Model Theory for Infinitary Logic

Model Theory for Infinitary Logic presents a systematic study of model theory extensions beyond first-order logic to include infinitely long formulas. The book addresses topics like compactness, completeness, and interpolation theorems for infinitary logics. The text establishes fundamental results about preservation and characterization theorems in model theory. Keisler introduces key concepts and methods through a sequence of progressively more complex logical systems. Core chapters examine ultraproducts, saturated models, and back-and-forth constructions in infinitary contexts. The treatment includes both classical results and then-recent developments in the field. This work represents an important bridge between traditional first-order model theory and more expansive logical frameworks. The theoretical foundations laid out continue to influence research in mathematical logic and set theory.

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.

Model Theory by David Marker This text covers the fundamentals of model theory with a focus on first-order logic and its applications to algebraic structures.

Set Theory: An Introduction to Independence Proofs by Kenneth Kunen The book presents advanced set theory methods with connections to model theory and infinitary logics.

Introduction to Higher Order Categorical Logic by J. Lambek and P. J. Scott The work connects category theory, lambda calculus, and infinitary logic systems through mathematical logic frameworks.

Sets and Classes: On the Work of Paul Bernays by Gert H. Müller This collection explores the foundations of infinite set theory and mathematical logic that underpin infinitary logic studies.

Introduction to Axiomatic Set Theory by Gaisi Takeuti and Wilson M. Zaring The text builds the mathematical foundations needed for understanding infinitary logic through set-theoretic methods.

🔹 First published in 1971, this book helped establish Model Theory as a distinct branch of mathematical logic, particularly focusing on languages that allow infinite conjunctions and disjunctions. 🔹 H. Jerome Keisler introduced the concept of "Keisler measures" in probability theory and model theory, which are now fundamental tools in the study of stable theories. 🔹 The book was part of the North-Holland Studies in Logic series, which played a crucial role in developing modern mathematical logic in the latter half of the 20th century. 🔹 Keisler's work on infinitary logic helped bridge the gap between classical first-order logic and set theory, influencing both fields significantly. 🔹 The author is also known for the Keisler-Shelah Isomorphism Theorem, a fundamental result in model theory that characterizes elementary equivalence of models in terms of isomorphic ultrapowers.