 Solution

ID: past-exam-of-the-model-theory-and-godel-s-incompleteness-theorems-course-of-the-university-of-oxford/2008/b/6/i/solution

Let

L_{A}

expand

L

by a constant

c_{a}

for each

a \in A

, and let

A_{A}

be the canonical expansion interpreting

c_{a}

a

. Its diagram of a structure is the set of all atomic and negated atomic

L_{A}

-sentences true in

A_{A}

. Its elementary diagram is the set of all

L_{A}

-sentences true there:

Diag (A) = {θ : θ atomic or negated atomic, A_{A} ⊨ θ},

CDiag (A) = Th (A_{A}) .

A \subseteq B

means that

A \subseteq B

, constants have the same interpretations, the operations of

B

restrict to those of

A

, and each relation restricts to the corresponding relation on

A

A

A ≼ B ⟺ A ⊨ ϕ (a) ⟺ B ⊨ ϕ (a) for every ϕ, a \in A .

Here the right-hand condition includes the substructure requirements by applying it to atomic logical formulas and their negations.

 New to topics? Read the docs here!