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

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

• Have you seen that note I wrote with this information and I keep linking to it in comments to questions like this? – Asaf Karagila Mar 9 '18 at 11:20
• @AsafKaragila: no. I don't. I don't know what are you talking about, sorry – Dog_69 Mar 9 '18 at 11:23
• This comment, for example. – Asaf Karagila Mar 9 '18 at 11:31
• Let me also point out that you're essentially looking for a book about set theory. But you're not telling us what do you already know, and what is your endgame goal. This would make the main difference between recommending something more heavy weight or something more lightweight. – Asaf Karagila Mar 9 '18 at 11:32
• Ok. I'll edit the question. – Dog_69 Mar 9 '18 at 11:44

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.

• By Thomas Jech I suppose you mean Karel Hrbacek Thomas Jech, Introduction to Set Theory, Third Edition, Revised, and Expanded (Pure and Applied Mathematics (Marcel Dekker)), don't you? Maybe I should add this comment, I have been looking at it and I don't like so much. I prefer Enderton. Respect to Kenneth, I didn't know him. I'll try to get the book. – Dog_69 Mar 9 '18 at 10:49
• @Dog_69 Pretty sure he means Jech: Set Theory. It's invaluable but quite densely written. Kunen's book has a much easier pace -- maybe better suited to self-study. – Stefan Mesken Mar 9 '18 at 11:21
• Ok. What happens is that the book written together with Hrbacek is also the third edition ''Revised and expanded''. Thanks @StefanMesken. By the way, I had forgotten include cofinality as required topic. Jech's book talks about that. Thanks. – Dog_69 Mar 9 '18 at 11:25
• @Dog_69 - I added links in order to avoid ambiguity. – Taroccoesbrocco Mar 9 '18 at 11:40