The Compactness Theorem

The Compactness Theorem in logic states that if every finite subset of a set of logical statements (or axioms) can be true at the same time, then the entire set—no matter how large or complex—can also be true together. Essentially, consistency in small parts implies consistency overall. This means that for certain logical systems, verifying smaller pieces suffices to ensure the entire system doesn't contain contradictions, making it a powerful tool for understanding the structure and consistency of mathematical and logical theories.