Definition
The compactness theorem is a fundamental result in (classical) mathematical logic stating that a set of first-order sentences has a model if and only if every finite subset has a model.
Formally, let be a set of first-order sentences. Then:
Implications
The compactness theorem has several important consequences:
- Infinite models: If a theory has arbitrarily large finite models, it has an infinite model
- Non-standard models: Used to construct non-standard models of arithmetic
- Preservation: Properties preserved under unions of chains are finitely axiomatizable
Proof Approaches
The compactness theorem can be proven via:
- Gödel’s completeness theorem: Syntactic consistency implies semantic satisfiability
- Ultraproducts: Using model-theoretic constructions
- Tychonoff’s theorem: Via topological methods
Applications
- Constructing non-standard models of theories
- Showing certain properties are not first-order definable
- Graph theory (existence of infinite graphs with certain properties)
- Algebraic applications
Related Concepts
- Model Theory: The field where compactness is fundamental
- First-Order Logic (Classical): The logic to which compactness applies
- Gödel’s Completeness Theorem: Related foundational result
- Satisfiability: The property preserved by compactness