This work deals with set-theoretic independence results, in particular for problems on the continuum. It is designed to enable a researcher interested in an independence result of the appropriate kind, to have much of the work done for him, thus allowing him to quote a general result.
Focusing on abstract elementary classes, the book explores a framework for structures sharing a common vocabulary, such as rings or fields. It introduces a partial order that extends the concept of substructures, ensuring closure under isomorphism and maintaining isomorphic relations among generalized substructures. Additionally, it emphasizes the importance of fundamental properties typical of elementary classes, including the closure under unions of increasing chains of substructures, providing a comprehensive understanding of this mathematical concept.
Focusing on the nuances of complete first-order theories, this book explores the distinctions between understandable and complex theories, primarily those that are countable. It builds on foundational concepts such as first-order logic, Gödel's completeness theorem, and naive set theory, making it suitable for readers with a background in model theory. By examining these dividing lines, the text aims to enhance the reader's understanding of the structure and nature of these mathematical theories.