Model Theory

Module aims

Build models from syntactic data and transfer properties along elementary maps, keeping language and theory distinct.

Course outline

1. 01 Languages and structures 20h

   Learning result Interpret a language in a structure and evaluate a formula.

   Formative check Four-option check

2. 02 Theories and satisfaction 25h

   Learning result Separate a theory from one of its models; give two non-isomorphic models of the same theory if the syllabus example allows.

   Formative check Counterexample

3. 03 Compactness 25h

   Learning result Apply compactness to obtain a model of an infinite theory from finite fragments.

   Formative check Written derivation

4. 04 Löwenheim–Skolem (taught grain) 20h

   Learning result State a downward or upward form with cardinality hypotheses.

   Formative check Four-option check

5. 05 Elementary maps 25h

   Learning result Check elementarity on a small example and say what isomorphism would add.

   Formative check Written derivation

6. 06 Synthesis 35h

   Learning result Write one compactness-plus-structure argument with the language named.

   Formative check Written derivation

Learning results

Define a structure for a language, check a theory’s axioms in an example, and apply compactness or Löwenheim–Skolem at synopsis grain.

Subject skills

Languages, structures, elementary equivalence, compactness, types at taught grain.

Transferable skills

Keep syntax and semantics labelled; do not treat a single model as the theory.