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I want to learn the following main objects in Set Theory.

  1. Zermelo–Fraenkel axioms of set theory

  2. Axiom of choice,

  3. Zorn's lemma

  4. Well ordering principle

  5. the equivalence of (2,3,4).

  6. Modern changes in way of thinking in set theory after emergence of Category theory.

Of course Halmos' book Naive Set Theory could be a good reference for it.

But, I would like to see more expository and quite recently written book, which will include (6).

I am not looking for a book, which puts above things in "Definition-theorem-proof" type; but also gives more discussions on above objects.

Can one suggest the modern book for it?

Maths Rahul
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    Many modern textbooks available: Jech, Kanamori, Ralf Schindler. For a more general point of view, ss Pavel Pudlak, Logical Foundations of Mathematics and Computational Complexity (Springer, 2013) and Kenneth Kunen, The Foundations of Mathematics (College Pub. 2009). – Mauro ALLEGRANZA Sep 30 '21 at 13:58
  • Thanks for suggestion. (I am not able to visit library due to covid. The only source I have is take suggestions online from experts on MathStack, hence I put this question.) – Maths Rahul Sep 30 '21 at 14:02
  • Since you know your background knowledge, capability of buying or borrowing books, interests, depth to which you wish to pursue this topic, etc. better than anyone here, see the various suggestions at Textbooks on set theory and Modern reference on logic-set theory-foundation, INCLUDING the comments, AND the linked and related questions on the right-web-page-side of each of these two questions, AND the linked and related questions for any of the "linked and related questions" you find relevant. – Dave L. Renfro Sep 30 '21 at 14:57
  • I am not worrying of buying books; but "exploring" books for these topics in library. – Maths Rahul Sep 30 '21 at 15:04
  • "exploring" books for these topics in library --- ?? Your earlier comment said that you are not able to visit a library. – Dave L. Renfro Sep 30 '21 at 15:40
  • I recommend to take a look at Halbeisen's "Combinatorial Set Theory". The book offer a lot of background discussion, it provides a good introduction to axiomatic set theory and has a 40-page long chapter devoted to equivalents of the Axiom of Choice. Best of all, you don't need to go to your library, since it's available online: https://people.math.ethz.ch/~halorenz/publications/pdf/cst.pdf – Vsotvep Sep 30 '21 at 16:06
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    As for your 6th point: I'm not aware of books about set theory that employ tools from category theory. I'm barely even aware of any research in set theory using category theory. It seems to me that the emergence of category theory did not have a big impact in the way of thinking in set theory. See also https://mathoverflow.net/questions/7197/how-can-category-theory-help-my-research-in-set-theory – Vsotvep Sep 30 '21 at 16:09
  • @Dave: Sorry for confusion; yes, I am unable to visit library and explore books for above topics. I am trying to look it online; but in the online sources, to go through "proper specific sources" which experts know, I am taking help and suggestions through mathstackexchange. – Maths Rahul Sep 30 '21 at 16:10
  • See http://karagila.org/wp-content/uploads/2016/01/ests-wh.pdf – Asaf Karagila Oct 02 '21 at 11:44

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