Questions: Basic Model Theory

Question 1

Two structures M and N are elementarily equivalent if and only if:

• A. There exists an isomorphism between M and N
• B. They satisfy exactly the same set of first-order sentences
• C. Every quantifier-free sentence true in M is also true in N
• D. They are both models of the same complete theory and have the same cardinality

Answer: (B)

Elementary equivalence (M ≡ N) means M and N agree on all first-order sentences — not just quantifier-free ones, and regardless of cardinality. Isomorphism (option A) implies elementary equivalence but is strictly stronger: non-isomorphic structures can be elementarily equivalent. Option C (only quantifier-free sentences) is too weak. Option D is close but cardinality alone is not sufficient — two structures of the same cardinality can be models of the same complete theory without being elementarily equivalent if the theory is not complete.

Question 2

A complete theory can have multiple non-isomorphic models.

• T. True
• F. False

Answer: True

Completeness means that for every sentence φ, either φ or ¬φ is a theorem of the theory — there is no sentence the theory leaves undecided. But this says nothing about isomorphism: different cardinalities yield different models. For example, the theory of dense linear orders without endpoints (DLO) is complete, yet it has countable models (like ℚ) and uncountable models (like ℝ) that are not isomorphic. Categoricity in a specific cardinality κ is the stronger property that all models of size κ are isomorphic.

Question 3

What does it mean for a theory to be categorical in cardinality κ, and why does this not imply completeness in general?

Model answer: A theory T is κ-categorical if all models of T of cardinality κ are isomorphic. This does not automatically imply completeness, but by the Łoś–Vaught test: if T is κ-categorical for some κ, has no finite models, and is consistent, then T is complete. Without those conditions, κ-categoricity alone does not force completeness.

The subtlety is that categoricity pins down the theory's models at one cardinality, but first-order logic cannot fix cardinality. A sentence true in all countable models might fail in an uncountable one unless completeness (every sentence decided) already holds. The Łoś–Vaught test is the bridge connecting these two properties.