Sets Models and Proofs 1st Edition by Ieke Moerdijk, Jaap van Oosten ISBN 9783319924144 3319924141

This textbook provides a concise and self-contained introduction to mathematical logic, with a focus on the fundamental topics in first-order logic and model theory. Including examples from several areas of mathematics (algebra, linear algebra and analysis), the book illustrates the relevance and usefulness of logic in the study of these subject areas. The authors start with an exposition of set theory and the axiom of choice as used in everyday mathematics. Proceeding at a gentle pace, they go on to present some of the first important results in model theory, followed by a careful exposition of Gentzen-style natural deduction and a detailed proof of Gödel’s completeness theorem for first-order logic. The book then explores the formal axiom system of Zermelo and Fraenkel before concluding with an extensive list of suggestions for further study. The present volume is primarily aimed at mathematics students who are already familiar with basic analysis, algebra and linear algebra. It contains numerous exercises of varying difficulty and can be used for self-study, though it is ideally suited as a text for a one-semester university course in the second or third year.

Sets Models and Proofs 1st Edition Table of contents:

• Chapter 1: Basic Set Theory

  • Sets, relations, and functions; basic set-theoretic operations and properties.

• Chapter 2: Propositional Logic and Proof Techniques

  • Introduction to propositional logic, truth tables, logical equivalence, and formal proof systems.

• Chapter 3: First-Order Logic

  • Quantifiers, predicates, and the syntax of first-order logic.

• Chapter 4: Models and Interpretations

  • What models are, and how they are used to interpret logical systems and structures.

• Chapter 5: Completeness and Soundness

  • Formal proofs, the completeness theorem, and the relationship between syntax and semantics.

• Chapter 6: Set-theoretic Models

  • More advanced topics in set theory, such as the axioms of Zermelo-Fraenkel set theory (ZF), and constructing set-theoretic models.

• Chapter 7: Cardinality and Infinite Sets

  • Cardinal numbers, comparing the sizes of infinite sets, and the concept of countability.

• Chapter 8: Proofs in Set Theory

  • Detailed examples of proofs within set theory, including properties of relations, functions, and theorems in set theory.

• Chapter 9: Model Theory and its Applications

  • Exploring model theory in more detail and its application to understanding the logical structure of mathematical theories.

• Chapter 10: Advanced Topics in Logic

  • Additional topics such as Gödel’s incompleteness theorems, set-theoretic independence, and more.

• Conclusion: The Future of Logic and Set Theory

  • Reflection on how set theory and logic are foundational to modern mathematics and their ongoing development.

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