The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (Springer Monographs in Mathematics) by Akihiro Kanamori (PDF)

Description

This is the softcover reprint of the very popular hardcover edition. The theory of large cardinals is currently a broad mainstream of modern set theory, the main area of investigation for the analysis of the relative consistency of mathematical propositions and possible new axioms for mathematics. The first of a projected multi-volume series, this book provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research. A “genetic” approach is taken, presenting the subject in the context of its historical development. With hindsight the consequential avenues are pursued and the most elegant or accessible expositions given. With open questions and speculations provided throughout the reader should not only come to appreciate the scope and coherence of the overall enterprise but also become prepared to pursue research in several specific areas by studying the relevant sections.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is for set theorists, budding set theorists, and mathematicians with an avid interest in large cardinal theory.Kanamori’s book updates and for the most part replaces his two earlier well-known surveys that he co-authored with Magidor, Reinhardt, and Solovay. While most of that earlier material does appear in this new book, he also includes recent developments in those same areas as well as a great deal of new material that emerged in the 1980s (most notably, the profound connection between large cardinals and descriptive set theory).Well, as a researcher in the theory of large cardinals, I feel Kanamori’s book is unquestionably a “must-have”. Since I got the book, I have used it as an important reference in every paper I’ve written. It’s filled with fine points, excellently explained, concerning virtually every area of importance in large cardinal research. And so far, I haven’t found any errors (needless to say, this is quite phenomenal for a book of this size and technical depth).Here’s an overview of the topics covered: Weak compactness, partitions, trees, and 0#. Forcing and sets of reals (introducing descriptive set theory and forcing in an excellent way). Saturated ideals, measurability and forcing, iterated ultrapowers. Supercompacts and strong cardinals, extendibles, almost huge and huge cardinals, axioms I_3 to I_0, and combinatorics of P_{kappa}{lambda}. He concludes with a treatment of the celebrated Martin-Steel-Woodin results on the consistency of PD and AD with many Woodin cardinals.

⭐I’m a graduate student in set theory and I’m finding Kanamori an excellent follow-up to Kunen. The book manages to combine detailed technical exposition with historical insight which is actually useful to understanding the material (not just a list of dates) and gives one a “feel” for the subject.Occasional excersises are contained which are good to help check if you’re keeping up (though sometimes the hints are a little too complete: it might be better if these were relegated to an appendix). More exercises would have improved this book.I believe this is pretty much the only book in which much of this material is collected together, so it’s pretty much essential to any-one seriously interested in Set Theory. I await the promised second and third volumes with anticipation!

⭐If you have background in set theory the book is ok if not then it’s difficult reading

⭐A must have for set theorists. Does a better job than Jech. It’s a little weird how you get a sense of the set theorists as people, but the historical stuff is otherwise really great.

⭐Assumes set theory through forcing and some model theory. He integrates a lot of historical information which is interesting but greatly adds to the reading without adding to the mathematical understanding. The semester is short and I want to learn what I need and get on with life. While it does have more details than Jech, I like Jech’s style better and not just because it is shorter. (The 2e is just a corrected reprint of 1e.)

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