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I am looking for a nice introductory book to read to learn and master ordinals and cardinals. Please help me!

Tahina_RAK
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  • I need it basically to prove facts in general topology such as properties of normal spaces – Tahina_RAK Jul 06 '15 at 14:29
  • Jech - Set theory seems the best to me – Rubertos Jul 06 '15 at 14:29
  • @Rubertos: advice from topologist would be valuable as well since they may know what do I need – Tahina_RAK Jul 06 '15 at 14:31
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    I don't see where you need ordinals and cardinals to prove properties of normal spaces. If I remember correctly, there is no use of ordinals and cardinals in general topology untill you get through really complex structures such as paracompact etc. – Rubertos Jul 06 '15 at 14:34
  • Moreover, I highly recommend you to just study "elementary set theory" before you study general topology. – Rubertos Jul 06 '15 at 14:34
  • @Rubertos: Jone's lemma is one of the most used results to prove that product of normal spaces is not normal. – Tahina_RAK Jul 06 '15 at 14:36
  • Yes you always need some sort of axiom of choice to prove properties of "infinite" products.. Still, you don't need to know cardinals and ordinals to prove that.. – Rubertos Jul 06 '15 at 14:41
  • I do agree with Rubertos on the fact that you have to cover quite advanced topology topics to require ordinals and cardinals knowledge. Maybe you are studying order topologies? I like the book "Set Theory for the Working Mathematician" if you want to stick to Set Theory for Math. Basic Set Theory (from Azriel Levy) is another good introduction on Set Theory. But it focusing on Set Theory rather than on Math application. – mathcounterexamples.net Jul 06 '15 at 14:42
  • And I think your post should be more specific. If you are studying "elementary general topology", then you just need "elementary set theory". But if you are studying "set-theoretic topology" (which is for high level mathematicians) such as that in the book by Kunen, then you shall definitely need to know set theory very well.. – Rubertos Jul 06 '15 at 14:44
  • I'm guessing that the first few chapters or an appendix of your textbook with cover all the set theory you will need without going needlessly down the rabbit hole of ordinals, cardinals and ZFC. – Dan Christensen Jul 22 '15 at 13:36

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Since you seem to be reading handbook of set theoretic topology, it might be a good idea to read Kunen's chapter in the handbook of mathematical logic. If you are feeling "red bull" motivated after that, then you can attempt first two chapters of Kunen's book and also the problems in Komjath, Totik's problem book.

hot_queen
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    I glanced over the comments about an hour ago, and again just now when I noticed you posted an answer, and I can find nothing Jessy has said that indicates he/she is reading the Handbook of Set Theoretic Topology. What am I missing? – Dave L. Renfro Jul 07 '15 at 15:42
  • Jessy posted the following a while back: http://math.stackexchange.com/questions/1346809/why-separable-normal-space-has-only-continuum-many-different-open-subsets – hot_queen Jul 07 '15 at 15:49
  • @DaveL.Renfro You seem to know quite a lot about set theoretic real analysis, maybe you know the answer to this question?: http://mathoverflow.net/questions/210854/a-question-of-erdos-on-entire-functions – hot_queen Jul 07 '15 at 15:54
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    Actually, for these kinds of things I'm mostly just vaguely aware of various results and avenues people have pursued, and even then it's mostly 1910s through 1930s classical things. I did some looking around anyway and didn't find anything, but I sent an email to someone who's as likely as anyone to know about it. – Dave L. Renfro Jul 07 '15 at 17:11
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    Maybe you should clarify "Kunen's book". Most people think of the first edition of his textbook Set Theory, but he has also written a "prequel", The Foundations of Mathematics, and the current edition of Set Theory has been revised and the first two chapters may not contain what you think they do. – Nate Eldredge Jul 07 '15 at 17:28
  • I meant Kunen's first book (1980). – hot_queen Jul 07 '15 at 18:56
  • Dave, I am curious if you asked Kunen about it because I did that. – hot_queen Jul 07 '15 at 21:40
  • No, not Kunen! I've never met him, or even (that I can recall) exchanged emails with him before. I asked Krzysztof Ciesielski, and thought about asking Arnold Miller (who I actually have met, or at least have "encountered" in person, and exchanged emails with a number of years ago) but I didn't, so if you're up to it you might try asking Miller. – Dave L. Renfro Jul 09 '15 at 21:11