Samuel Jansson: The Compactness Theorem: Proofs and Applications

Abstract: The Compactness Theorem is a landmark result in mathematical logic that bridges the gap between the finite and the infinite. It states that a vast collection of logical statements can be true simultaneously if and only if every finite selection of those statements is consistent on its own. This thesis provides a comprehensive study of the theorem, presenting three distinct proofs: an inductive approach for the countable case, a general proof via Zorn’s Lemma, and a model-theoretic construction using ultraproducts. By exploring its applications in logic, topology, and combinatorics, this work demonstrates the theorem’s versatility and its ability to solve problems across seemingly unrelated branches of mathematics