Questions: Model Completeness and the Model Completeness Test

Question 1

A theory T is model-complete. What can you directly conclude?

• A. Every model of T satisfies exactly the same sentences — T decides every sentence
• B. Every embedding of one model of T into another is an elementary embedding
• C. All models of T are isomorphic to each other
• D. T is complete and has quantifier elimination

Answer: (B)

Model completeness means precisely that every embedding between models of T is elementary — the submodel and the extending model agree on all first-order sentences with parameters from the submodel. Option A describes completeness (a different property). Option C would follow only if T were also categorical. Option D conflates model completeness with the stronger property of quantifier elimination, which implies model completeness but is not equivalent to it.

Question 2

You are trying to determine whether a theory T is model-complete. You verify that for every pair of models M ⊆ N of T, every existential sentence with parameters from M that is true in N is already true in M. What have you established?

• A. T is complete — it decides every sentence
• B. T is model-complete by Robinson's test
• C. T admits quantifier elimination
• D. All models of T are elementarily equivalent

Answer: (B)

This is Robinson's model completeness test: T is model-complete if and only if for every M ⊆ N ⊨ T, every existential sentence true in N with parameters from M is already true in M. The test checks whether extending a model can 'create' new witnesses for existential claims about old elements — if not, embeddings are elementary. This does not establish completeness (option A), which requires T to decide every sentence outright, nor quantifier elimination (option C), which is a stronger property.

Question 3

A model-complete theory is expected to be complete — if most embedding between its models is elementary, then most its models satisfy the same sentences.

• T. True
• F. False

Answer: False

Model completeness and completeness are independent properties. Model completeness governs how models extend each other (embeddings are elementary), while completeness means the theory decides every sentence (T ⊨ φ or T ⊨ ¬φ for all φ). Algebraically closed fields of different transcendence degrees are models of ACF (fixed characteristic) that are not isomorphic, and ACF is model-complete. However, without fixing the characteristic, ACF has models of different characteristics satisfying different sentences — so the theory without a fixed characteristic is model-complete but not complete.

Question 4

If T is model-complete, then every formula is equivalent modulo T to a universal formula (one using only universal quantifiers over a quantifier-free matrix).

• T. True
• F. False

Answer: True

This is one of the central equivalent characterizations of model completeness. Universal formulas are preserved under passing to substructures, and if every formula is universally equivalent, then embeddings automatically preserve all formulas in both directions — exactly the condition for being elementary. This equivalence is what makes model completeness analyzable at the formula level: you do not need to check all embeddings directly if you can show every definable property is universally expressible.

Question 5

What is the key distinction between a theory being 'model-complete' and being 'complete,' and why does the example of algebraically closed fields illustrate the difference?

Model answer: A complete theory decides every sentence (for every φ, either T ⊨ φ or T ⊨ ¬φ), while a model-complete theory only guarantees that embeddings between its models are elementary. ACF without a specified characteristic is model-complete — any embedding of one algebraically closed field into another is elementary — but it is not complete, because models of characteristic 0 (like ℂ) and characteristic p satisfy different sentences. Fixing the characteristic (ACF_p) makes it both complete and model-complete.

The confusion arises because both properties use the word 'complete,' but they describe different structural features. Completeness is a property of the theory's deductive power over sentences; model completeness is a property of how models relate under embedding. A theory can have either, both, or neither. ACF_p (fixed characteristic) has both; Presburger arithmetic is complete but not model-complete; some theories are model-complete but have multiple incompatible completions.