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I know that the question about logic, axiomatic set theory and continuum hypothesis (CH) textbooks was asked many times here. Say, the following link contains a huge list of introductory set-theoretic books:

Textbooks on set theory

My question is somehow concrete and comes after a month of attempts to study the independence of CH. Mainly, I've tried to learn Kunen's and also Jech's "Set theory". The first chapters in both books contained basic information on the ordinals and cardinals. Though the material is familiar to me (from Kuratowski's "Set theory"), it seemed that almost all the proofs are not complete. Thus, I didn't risk to go on reading these books. I've also tried to take a glance at another books (including all the books from the link above) and wasn't satisfied again. My ideal book is the following:

  1. Contains all the proofs leading to the independence of CH and helping to understand the forcing technique
  2. The material has to be introduced in a clear way without philosophy (alas, that's why Cohen's original book doesn't fit)
  3. Reflects the classic approach based on introducing ZFC (that's why Halbeisen's "Combinatorial set theory" doesn't fit)
  4. Is concentrated on the topic (say, it may not contain proofs from model theory, or extra-advanced set theory; also, I'm not interested in applications to analysis, that's why Ciesielski's book isn't the best variant)
  5. Reflects necessary (for the further material) results (and their proofs) on the ordinals and cardinals

I understand that I have to make some compromise, or to work not with just one book. That's why I'd like to ask the opinion of those who had the similar problem and, finally, managed to understand the independence proof. What book or books were finally useful for you? Thanks in advance!

Bertrand Haskell
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    I don't know if you read French, but Théorie des ensembles from Patrick Dehornoy would almost cover your requirements. Its intent is to prove independence of AC and CH with detailed proves from the beginning of set theory. – mathcounterexamples.net Sep 21 '20 at 08:23
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    So Jech is not good enough? – markvs Sep 21 '20 at 08:23
  • @mathcounterexamples.net thanks a lot, I didn't hear about it before. Alas, I'm not good with French. Will try to find the translated version. – Bertrand Haskell Sep 21 '20 at 08:30
  • @JCAA Actually, two of Jech's books I've tried to read are well-organizied. Probably, I'll continue to work with them but now they're a bit advanced for me since not all the proofs are clear. – Bertrand Haskell Sep 21 '20 at 08:33
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    @ElmarGuseinov Unfortunately, I don't think that an English translation exists yet. – mathcounterexamples.net Sep 21 '20 at 08:37
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    I'm confused, what's not classical in Halbeisen? Also, "proofs seem incomplete" is a bad excuse for not reading a book. You're here to learn, not to read a novel. Learning happens by filling in the gaps yourself. Also, Halbeisen's book is an excellent introduction to forcing. – Asaf Karagila Sep 21 '20 at 08:44
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    I don't think Jech is the "best book". To be frank, I don't like his approach to forcing, and if you only read up to the point of forcing the negation of CH, you will miss out a lot of good and important information about forcing. But the overall question is what do you want to do with this? Is this just a goalpost in the study of set theory, or are you just looking to check a box on your bucket list before you move to studying algebra or whatnot? Different goals will have different book recommendations. – Asaf Karagila Sep 21 '20 at 09:02
  • @AsafKaragila I actually have the giant plans. First, I think that techniques like forcing and theorems like Godel's first incompleteness theorem are practically basic in math. It's seems to me a bit strange when someone tries to prove something even not thinking about whether the conjecture is provable at all. I'd like to create a maximally clear system (or a variant of such a system) that generates consistent sentences and check whether this list intersects with some known list of open problems (say, in group, or number theory). I'm not going to be a set-theorist but think it's the future. – Bertrand Haskell Sep 21 '20 at 09:12
  • Which Kunen book are you referring to? – DanielWainfleet Sep 21 '20 at 16:11
  • @DanielWainfleet Set theory: an introduction to independence proofs – Bertrand Haskell Sep 21 '20 at 16:13
  • Cohen's original book is philosophical??? – Sam Nov 06 '20 at 19:09

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