Connections with Model Theory
The Compactness Theorem via Ultraproducts
The theorem that a finitely satisfiable theory has a model, proved algebraically by an ultraproduct construction, and its consequences.
Learning objectives
- State compactness in both forms
- Follow the ultraproduct proof
- Derive the standard consequences
The statement
A set Σ of first-order sentences has a model if and only if every finite subset of Σ has a model.
If Σ ⊧ Φ then Σ0 ⊧ Φ for some finite Σ0 ⊆ Σ.
The two forms are equivalent: apply the first to Σ ∪ {¬Φ}.
The ultraproduct proof
- Let I be the set of finite subsets of Σ. For each i ∈ I, choose a model Ai of i, which exists by hypothesis.
- For each σ ∈ Σ, let Jσ = {i ∈ I : σ ∈ i}.
- The family {Jσ} has the finite intersection property, since Jσ1 ∩ … ∩ Jσn contains the finite set {σ1,…,σn}.
- By BPI, extend the generated filter to an ultrafilter U.
- By Łoś's theorem, the ultraproduct ∏i Ai/U satisfies each σ, since Jσ ∈ U and every Ai with i ∈ Jσ models σ.
No proof theory is used — no derivations, no Gödel completeness theorem. The model is constructed directly. This is why compactness sits naturally in a universal algebra text.
Consequences
| Application | Argument |
|---|---|
| Upward Löwenheim–Skolem | Add κ distinct constants; every finite subset is satisfiable |
| Non-standard models of arithmetic | Add a constant c with axioms c > n for each numeral |
| Non-standard analysis | Add an infinitesimal; every finite subset of the axioms is satisfiable |
| Finiteness is not first-order | The theory of a structure plus 'at least n elements' for all n is finitely satisfiable |
| Well-ordering is not first-order | Add a descending chain of constants |
| Torsion groups are not elementary | Add an element of order exceeding every bound |
Every application follows the same shape: add new constants with axioms saying something extreme, observe that any finite subset only demands a finite amount and is therefore satisfiable in a suitable existing structure, then apply compactness. Recognising the shape makes the applications routine.
Equivalences
Compactness is equivalent, over ZF, to the Boolean prime ideal theorem, which is strictly weaker than full choice.
The set of complete theories in a language carries a natural topology — the Stone topology of the Lindenbaum algebra — and the compactness theorem is exactly the statement that this space is compact. The name is not an analogy but an identification.
Frequently asked questions
Does compactness fail for infinitary logic?
Yes. Logic allowing infinite conjunctions can express finiteness and well-ordering, and compactness fails. Compactness is a distinguishing feature of first-order logic, and Lindström's theorem shows it is essentially characteristic.
Is the compactness theorem constructive?
No — it needs BPI, which is not provable in ZF. There are ZF models where the theorem fails.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section V.2, book pages 243-246.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.