The Higher Infinite

Overview
The Higher Infinite is a scholarly monograph authored by Akihiro Kanamori, first published in 1994 (revised edition 2003) by Springer. The work provides a comprehensive historical and technical account of large cardinal axioms—strong axioms of infinity—in the context of modern set theory. It is widely cited as a standard reference for researchers and graduate students studying the development and implications of large cardinal hypotheses.

Author
Akihiro Kanamori is a Japanese mathematician who has made significant contributions to the theory of large cardinals and forcing. He holds a professorship at the University of California, Irvine, and has authored numerous papers and books in set theory.

Publication Details

• Title: The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings
• Publisher: Springer-Verlag
• First Edition: 1994 (Hardcover)
• Revised Edition: 2003 (Second edition, expanded material)
• ISBN: 978-0387951994 (first edition), 978-0387951995 (second edition)
• Pages: Approximately 560 (second edition)

Content Summary
The book is organized into three main parts:

1. Historical Foundations – Traces the origins of infinitary concepts from Cantor’s work on transfinite numbers through early developments in measurable and inaccessible cardinals.
2. Core Large Cardinal Notions – Provides rigorous definitions, basic properties, and interrelations of key large cardinal notions such as inaccessible, Mahlo, weakly compact, measurable, strong, supercompact, and huge cardinals.
3. Advanced Topics and Applications – Discusses modern topics including inner model theory, determinacy, Woodin cardinals, and the role of large cardinals in proving consistency results (e.g., the consistency of the Suslin hypothesis, the non‑existence of certain definable well‑orders of the reals).

Throughout, Kanamori emphasizes both the combinatorial and structural aspects of large cardinal axioms, and he includes extensive references to primary literature, facilitating further research.

Significance in Set Theory
The Higher Infinite is recognized for several reasons:

• Comprehensiveness: It systematically surveys the evolution of large cardinal concepts, integrating historical narrative with technical depth.
• Pedagogical Value: The exposition balances rigorous proofs with intuitive explanations, making it a common textbook for graduate courses on advanced set theory.
• Reference Standard: Researchers cite the book when summarizing known results about large cardinals, their relative consistency strengths, and their connections to descriptive set theory and inner model theory.

Reception
The work has received positive reviews in mathematical literature. Reviewers have praised its thoroughness and clarity, noting that it fills a gap between introductory set theory texts and specialized research monographs. The second edition incorporated new developments up to the early 2000s, such as progress on the core model induction and advances in determinacy.

Related Concepts

• Large Cardinals: Strong axioms of infinity extending beyond Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC).
• Inner Model Theory: Studies canonical models of set theory containing large cardinals.
• Determinacy: Principles asserting that certain infinite games are determined; many determinacy results are equiconsistent with large cardinal axioms described in the book.

Further Reading

• Thomas Jech, Set Theory (3rd ed., 2003) – Provides background on foundational concepts referenced in The Higher Infinite.
• Joel D. Hamkins, “The Set-Theoretic Multiverse” (2012) – Discusses philosophical implications of large cardinals and their place within multiple set-theoretic universes.

External Links

• Publisher’s page (Springer): https://www.springer.com/gp/book/9780387951994
• Author’s faculty page at UC Irvine (for additional publications).