b∈B there exists an automorphism f on B such that f(c) =c for all c∈C and f(b)∈A thenAB.
(b) The converse does not hold.
Exercise 2 4 Points
Let T ⊆ FO(τ) be a theory and let ϕ, ψ ∈ FO(τ) be sentences. Prove that the following statements are equivalent.
(a) There is a Π1-sentenceϑ∈FO(τ) such thatT |=ϕ→ϑand T |=ϑ→ψ.
(b) For all modelsA,B of T withA⊆B,B|=ϕ impliesA|=ψ.
Hint: Consider the set{ϑ∈FO(τ) |ϑis a Π1-sentence with T |=ϕ→ϑ}.
Exercise 3 5 Points
Prove that a theoryT ⊆FO(τ) is model complete if and only if for each formulaϕ(x)∈FO(τ) there is some Σ1-formula ψ(x)∈FO(τ) withT |=∀x(ϕ(x)↔ψ(x)).
Hint: Consider the set Φ :={ψ(x)∈FO(τ)| ψ∈Σ1 and T |=∀x(ψ(x)→ϕ(x))}.
Exercise 4 3 + 4 + 4 Points
LetAbe aτ-structure and letB ⊆A. Ann-typepofAoverB is aprincipal typeif there exists a formula ϕ(¯x)∈p such that AB |=∀¯x(ϕ(¯x)→ψ(¯x)) for all ψ(¯x)∈p.
(a) Letp be a complete type ofA overB that is realised by a tuple ¯b⊆B. Prove that pis a principal type.
(b) Show that all principal types ofA overB are realised in A.
(c) LetA and B be two τ-structures withA⊆B. Prove that A4B holds if and only if all principal types ofB overA are realised inA.
http://logic.rwth-aachen.de/Teaching/MaLo2-WS10
