Looking for a good book to read about limit ordinals, aleph and beth numbers and AC

Question

I have read it the Appendix of Kelley's book General Topology for a self-teaching introduction to Set Theory. That means: I know the general definition of ordinals, and thus the definition of naturals and cardinals. I can know that not ordinal has a predecesor, and that successor ordinals and cardinals may be different. Hence I manage with the definitions of partial, orders, total orders, linear orders... And theoretically with maps between ordered structures. I mean, I think I have undertood (mor or less) the Appendix of Kelley's books (only I have read until theorem 174, if $x$ is an infinite cardinal then $\mbox{card }x+1 = \mbox{card} x$) . And that is all my basis.

So I would like to continue studying it, focus on limit ordinals and cardinals (for example in the well-known definiton $\lambda = \bigcup_{\kappa\in\lambda}\kappa$), cofinality, aleph and beth numbers, the importance of the Axiom of Choice in their definitions, the Continuum Hypothesis and the generalized CH, etc. And I would like to understand the difference between induction, strong induction and transfinite induction principles, because for example, the only difference I can see between strong induction and transfinite induction explained by Enderton is that the first consider only $\omega$ when the second is in every well-ordered set. But the proof is the same. And for me there is no difference between induction and stron induction.

I think Enderton's book discuss alll of these topics, but I his approach is different from Kelley's ones and I don't feel much comfortable. I'll read it if you think is the best book but I don't like it much.

For me Bourbaki is always an option, but I've read his Theory of sets is not very good. They develop their own theory, quite strange and very out of phase nowadays.

Thanks

Answer 1

I suggest two references:

Kenneth Kunen: Set Theory (Studies in Logic: Mathematical Logic and Foundations, new edition), 2011.

Thomas Jech: Set Theory (3rd Edition), 2006.

Both of them have some exercises in order to get some more deep insight about the theory.