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Is there a good book naive set theory which prove important theorems and propositions like:

  • The rational numbers are countable
  • The real numbers are not countable
  • $card \ (0,1)=card\ \mathbb R$
  • The union of countable sets are countable
  • Schröder–Bernstein theorem
  • Other interesting theorems

I would like to find a book not so basic as high school set theory books and not so advanced as Naive set theory by Halmos (despide the name this book is not a naive set theory book).

Thanks

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    $\begingroup$ I learned set theory from Axiomatic Set Theory, Takeuti & Zaring, but it may be too rigorous for your liking. Enderton's "Elements of Set Theory" is a bit easier, perhaps it might work for you. (Also, your selection of theorems is mostly about cardinality, which will usually be within just a few chapters in your average set theory textbook.) $\endgroup$
    – Mario Carneiro
    Apr 1, 2015 at 1:54

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I think the first half of Irving Kaplansky's book Set Theory and Metric Spaces fits what you're looking for. It covers all these topics, and just a little more, but nowhere near as much as the typical books one finds mentioned, such as Enderton's book or Hrbacek/Jech's book. It's a little below the level of Halmos, although not so much that it would be little more than a guide for how to work with unions and intersections of sets (high school level stuff, I would suppose). It's also very well written and not obsessed with mathematical formalism -- there are plenty of well written verbal explanations. Scattered throughout the book are also some very interesting "notes to the reader", such as how differently a typical mathematician tends to view the axiom of choice, Zorn's lemma, and the well-ordering principle despite the fact that are mathematically equivalent (and Kaplansky proves their equivalence in great detail). I seem to recall that there is also a short discussion about some really big cardinals, such cardinals that (when also viewed as ordinals) satisfy ${\aleph}_{\alpha} = {\alpha}.$

A bit longer, but still in the same spirit as Kaplansky's treatment, is Classic Set Theory: For Guided Independent Study by Derek Goldrei.

You might also want to look at Schaum's Outline of Set Theory and Related Topics by Seymour Lipschutz, especially if you want to see a lot of worked problems.

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Perhaps not as comprehensive as you need, but Smullyan's {\it Satan, Cantor, and Infinity} discusses many of these topics in an understandable yet sophisticated way.

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Velleman's How To Prove It proves each of these things as well as many more interesting theorems. However, if you're already very familiar with reading and writing proofs then it might be a slow read.

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The book on "Topology" by James Dugundji is very good; the first chapter is "Elementary Set Theory", the second is "Ordinals and Cardinals", covering what you need (and in particular your last point).

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