Best Entry Books for Mathematical Logic in Pure Mathematics

When delving into the fascinating world of pure mathematics, mathematical logic is an essential component. Whether you're a beginner or seeking a more comprehensive introduction, the right textbook can make a significant difference in your understanding and mastery of the subject.

Foundational Concepts and Accessible Learning

For those interested in pure mathematics, a highly recommended entry book for mathematical logic is "How to Prove It: A Structured Approach" by Daniel J. Velleman. This book is excellent for beginners as it covers the foundational concepts of logic, proofs, and set theory in a clear and accessible manner. It provides numerous examples and exercises to help reinforce understanding.

Formal Introduction with Challenges

For a more formal introduction to logic and its applications in mathematics, consider "Introduction to Mathematical Logic" by Elliott Mendelson. While it may be more challenging than Velleman's book, it provides a solid grounding in the subject, making it an invaluable resource for those ready to tackle more rigorous material.

Comprehensive Text with Clarity

Another great option is "Mathematical Logic" by H. D. Ebbinghaus, J. Flum, and W. Thomas. This book covers both propositional and predicate logic with clarity, making it a worthwhile choice for those who want a thorough understanding of the subject. Additionally, it includes an introductory chapter on natural deduction and a discussion on intuitionistic logic.

Alternative Recommendations

A book I found particularly worthwhile is "Formal Number Theory and Computability - A Workbook" by Alec Fisher. This workbook offers a unique approach focused on computational logic and problem-solving, which can be very beneficial for students who learn by doing.

Best Methods of Proof and Writing Proofs

While books on mathematical logic can be helpful, books specifically on the basic methods of proof and how to write proofs might be a better choice for beginners. One such book is "Logic and Structure" by Dirk van Dalen. This book covers propositional and first-order logic and has a good introduction to natural deduction. Van Dalen's writing style is particularly well-regarded for its clarity and depth.

Solid Foundation for Further Studies

A well-written and accessible option that delves deeper into set theory and logic is "Set Theory and Logic" by Robert L. Stoll. While it may be a bit below the level you are looking for, the first few chapters provide a good introduction to logic and computability. The use of universal register machines as intuitive equivalents to Turing machines is a unique and valuable feature.

Comprehensive Coverage with Depth

Lastly, "Introduction to Mathematics" by Kenneth Kunen can be a bit daunting due to its comprehensive coverage of topics such as recursion, the Turing-Church thesis, and G?del's incompleteness theorems. However, the first few chapters are a rewarding introduction to logic and computability.

The best book is going to depend on your particular interests and is highly subjective. Whether you prefer a more accessible introduction, a formal and comprehensive coverage, or a solid foundation for your studies, these texts will help you build a strong foundation in mathematical logic, which is essential for further studies in pure mathematics.