Like other books in the Cambridge *Elements* series, this John Burgess’s contribution on *Set Theory* is about seventy, not-very-packed, pages (perhaps 30K words?). More than an encyclopedia article, or a handbook chapter, but half the length of a short book like my *Gödel Without Tears*. Books in the series are aimed at providing “a dynamic reference resource for graduate students [and] researchers”. And that’s already a tall order for a book on *this* topic, at any rate: for most graduate students in philosophy (even if logic-minded) are likely to be pretty much beginners when it comes to tackling some set theory — and a book accessible to such beginners isn’t likely to also to be of much interest to researchers.

OK, forget the impossible prospectus, and let me try to assess the book in its own terms. First, I certainly enjoyed a quick read. It is engagingly written. And at various points in the later pages Burgess very helpfully put some order into my fragmentary knowledge, or offered genuinely illuminating remarks. So I endorse the comment added below, “I especially liked the second half — on ‘higher set theory’ — and the picture it gives of the various subject areas (descriptive set theory, continuum questions, combinatorial set theory) and techniques (large cardinals, forcing, inner models, infinite games, …) and of how they’re interrelated. I can’t recall anything else that gives as good an overview so briefly.” However, although set theory isn’t my special thing, I didn’t exactly come to this innocent of prior knowledge. And I do have to doubt whether later pages of the book will really be accessible to many of the intended student audience. OK, it may be — for one example — that all the materials have officially been given to understand e.g. the Levy Reflection Principle on p. 55: but I suspect that a significant amount of mathematical maturity, as they say, would be needed to really appreciate what’s going on.

In headline terms, then, I don’t think that the book as a whole would work as advertised for many students. Still, the first half *does* make a nice motivating introduction, but one to be followed by (or read in conjunction with) a standard accessible introduction to set theory like Goldrei or Enderton. And *then *the enthusiast can return to try reading from §8 “Topics in Higher Set Theory” onwards to get a first overview of a few further more advanced topics, with a hope of getting a first inkling of what some of the interesting issues might be, before tackling a second-level set theory text.

Rowsety MoidI finished reading it Friday. I especially liked the 2nd half — on ‘higher set theory’ — and the picture it gives of the various subject areas (descriptive set theory, continuum questions, combinatorial set theory) and techniques (large cardinals, forcing, inner models, infinite games, …) and of how they’re interrelated. I can’t recall anything else that gives as good an overview so briefly. And I don’t think it’s necessary to understand it all to get a useful feel for what’s going on.

I don’t

quiteagree with your reading advice. I think that if someone would benefit from reading a standard accessible introduction such as Goldrei, Enderton, or Button, it would be better for them to read it before,or along with,the first part of Burgess. For even the first part moves briskly and, as Burgess says at the beginning of what we’re calling part 2 (p 41):However, I think there are some relatively minor but still significant problems with the book

as a bookand with the series.* There are quite a few typographical and other minor errors, such as having ‘R’ twice when one of them should be ‘S’ (Table 11, p 18) and misspelling an author’s name (Natalie Wolchover appears as ‘Wachover’ on pages 66 and 73).

* The typesetting doesn’t correctly handle cases such as aleph-with-a-subscript as a superscript (2 to the aleph-0, for example).

* The physical cover material. I’m not a fan of the slightly soft, matt finish that’s now so often used. (Why

isis used, btw? I would appreciate any clues.) It holds fingerprints likeforever.* The price. £15 is a lot for so short a book. Imagine buying 3 of 4 books in the series, which I suspect some of us will do. That’s £45 or £60 for what could be one ordinary-sized book.

And Bell’s

Higher-Order Logic and Type Theoryin the Elements series, for example, is 88 pages for £15; the Dover edition of hisToposes and Local Set Theories: An Introduction(a not completely unrelated book) is 288 pages for a similar price.Peter SmithI very much agree, and have added your main comment into my too-rushed post!

As to pricing policy, it’s a particular mystery why the

Kindleedition is so expensive. I guess many students, though, will have ready access via their library’s Cambridge Core subscription; and the rest will find the book on that trusty PDF repository of which we do not speak.Aron TI get the Element (and other Cambridge) books for free as an alumni since my university gives us access to Cambridge Core. People should check their alumni benefits, because I would be surprised if my university is the only one that offers that benefit.

Mark Strange8.2 Continuum Theory Page 45.

“The main alternatives to CH considered have

been that c = ℵ2 and that c is a fixed point of the alephs, a κ such that κ = ℵκ.

One isolated result proved early is König’s theorem that c = ℵω:

But after this

there were a several of decades of lack of progress.

The reason why emerged in the middle 1900s. Gödel’s First Incompleteness

theorem tells us any reasonably strong consistent mathematical axiom system T

will leave some Ψ undecidable, neither provable nor disprovable.”

OUCH