Questions: Consequences and Applications of the Compactness Theorem

Question 1

You want to prove that no first-order theory can force all its models to be finite. Which technique achieves this?

• A. Show that any finite model can be extended by adding new elements, so no theory prevents extension
• B. Take the theory Σ, add sentences 'there exist at least n distinct elements' for each n, argue every finite subset has a model, then invoke compactness to get an infinite model
• C. Invoke the Löwenheim-Skolem theorem directly, which states that all first-order theories have infinite models
• D. Show the theory is consistent, which by Gödel's completeness theorem implies it has a model of every infinite cardinality

Answer: (B)

The standard compactness argument for non-expressibility of finiteness: take any theory Σ purporting to characterize finite structures. Augment it with sentences φ_n asserting 'there exist at least n distinct elements' for each n ≥ 1. Every finite subset of the augmented theory has a model (take a finite structure large enough to satisfy the finitely many φ_n in that subset). By compactness, the full augmented theory has a model — which satisfies all φ_n, so it is infinite. This contradicts the assumption that Σ forces finiteness. The key move is arguing that every *finite* subset has a model before invoking compactness.

Question 2

Non-standard models of Peano arithmetic (PA) exist that contain 'infinite integers.' How does compactness establish their existence?

• A. By showing PA is inconsistent, which by completeness implies models of all sizes
• B. By adding a constant c and sentences 'c > n' for all standard n; every finite subset has a model in ℕ, so compactness gives a full model where c exceeds all standard naturals
• C. By the Löwenheim-Skolem theorem, which directly constructs non-standard models of any consistent theory
• D. By showing that PA's axioms are satisfiable in ℕ, then extending ℕ with infinitely many new elements

Answer: (B)

The construction: augment PA with a new constant c and sentences 'c > 0', 'c > 1', 'c > 2', ... Every finite subset involves finitely many sentences 'c > k_max', and ℕ serves as a model with c = k_max + 1. Compactness then delivers a model of the *entire* augmented theory — but in this model, c must be greater than every standard natural number 0, 1, 2, 3, ... It is an 'infinite integer.' This model satisfies all of PA's axioms (so it is a genuine model of arithmetic) but contains elements invisible in ℕ.

Question 3

The compactness theorem implies that any property of structures that can be expressed by first-order sentences can be checked by examining only finite subsets of those sentences.

• T. True
• F. False

Answer: True

This is precisely what compactness says: a set of sentences is satisfiable (has a model) iff every finite subset is satisfiable. Any first-order entailment or unsatisfiability reduces to finite evidence. This is what makes first-order logic amenable to sound, complete, and recursively enumerable proof systems — the gap between syntax (proofs) and semantics (models) is bridged finitely. The flip side is the limitation: properties that require 'infinite witness' — well-foundedness, finiteness, Archimedean properties — cannot be captured by first-order sentences.

Question 4

The compactness theorem holds for second-order logic just as it does for first-order logic.

• T. True
• F. False

Answer: False

Compactness fails for second-order logic, which is precisely why second-order logic is more expressive. In second-order logic, you can quantify over sets of elements, and this added power allows you to express finiteness, well-foundedness, and other properties that first-order logic cannot. The Peano categoricity theorem — that the natural numbers are (up to isomorphism) the unique second-order model of the Peano axioms — is impossible in first-order logic precisely because compactness forces non-standard models to exist. Compactness is not a theorem of all logics; it is specific to first-order logic.

Question 5

Explain the key logical move that makes every compactness argument work — what do you establish first, and why does that let you invoke compactness?

Model answer: Every compactness argument first shows that every *finite* subset of some infinite set of sentences has a model. Since any finite set of constraints can be satisfied (by a structure large enough or rich enough), you establish finite satisfiability. Compactness then guarantees that the *infinite* set also has a model — even though no single finite structure may satisfy all the sentences simultaneously. The gap between 'every finite piece works' and 'the whole thing works' is exactly what compactness bridges.

The technique always has this shape: (1) define an infinite theory by adding infinitely many sentences to a base theory; (2) argue that any finite sub-collection has a model by explicit construction; (3) invoke compactness. The resulting model satisfies all the infinite collection's sentences, which often forces it to have properties (being infinite, containing non-standard elements) that no single finite model could realize. Compactness is a transfer theorem: finite satisfiability implies global satisfiability.