Mathematical Logic and Computability

Mathematical Logic and Computability provides a foundational introduction to mathematical logic, proof theory, and computability theory at the undergraduate level. The textbook covers propositional calculus, first-order logic, formal axiomatic theories, and Turing machines. The book follows a structured progression from basic logical concepts through increasingly complex mathematical structures and theorems. Through worked examples and exercises, students learn methods of formal mathematical reasoning and proof construction. Each chapter builds upon previous material while introducing new logical frameworks and computational concepts. The treatment includes Gödel's completeness and incompleteness theorems, as well as fundamental results about recursive functions and decidability. This text links pure mathematical logic with practical questions about what can and cannot be computed, exploring deep connections between abstract reasoning systems and the foundations of computer science. The material bridges theoretical mathematics and applied computational approaches.

There are not enough internet reviews to create a summary of this book. Instead, here is a summary of reviews of H. Jerome Keisler's overall work: Students and mathematicians who have used Keisler's "Elementary Calculus" textbook appreciate the intuitive infinitesimal approach compared to traditional epsilon-delta methods. Several reviews on math forums mention this makes calculus concepts clearer for first-time learners. Readers liked: - Clear explanations of non-standard analysis - Historical context provided alongside concepts - Thorough problem sets with solutions - Free digital availability of the textbook Readers disliked: - Limited availability of physical copies - Some exercises lack intermediate steps - Advanced prerequisites needed for later chapters On Goodreads, his books average 4.2/5 stars across 15 reviews. Math.StackExchange users frequently recommend his calculus text for self-study. One reviewer noted: "Finally understood limits thanks to the infinitesimal approach." Another stated: "The rigor of epsilon-delta proofs with the intuition of infinitesimals." Amazon reviews (12 total) focus on the text's value for mathematics students, though some mention it's less suitable for applied sciences.

Introduction to Mathematical Logic by Elliott Mendelson This text covers propositional logic, first-order logic, formal theories, and Gödel's incompleteness theorems with a focus on mathematical rigor and foundational concepts.

A Mathematical Introduction to Logic by Herbert B. Enderton The book presents logic from a mathematical perspective with detailed treatments of model theory, recursion theory, and incompleteness.

Computability Theory by S. Barry Cooper This work connects mathematical logic to computation theory through explorations of recursive functions, Turing machines, and degrees of unsolvability.

Set Theory and Logic by Robert R. Stoll The text bridges set theory with mathematical logic through systematic development of formal systems and proof methods.

Logic and Structure by Dirk van Dalen The book develops mathematical logic from first principles through completeness theorems, model theory, and applications to algebra.

🔵 The author, H. Jerome Keisler, is known for developing an approach to mathematical analysis using infinitesimals, called Non-Standard Analysis, which provides an alternative way to teach calculus. 🔵 This textbook uniquely combines mathematical logic with computability theory, making connections between Turing machines, recursive functions, and formal proofs that many other texts treat separately. 🔵 While published in 1996, the book remains relevant today because it addresses fundamental questions about what computers can and cannot compute—questions that are increasingly important in the age of AI and quantum computing. 🔵 Keisler developed much of the material while teaching at the University of Wisconsin-Madison, where he became a professor at the remarkably young age of 26. 🔵 The book includes original historical notes about logicians like Kurt Gödel, Alan Turing, and Alonzo Church, whose work in the 1930s laid the foundation for modern computer science.