The Simon compactness theorem is a result in mathematical logic that extends the concept of compactness from propositional logic to first-order logic. It states that if every finite subset of a set of first-order sentences has a model, then there exists a model for the entire set. This theorem is particularly useful in model theory, as it allows for the construction of models for infinite sets of sentences. This theorem is named after Hugh Simon, who contributed to the field of logic and model theory. The compactness theorem is foundational in understanding the relationships between syntax and semantics in logic, providing insights into how infinite structures can be analyzed through finite approximations.