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I'm looking for a book to self-study axiomatic set theory, and heard this was a classic. What are the main prerequisites for this text? My knowledge of set theory isn't too great. Probably the only time I came across nontrivial set theory was when I read the proof that every nonzero ring has a maximal ideal (Zorn's Lemma).

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    $\begingroup$ Why do you want to study set theory? Do you have a specific purpose in mind? Also, I don't think Jech has any specific mathematical prerequisites other than sufficient mathematical maturity. $\endgroup$
    – Potato
    May 12, 2013 at 8:04
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    $\begingroup$ @Potato: Why not study set theory? Does it have to have a purpose? $\endgroup$
    – Asaf Karagila
    May 12, 2013 at 8:15
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    $\begingroup$ @AsafKaragila The purpose could of course just be pleasure. However, if the asker has some other purpose in mind (and I think many people who ask such questions do -- who in their right mind would find set theory interesting in its own right? :P ), that would be nice to know, because we could give more targeted advice. $\endgroup$
    – Potato
    May 12, 2013 at 8:20

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Reading Jech generally requires no prerequisites, but it is good to be familiar with the basics of logic and very naive set theory before reading it.

The first part of the book, which is the introductory part, covers a lot of basic topics in modern set theory, and most of the proofs are relatively complete. However Jech often skips small and minor steps, leaving them as exercises. If you are not used to this sort of writing it may be hard to read the book in full, on the other hand it may give you the push to fill in the details.

My best advice is to find someone to guide you through the book, so you could get an additional structure and help when reading it. You can ask questions here, but I'm not sure if that would be completely enough.

If you prefer to start with a different reading instead there are the books in Arthur's post as well as:

  1. Kunen's book, Set Theory (which has a 2011 edition).
  2. Halbeisen's book, Combinatorial Set Theory: With a Gentle Introduction to Forcing (Springer Monographs in Mathematics) (which is available freely on the author's website as for 2013).
  3. Levy's book, Basic Set Theory.
  4. Enderton's book, Elements of Set Theory.

I did not read the last two books, but I did hear a lot of good things about them from people in whom I have great trust on the matter. Note that each book is aimed to cover slightly different material in a slightly different approach. It all depends on what level of set theory you are aiming to have at the end of the book.

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Jech's tome (at least the Third Millennium Edition with which I am familiar) does start out at the very basics, going through each of the axioms of $\sf{ZFC}$ is quite a bit of detail. To a certain extent it is quite self-contained.

However, at the very least it would be beneficial to have seen some basic first-order logic (including Gödel's Completeness and Incompleteness Theorems).

Additionally, it is quite brisk, eventually leading to topics that are at the forefront of current research (for example, some exercises have as references articles written this century). I would not recommend this as a first serious set-theory text, especially for self-study. In my opinion, its main purpose is as more of a reference than a text book.

I find that the two volume Discovering Modern Set Theory by W. Just and M. Weese is a very apt introduction to set-theory. The first volume is fittingly entitled "The Basics" and the second "Set-Theoretic Tools for Every Mathematician". They do not cover forcing, but do cover the related concept of Martin's Axiom.

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Hrbacek & Jech, Introduction to Set Theory, is a substantially gentler introduction than Jech, Kunen, or Halbeisen, but it still gets as far as discussing $\diamondsuit$ and Martin’s Axiom. (I’m not familiar with the others that have been mentioned so far.)

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I studied set theory by myself too, so my experience might be relevant to you. I was aware of Jech's book when I became interested to read up on the subject, but I choose not to use it for self study. I thought its overwhelming depth would take away the fascination and curiosity I had. It must be a good reference textbook by looking at its contents and structure, but I would only recommend it for self study if you have someone to guide you through it, as Asaf has advised already.

The two books I ended up with and really loved are Naive Set Theory by Paul Halmos, and Axiomatic Set Theory by Patrick Suppes. Do not be mislead by the title of Halmos' book, it is not that naive. It is rather axiomatic, he is just not using ZFC explicitly.

Both authors have a very good narrative style. They put things in historical context considering the developments of set theory, and emphasise the importance of different proofs in a very mind-opening way. The exercises are very instructive too, giving you the playground to try what you have just read.

I remember how much I was stunned for months by the powerful construct for families in Halmos' book. Suppes' family construct was a bit different, significantly less powerful I would say. By the time I got to the cardinal and ordinal numbers, $\varepsilon_0$, and friends I was a totally different person. Anyhow, both authors have a fascinating style, and they complemented each other very well for my purpose, i.e. to blow my mind.

Someone mentioned above that a prerequisite to set theory could be a good introductory textbook on logic. Suppes in fact has one, that is how I got to know his style even before reading his set theory book. I was not disappointed with either.

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Jech begins his book with a list of all ZFC axioms in their most barebones form.

What I mean by "barebones" here is that Jech does not bother to define any of the terms or notation he uses to enunciate these axioms. In some axioms, he even refers to unspecified variables without saying what those variables stand for. For example, here's the very first axiom in this list:

1.1. Axiom of Extensionality. If $X$ and $Y$ have the same elements then $X = Y$.

Jech clearly expects that the reader knows what $X$ and $Y$ are. More generally, he expects the reader to be thoroughly familiar with this list of axioms, with what they say, as well as with what they do not say.

Furthermore, as I describe in another post, Jech's exposition will be difficult for anyone who is not completely fluent with classes (as opposed to sets).

After having struggled through the first two chapters of Jech's book, I am now pretty convinced that Jech was writing to those with a very thorough prior training in set theory.

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