H. Jerome Keisler

H. Jerome Keisler is an American mathematician and logician known for his work in model theory and non-standard analysis. As professor emeritus at the University of Wisconsin-Madison, he made significant contributions to mathematical logic and authored influential textbooks in the field. Keisler developed key techniques in model theory, particularly ultraproducts and the Keisler-Shelah isomorphism theorem. His work helped establish connections between model theory and other areas of mathematics, including algebra and analysis. His textbook "Elementary Calculus: An Infinitesimal Approach" presented calculus using non-standard analysis, providing an alternative to the traditional epsilon-delta method. This approach utilized infinitesimals in a mathematically rigorous way, building on Abraham Robinson's work in non-standard analysis. Keisler's research publications and educational materials have influenced generations of mathematicians and students. His contributions to mathematical logic earned him recognition in the academic community, including election as a member of the American Academy of Arts and Sciences.

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.

Elementary Calculus: An Infinitesimal Approach (1976) A calculus textbook that presents the subject using non-standard analysis and infinitesimals rather than the traditional epsilon-delta approach, making calculus concepts more intuitive for beginners while maintaining mathematical rigor.

Model Theory (1977) A comprehensive textbook covering the fundamentals of model theory, including ultraproducts, elementary chains, and the theory of types, which became a standard reference in mathematical logic.

Foundations of Infinitesimal Calculus (1976) A mathematical text that provides the theoretical foundations for non-standard analysis and its application to calculus, including detailed proofs and mathematical structures underlying infinitesimal methods.

Mathematical Logic and Computability (1989) An introduction to mathematical logic covering propositional calculus, predicate calculus, computability theory, and Gödel's completeness and incompleteness theorems.

Model Theory for Infinitary Logic (1971) A specialized text exploring the extension of model theory to infinitary languages, including the study of infinitary formulas and their models.

Abraham Robinson pioneered non-standard analysis and developed the mathematical foundation that Keisler built upon in his calculus textbook. His work formalized infinitesimals using mathematical logic and model theory, creating a rigorous framework for their use in analysis.

Paul Cohen made fundamental contributions to mathematical logic and set theory, including his proof of the independence of the continuum hypothesis. His forcing technique revolutionized set theory and influenced model theory methods used by Keisler.

Michael Morley developed categoricity in uncountable cardinals and other core concepts in model theory that complemented Keisler's work. His theorem on categorical theories became a cornerstone of model theory and influenced Keisler's research on ultrapowers.

Saharon Shelah collaborated with Keisler on isomorphism theorems and advanced model theory through classification theory. His work on stability theory and the development of abstract elementary classes extended many of the model-theoretic concepts Keisler worked with.

Jon Barwise combined mathematical logic with set theory and made contributions to infinitary logic and admissible sets. His work on abstract model theory connected with Keisler's research on ultraproducts and back-and-forth constructions.