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I want to do a survey of textbooks in set theory. Amazon returns 3582 books for the keywords "set theory". A small somewhat random selection with number of references in Google scholar is the following.

Two questions:

  • What textbooks in set-theory are considered standard?

  • What is a good place to aggregate standard references (of textbooks)?

(I'm only interested in text-books, which as a format already provide a kind of standardization)

Textbooks on set theory (author: title (year) #Google Scholar citations )

  • Thomas J. Jech: Set Theory 3rd Edition (1978) #2441
  • Kenneth Kunen: Set Theory (1980) #1881
  • P. R. Halmos: Naive Set Theory (1974) #1079
  • AA Fraenkel, Y Bar-Hillel, A Levy: Foundations of set theory (1973) #696
  • K Kuratowski, A Mostowski, M Mączyński: Set theory (1976) #609
  • Suppes: Axiomatic set theory (1972) #589
  • Quine: Set theory and its logic (1969) #442
  • A Levy: Basic set theory (1979) #442
  • Herbert B. Enderton: Elements of Set Theory (1977) #349
  • Karel Hrbacek, Thomas J. Jech: Introduction to set theory (1999) #284
  • RR Stoll: Set theory and logic (1979) #281
  • N. Bourbaki: theory of sets (1970) #211
  • K Devlin: The joy of sets: fundamentals of contemporary set theory (1994) #182
  • Ciesielski: Set theory for the working mathematician (1997) #156
  • YN Moschovakis: Notes on set theory (1994) #138
  • FW Lawvere: Sets for mathematics (2003) #118
  • G Takeuti, WM Zaring: Introduction to axiomatic set theory (1982) #106
  • Kaplansky: Set Theory and Metric Spaces (2001) #103
  • Potter: Set Theory and Its Philosophy: A Critical Introduction (2004) #91
  • PT Johnstone: Notes on logic and set theory (1987) #87
  • Judith Roitman: Introduction to Modern Set Theory (1990) #47
  • Derek Goldrei: Set Theory: For Guided Independent Study (1996) #31
  • Basic set theory: A Shen, NK Vereshchagin (2002) #14
  • Ralf Schindler: Set Theory (2014) #13
  • Foreman, Kanamori: Handbook of Set Theory (2009) #38

Note: Please feel free to add textbooks to the list, as long as they are not just on a smaller subset of set theory. (Some of the dates might not be the first published edition.)

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(Excerpted from an earlier version of a study guide to logic texts more generally -- you will find the latest version here: http://www.logicmatters.net/students/tyl/)

Mere lists are fairly uninteresting and unhelpful. So let's be a bit more selective!

We should certainly distinguish books covering the elements of set theory – the beginnings that anyone really ought to know about – from those that take on advanced topics such as ‘large cardinals’, proofs using forcing, etc.

On the elements, two excellent standard ‘entry level’ treatments are

  • Herbert B. Enderton, The Elements of Set Theory (Academic Press, 1997) is particularly clear in marking off the informal development of the theory of sets, cardinals, ordinals etc. (guided by the conception of sets as constructed in a cumulative hierarchy) and the formal axiomatization of ZFC. It is also particularly good and non-confusing about what is involved in (apparent) talk of classes which are too big to be sets – something that can mystify beginners.

  • Derek Goldrei, Classic Set Theory (Chapman & Hall/CRC 1996) is written by a staff tutor at the Open University in the UK and has the subtitle ‘For guided independent study’. It is as you might expect extremely clear, and is indeed very well-structured for independent reading.

Still starting from scratch, and initially only half a notch up in sophistication, we find two more really nice books (also widely enough used to be considered "standard", whatever exactly that means):

  • Karel Hrbacek and Thomas Jech, Introduction to Set Theory (Marcel Dekker, 3rd edition 1999). This goes a bit further than Enderton or Goldrei (more so in the 3rd edition than earlier ones). The final chapter gives a remarkably accessible glimpse ahead towards large cardinal axioms and independence proofs.

  • Yiannis Moschovakis, Notes on Set Theory (Springer, 2nd edition 2006). A slightly more individual path through the material than the previously books mentioned, again with glimpses ahead and again attractively written.

My next recommendation might come as a bit of surprise, as it is something of a ‘blast from the past’: but don’t ignore old classics: they can have a lot to teach us even if we have read the modern books:

  • Abraham Fraenkel, Yehoshua Bar-Hillel and Azriel Levy, Foundations of Set-Theory (North- Holland, 2nd edition 1973). This puts the development of our canonical ZFC set theory into some context, and also discusses alternative approaches. It really is attractively readable. I’m not an enthusiast for history for history’s sake: but it is very much worth knowing the stories that unfold here.

One intriguing feature of that last book is that it doesn’t at all emphasize the ‘cumulative hierarchy’ – the picture of the universe of sets as built up in a hierarchy of levels, each level containing all the sets at previous levels plus new ones (so the levels are cumulative). This picture – nowadays familiar to every beginner – comes to the foreground again in

  • Michael Potter, Set Theory and Its Philosophy (OUP, 2004). For mathematicians concerned with foundational issues this surely is – at some stage – a ‘must read’, a unique blend of mathematical exposition (mostly about the level of Enderton, with a few glimpses beyond) and extensive conceptual commentary. Potter is presenting not straight ZFC but a very attractive variant due to Dana Scott whose axioms more directly encapsulate the idea of the cumulative hierarchy of sets.

Turning now to advanced topics Two books that choose themselves as classics are

  • Kenneth Kunen, Set Theory (North Holland, 1980), particularly for independence proofs.

  • Thomas Jech, Set Theory: The Third Millenium Edition (Springer 2003), for everything.

And then there are some wonderful advanced books with narrower focus (like Bell's on Set Theory: Boolean Valued Models and Independence Proofs). But this is already long enough and in fact, if you can cope with Jech's bible, you'll be able to find your own way around the copious literature!

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  • $\begingroup$ There is a book by Fraenkel named Abstract Set Theory. Do you have any suggestion regarding it (I mentioned this since you mentioned Fraenkel's other masterpiece in this answer)? $\endgroup$ – user170039 Nov 6 '18 at 3:57
  • $\begingroup$ There is an old set theory book by D.Monk,and a few days ago I realized he has on his website what he calls Lectures on set theory (also solutions for exercises) which seems to me like a new book, which, in terms of topics covered, is comparable to Jech and Kunen's texts (and may be preferable to some readers depending on taste). I do not know if it has officially been (or will be) published, the pdf I see is marked March 11, 2019. See math.colorado.edu/~monkd and in particular euclid.colorado.edu/~monkd/setth.pdf @RParadox $\endgroup$ – Mirko Sep 16 '19 at 2:19
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Here are some books not included in your list.

  • Kunen has completely rewritten his text Set Theory: An Introduction to Independence Proofs. See Amazon. It contains a lot of new material.

  • Holz, Steffens, Weitz, Introduction to Cardinal Arithmetic. The first chapter (about 100 pages) of this book is a very good introduction to set theory. One of the best I have ever seen.

  • Just and Weese have a two volume introduction published by the AMS. The second volume is a very good second course if you like their conversational style.

  • Drake, Set Theory, An Introduction to Large Cardinals. Contains introductory material as well as some advanced topics.

  • Drake, Singh, Intermediate Set Theory. If I recall correctly, this book contains a detailed development of set theory and constructibility.

  • There is a new Dover edition of Smullyan, Fitting, Set Theory and the Continuum Problem. This book has a non-standard approach to different topics.

  • The new Dover edition of Lévy's Basic Set Theory contains an errata not available in the old version.

  • Schimmerling's new book, A Course on Set Theory, looks like a nice and compact introduction.

  • Henle, An Outline of Set Theory is a problem-oriented text. It has a section on Goodstein's theorem.

Five classic texts still relevant today:

  • Sierpiński, Cardinal and Ordinal Numbers. A very rich collection of results on ordinal and cardinal arithmetic.

  • Kuratowski, Mostowski, Set Theory. The old bible of set theory?

  • Heinz Bachmann, Transfinite Zahlen. Unfortunately, I don't know any German. As far as I understand, this book contains some results not found in Sierpinski's book.

  • Cohen, Set Theory and the Continuum Hypothesis. I guess the last chapter on forcing is quite dated. But the previous chapters are insightful.

  • Erdős, Hajnal, Máté, Rado, Combinatorial Set Theory: Partition Relations for Cardinals. This is a more specialized book with a quick review of basics. It looks like this book is still published: Amazon link.

There are some online texts available as well. I remember seeing notes by Steve Jackson, J. Donald Monk and Sy Friedman among others.

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  • $\begingroup$ thanks, I will add some of them. $\endgroup$ – RParadox Dec 7 '12 at 10:23
  • $\begingroup$ Just wonder if anyone could compare Kunen's "Set Theory", the latest edition (2011) vs the previous one (1983)? $\endgroup$ – athos May 12 '14 at 10:03
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I used the book Set Theory by Andras Hajnal and Peter Hamburger and got the impression (since I was taking the class during a program in Hungary) that it was a common book there. It has a good introduction to naive set theory and a lot of more advanced topics in combinatorial set theory as well.

A link to the book is here.

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P.R.Halmos: Naive Set Theory (1974)

An excellent "Outline of the elements of naive set theory" as the author himself describes the book. The purpose of the book is to equip the beginning student of advanced mathematics with the necessary minimum of set theory "with minimum of philosophic discourse and logical formalism". This means the following: the axioms of set theory are stated and discussed in the most gentle way, followed by an account of all classical topics treated under the label of set theory in the same manner. This is a great place to start for anyone.

Suppes: Axiomatic Set Theory (1960)

If after reading Halmos, you develop an appetite for a more formal treatment (of ZF theory), Suppes (1960) is a great companion to continue this journey with (as recommended in Halmos (1960)). Be prepared for some logical formalism and great historical references.

Both books are available from Dover, and it is not a bad idea to get the original printing by van Nostrand of either. Have fun!

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Frank Drake's book, "Set Theory: An Introduction to Large Cardinals" (already mentioned by Ali Kare) deserves your attention even if you're not particularly interested in large cardinals. Despite its subtitle, it contains very nice presentations of a lot of general set-theoretic background before getting to the large cardinals.

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  • $\begingroup$ +1 I guess I'll have to read the book to know why cardinals deserve attention $\endgroup$ – T. Webster Jul 2 '13 at 0:27
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I don't know if it is a standard or such, but with Judith Roitman's Introduction to Modern Set Theory (1990). I had a solid book for my studies.

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    $\begingroup$ And Judy’s text is freely available on her website in a form that she describes as ‘extensively revised’. (Did you have the course from her?) $\endgroup$ – Brian M. Scott Dec 6 '12 at 0:21
  • $\begingroup$ @Brian Wow, I didn't know professor Roitman had revised and expanded her textbook. I'm a little disappointed she didn't take the opportunity to put some mathematical logic and it's applications to more sophisticated set theory into the revision,but it's really good to know since a good advanced text in the subject is hard to find. Jech is just too damn big and expensive to be practical for any except graduate students looking to do research in set theory. $\endgroup$ – Mathemagician1234 Mar 16 '13 at 7:46
  • $\begingroup$ The link above is obsolete but the text is still available here. $\endgroup$ – mathematrucker Jan 5 '20 at 22:34
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This might be considered "Baby Kunin," although it is described as a first year graduate course level. It has set theory and logic:

"Foundations of Mathematics." https://www.math.wisc.edu/~miller/old/m771-10/kunen770.pdf

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I just posted this as a comment to one of the existing answers, but then thought it might deserve to be expanded and published as a separate answer.

There is an old set theory book by D.Monk,and a few days ago I realized he has on his website what he calls Lectures on set theory (also solutions for exercises) which seems to me like a new book, which, in terms of topics covered, is comparable to Jech and Kunen's texts (and may be preferable to some readers depending on taste). (Also, as seen below, it contains fairly recent results, e.g. a proof that $\frak p=\frak t$ (as well as PCF, which perhaps no longer counts as a recent topic, but is good to see in a book form).) I do not know if it has officially been (or will be) published, the pdf I see is marked March 11, 2019. See http://math.colorado.edu/~monkd/ and in particular http://euclid.colorado.edu/~monkd/setth.pdf

Update, I had overlooked a bigger and newer file on the same web page
http://euclid.colorado.edu/~monkd/full.pdf
NOTES ON SET THEORY J. Donald Monk September 14, 2019

Here are some excerpts (for the smaller, older file) :

Preface
Edition of March 11, 2019: chapter on $\frak p=\frak t$ rewritten.
Edition of August 9, 2017: chapter on proper forcing rewritten.
Edition of November 14, 2016: chapter on proper forcing changed; the proof of Theorem 28.5 was in error, and a new proof using a game is given (Theorem 28.33).
Edition of August 30, 2016: Proposition 27.21 corrected.

Some background on these notes:
0. The exercise solutions have not been carefully checked.
1. The axioms for first-order logic are due to Tarski.
2. The treatment of forcing follows Kunen, except for using Boolean values in the definition.
3. The proof of Hausdorff’s theorem in Chapter 17 follows Hausdorff’s original proof closely.
4. The treatment of proper forcing in Chapter 28 follows Jech to a large extent.
5. For PCF in chapters 30–32 we follow Abraham and Magidor.
6. Chapter 33 is based on Blass.
7. The proof that $\frak p=\frak t$ in Chapter 34 is based upon notes of Fremlin and a thesis of Roccasalvo.
8. The consistency proofs in Chapter 35 are partly from Kunen and partly from the author.

TABLE OF CONTENTS
LOGIC
1. Sentential logic .... 1
2. First-order logic .... 12
3. Proofs .... 24
4. The completeness theorem .... 42
ELEMENTARY SET THEORY
5. The axioms of set theory .... 67
6. Elementary set theory 70
7. Ordinals, I .... 75
8. Recursion .... 80
9. Ordinals, II .... 87
10. The axiom of choice. .... 106
11. The Banach-Tarski paradox .... 111
12. Cardinals .... 121
GENERIC SETS AND FORCING, I
13. Boolean algebras and forcing orders. .... 144
14. Models of set theory .... 160
15. Generic extensions and forcing . .... 186
16. Independence of CH .... 207
INFINITE COMBINATORICS
17. Linear orders .... 215
18. Trees .... 241
19. Clubs and stationary sets .... 253
20. Infinite combinatorics .... 267
21. Martin’s axiom .... 277
22. Large cardinals .... 288
23. Constructible sets. .... 311
GENERIC SETS AND FORCING, II
24. Powers of regular cardinals. .... 332
25. Isomorphims and $\neg$AC .... 340
26. Embeddings, iterated forcing, and Martin’s axiom .... 352
27. Various forcing orders. .... 369
28. Proper forcing .... 392
29. More examples of iterated forcing .... 412
PCF
30. Cofinality of posets .... 422
31. Basic properties of PCF .... 456
32. Main cofinality theorems. .... 478
CONTINUUM CARDINALS
33. $^\omega \omega$ and $\mathscr P(\omega)/fin$ .... 504
34. $\frak p=\frak t$ .... 521
35. Consistency results concerning $\mathscr P(\omega)/fin$ .... 556
APPENDICES
36. The integers .... 565
37. The rationals .... 572
38. The reals .... 578
INDEX OF SYMBOLS .... 588
INDEX OF WORDS .... 592

Update, contents for the bigger and newer file:
http://euclid.colorado.edu/~monkd/full.pdf

NOTES ON SET THEORY
J. Donald Monk September 14, 2019
A large portion of these notes, roughly Chapters 12–31, follows Kunen [2011].
TABLE OF CONTENTS
LOGIC
1. Sentential logic .... 1
2. First-order logic .... 16
3. Proofs .... 29
4. The completeness theorem .... 49
ELEMENTARY SET THEORY
5. The axioms of set theory .... 77
6. Elementary set theory .... 80
7. Ordinals, I .... 89
8. Recursion .... 94
9. Ordinals, II .... 103
10. The axiom of choice. .... 127
11. Cardinals .... 135
MODELS OF SET THEORY
12. The set-theoretical hierarchy .... 164
13. Absoluteness. .... 184
14. Checking the axioms .... 194
15. Reflection theorems. ... 200
16. Consistency of no inaccessibles. .... 207
17. Constructible sets. .... 208
INFINITE COMBINATORICS
18. Real numbers in set theory .... 225
19. The Cicho´n diagram .... 276
20. Continuum cardinals .... 295
21. Linear orders .... 320
22. Trees .... 358
23. Clubs and stationary sets .... 394
24. Infinite combinatorics ... 427
25. Martin’s axiom .... 438
26. Large cardinals .... 480
FORCING
27. Boolean algebras and forcing orders. .... 509
28. Generic extensions and forcing .... 542
29. Forcing and cardinal arithmetic .... 566
30. General theory of forcing .... 581
31. Iterated forcing .... 610
32. $\frak p=\frak t$ .... 694
PCF
33. Cofinality of posets .... 733
34. Basic properties of PCF .... 766
35. Main cofinality theorems .... 787
ADDITIONAL SECTIONS
36. Various forcing orders .... 813
37. More examples of iterated forcing .... 836
38. Consistency results concerning $\mathscr P(\omega)/fin$ .... 846
REAL NUMBERS
39. The integers .... 855
40. The rationals .... 862
41. The reals .... 868
Index of symbols .... 878
Index of words .... 885
Jan. 13, 2019: indices corrected
March 11, 2019: chapter on $\frak p=\frak t$ rewritten.
June 11, 2019: correction concerning $L[B]$.
Sept. 14, 2019: several corrections throughout, and new results at end of Chapter 11.

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