# Good books of naive set theory

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

• 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.) – Mario Carneiro Apr 1 '15 at 1:54

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 ${\aleph}_{\alpha} = {\alpha}.$