Questions: Compactness Theorem in Model Theory

Question 1

You want to show that an infinite set Σ of first-order sentences has a model. You verify that every finite subset of Σ has a model. What does the Compactness Theorem allow you to conclude?

• A. Σ has a finite model
• B. Σ has a model (possibly infinite), but you cannot determine its cardinality
• C. Σ is consistent but may still lack any model
• D. Nothing — compactness only applies to finite sentence sets

Answer: (B)

The Compactness Theorem states that Σ has a model if and only if every finite subset of Σ has a model. So verifying every finite subset is satisfiable is exactly sufficient to guarantee that Σ itself is satisfiable. The theorem does not guarantee a finite model — Σ may have only infinite models. This is the constructive power of compactness: you never have to check the infinite set directly, only its finite approximations.

Question 2

To construct a non-standard model of arithmetic with an element larger than every standard natural number, which compactness strategy is used?

• A. Add an axiom asserting the domain is uncountable, then apply compactness
• B. Add a constant c and sentences {c > 0, c > 1, c > 2, ...} to the arithmetic axioms; every finite subset is satisfiable, so by compactness the whole set has a model
• C. Prove the standard model satisfies all these sentences and extend it
• D. Use induction to show the natural numbers contain an infinite element

Answer: (B)

The strategy is: (1) Add a new constant symbol c to the language of arithmetic. (2) Add infinitely many sentences {c > 0, c > 1, c > 2, ...}. (3) Any finite subset only requires c to exceed finitely many standard naturals — satisfiable by taking c to be any large standard number. (4) Compactness guarantees a model for the whole set. In this model, c is an element larger than every standard natural — an infinite non-standard element. This construction cannot be done in the standard model because no standard natural number exceeds all others.

Question 3

If a first-order theory has any infinite model, the Compactness Theorem (combined with the Löwenheim-Skolem theorems) implies it has models of all infinite cardinalities.

• T. True
• F. False

Answer: True

This follows from combining compactness with the upward and downward Löwenheim-Skolem theorems (which themselves use compactness in their proofs). If a theory has an infinite model, it cannot pin down the exact cardinality of its models — it will have models of every infinite cardinality. This is a profound limitation of first-order logic: no first-order theory can uniquely characterize a specific infinite structure up to isomorphism, including the natural numbers. The rationals, reals, and complex numbers all share this non-categoricity at the first-order level.

Question 4

The Compactness Theorem guarantees that if nearly every finite subset of Σ has a model, then Σ has a finite model.

• T. True
• F. False

Answer: False

This is a critical misreading. Compactness guarantees that Σ has *some* model — it makes no claim about whether that model is finite or infinite. In fact, the canonical application of compactness is to construct *infinite* models with prescribed properties (like non-standard arithmetic). The theorem says: satisfiability of the whole set is equivalent to satisfiability of all finite subsets. What kind of model exists depends on what Σ says — if Σ asserts 'there are infinitely many elements,' the model compactness guarantees will be infinite.

Question 5

Use the Compactness Theorem to explain why 'the domain is infinite' cannot be expressed by a single first-order sentence.

Model answer: For each n, let σₙ be the first-order sentence 'there exist at least n distinct elements.' The set Γ = {σ₁, σ₂, σ₃, ...} expresses 'the domain is infinite' collectively, but each finite subset only requires the domain to have at least some finite number of elements — satisfiable by any sufficiently large finite structure. By compactness, Γ has a model, but so does Γ ∪ {some finite-model sentence} for any finite structure property. A single first-order sentence φ expressing 'the domain is infinite' would have to be satisfied by exactly the infinite structures — but any such φ, if satisfiable, has a model of every large enough finite size by the downward Löwenheim-Skolem theorem, contradicting the claim that only infinite models satisfy it.

More directly: if φ captured 'the domain is infinite,' then {φ} ∪ {there are at most n elements, for some large n} would be unsatisfiable — but any finite subset of this set is satisfiable. Compactness would then require the whole set to be satisfiable, contradiction. This argument shows first-order logic cannot separate the finite from the infinite.