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Foundations Good book to learn about set-theoretic infinity?

  1. May 11, 2015 #1
    I am looking for a good textbook covering cardinal and ordinal arithmetic suitable for self study. I'm a recently graduated undergrad (in mathematics) so I could probably handle up to intro graduate level material. I know most good set theory books might have a few chapters about these topics, but I'd especially be interested in a book that sets out to teach the reader specifically about the levels of infinity that come from studying the cardinality of sets. Any help would be greatly appreciated.
  2. jcsd
  3. May 11, 2015 #2
    I think the best book on the subject is Hrbacek and Jech: https://i.chzbgr.com/maxW500/8020280576/h5E34A50C/ It should cover exactly what you want. Of course it has many topics which are not about cardinals and ordinals, but that's the nature of a book on set theory. A good follow up book should be Jech's set theory book, but don't attempt to read it right now, it's too dense and unmotivated if you're new to set theory: https://www.amazon.com/Set-Theory-Millennium-Monographs-Mathematics/dp/3642078990

    A good contender with Hrbacek and Jech is Enderton: https://www.amazon.com/Elements-Set-Theory-Herbert-Enderton/dp/0122384407

    I don't know of a book which focuses specifically on ordinals and cardinals though.
    Last edited by a moderator: May 7, 2017
  4. May 12, 2015 #3


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    Hmm, some reasons why I dislike Enderton's set theory book:

    1. He uses "we" and "the" to say things that are not canonical. For example, he calls "the abstraction method" what surely everyone else calls a set comprehension, and he calls an "entrance requirement" what anyone else calls a predicate. He even calls naive set theory "baby set theory" initially.

    2. He says that the axiom system "leaves the primitive notion of set undefined" but how can that be true? If it was undefined, a set could be something with duplicate elements. But we know it can't have. So the axioms have a defining role. He even says it: "the axioms can be thought of as divulging partial information regarding the meaning of the primitive notions." Can they be thought of that way or do they, which is it? He seems to be on the side that they don't, they leave it undefined.

    I ám lenient with foreign authors who write in English but this guy is British and he should know better. He's too imprecise for me.

  5. May 12, 2015 #4


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    To answer the question, I completely agree on Hrbacek and Jech, supremely clear and accurate, surely it can't be beat.
  6. May 15, 2015 #5
    Thanks everyone for the advice!
    I have indeed begun using Hrbacek and Jech for my self study supplemented by some online lecture notes I found here: http://kaharris.org/teaching/582/index.html.
    Here's to a great summer of set theory! =)
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