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How is aleph-one defined?

  1. Jul 24, 2008 #1
    1. The problem statement, all variables and given/known data
    aleph-0 is defined as the cardinality as the integers right?

    How is aleph-one defined? Is the only definition as the next largest cardinal number after aleph-1?

    Annoyingly, my set theory book (kaplansky) doesn't seem to define it anywhere. Also the wikipedia article just confused me more.

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Jul 24, 2008 #2


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    Re: aleph-one

    How much do you know about ordinal numbers?
  4. Jul 24, 2008 #3
    Re: aleph-one

    I see now. The problem was that Kaplansky uses alpha-two on page 47 but defines ordinal numbers on page 57.
  5. Jul 25, 2008 #4
    Re: aleph-one

    Surely one of the real problems with defining aleph-1 is that what it is depends on what set theory you use. If you accept CH, then it is the cardinality of the real numbers. But it need not be. I have this idea that there may be infinitely many cardinals between aleph-0 and c (cardinality of the continuum, or 2^aleph-0) in some models, but it is just a feeling.
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